uk: Математика, природничі науки

description: A reconfigurable is an object or collection of objects whose trans- formation between various states defines its functionality or aes- thetic appeal. For example, consider a mechanical assembly com- posed of interlocking pieces, a transforming folding bicycle, or a space-saving arrangement of apartment furniture. Traditional computer-aided design tools are best intended for static objects. I will report on some of our recent attempts to create optimization-based tools for computational design of reconfigurable shapes. I will first discuss a prototype of a general purpose tool. Then I will dive into the curious and delightful special case of computational design of generalized Matryoshka, a.k.a. Russian Nesting Dolls.

uk: вхід без авторизації

name: From Reconfigurables to Matryoshka: Computational Design in Space-Time


Uri Ascher

type: Person

cs: Matematika

de: Mathematik

fi: matematiikka

uk: Математика

en: Mathematics

fr: Mathématiques

da: Matematik

es: Matemáticas

nl: Wiskunde

de: Mathematik, Naturwissenschaften

uk: Математика, природничі науки

en: Mathematics, Natural Sciences

description: Visual computing is a wide area that includes computer graphics and image processing, where the 'eye-norm' rules. I will briefly discuss two case studies involving numerical methods and analysis applied to this area. The first involves motion simulation and calibration of soft objects such as cloth, plants and skin. The governing elastodynamics PDE system, discretized in space already at the variational level using FEM,leads to a large, expensive to assemble, dynamical system in time, where the damped motion may mask highly oscillatory stiffness. Geometric integration ideas are making their way into visual computing research these days in search for more quantitative computations. The other case study involves some image processing problems where theremis a premium for local approaches that do not necessarily use underlying PDEs. Here the trend is in the other direction. I will demonstrate and discuss.

TIB AV-Portal

type: Service

type: WebContent

type: MediaObject

@language: en

cs: přihlášení není vyžadováno

de: keine Anmeldung erforderlich

uk: вхід без авторизації

en: no login required

nl: geen login vereist

name: Numerical Analysis in Visual Computing: not too little, not too much

name: Banff International Research Station (BIRS) for Mathematical Innovation and Discovery

type: Organization

cs: Video

de: Video

fi: Video

uk: Відео

en: Video

fr: Vidéo

da: Video

nl: Video

es: Video


uk: Математика, природничі науки

description: In this talk we will discuss strategies for general purpose software implementation of isogeometric analysis. More than a decade after the introduction of the latter, industrialization of the technology remains a major challenge, since it involves a new way of thinking (a paradigm shift) with respect to existing computer-aided design and finite element analysis software and requires a re-engineering of current design and simulation practices.

uk: вхід без авторизації

name: Versatile software for isogeometric simulations


uk: Математика, природничі науки

description: An brief overview of different surface representations and their associated analysis (FEM) functions.

uk: вхід без авторизації

name: Design & Analysis: manifolds with irregular layout


uk: Математика, природничі науки

description: The proximal splitting algorithms can iteratively approximate an unspecial vector among possibly infinitely many minimizers of a superposition of multiple nonsmooth convex functions. With elegant translations of the solution set, i.e., the set of all minimizers, into the fixed point sets of nonexpansive mappings, the hybrid steepest descent method allows further strategic selection of a most desirable vector among the solution set, by minimizing an additional convex function over the solution set. In this talk, we introduce fixed point theoretic interpretations of variety of proximal splitting algorithms and their enhancements by the hybrid steepest descent method with applications to recent advanced statistical estimation problems.

uk: вхід без авторизації

name: Hierarchical Convex Optimization with Proximal Splitting Operators


uk: Математика, природничі науки

description: We consider here a generalization of exact penalty functions approach to solution of variational inequalities. Instead of more common potential mappings, associated with the gradient field of a penalty function the oriented field of a sharp penalty mapping is considered. Linearly combined with the variational inequality operator it directs the iteration method toward approximate feasibility and optimality,which is unavoidably only approximate as the exact balance between these two acting forces can not being attained beforehand. Nevertheless the accuracy of the limit points can be controlled by the parameters of the scheme and can be made arbitrary high and in the dynamic variants of this idea can even guarantee the exact solution of a monotone variational inequality. If one is content with the finite accuracy, the notion of sharp penalty mappings gives an additional freedom in construction of iteration schemes for approximate solution of variational inequalities with controllable accuracy. Due to some advantages of iteration methods, such as low memory requirements, data splitting, parallel execution, etc. it may be of interest for solution of variational inequalities of high dimension in particular for transportation equilibrium problems. Some part of the talk is devoted to a new approach for studying convergence properties of iteration processes arising from the above.Typically convergence proofs are based on Lyapunov-like statements about relaxation properties of a related sequences of values of convergence indicators. There are several general nontrivial methods for establishing such properties, but the analysis of algorithms for obtaining approximate solutions of optimization and equilibrium problems excludes the very idea of such convergence so new techniques used toward the validation of iteration methods were developed and will be discussed in this report.

uk: вхід без авторизації

name: Sharp Penalty Mapping Approach to Approximate Solution of Monotone Variational Inequalities


Quasidense multifunctions

uk: Математика, природничі науки

description: Quasidensity is a concept that can be applied to subsets of $E \times E^*$, where $E$ is a nonzero real Banach space. Every closed quasidense monotone set is maximally monotone, but there exist maximally monotone sets that are not quasidense. The graph of the subdifferential of a proper, convex lower semicontinuous function on $E$ is quasidense. The graphs of certain subdifferentials of certain nonconvex functions are also quasidense. (This follows from joint work with Xianfu Wang.) The closed monotone quasidense sets have a number of very desirable properties, including a sum theorem and a parallel sum theorem, and so quasidensity satisfies the ideal calculus rules. We give five conditions equivalent to the statement that a closed monotone set be quasidense, but quasidensity seems to be the only concept of the six that extends easily to nonmonotone sets. There are also generalizations to general Banach spaces of the Brezis-Browder theorem on linear relations, but we will not discuss these in this talk.

uk: вхід без авторизації


uk: Математика, природничі науки

description: The Douglas-Rachford splitting method plays an important role in optimization. We modify the method so that it can be used to find the least norm solution and the least norm primal-dual solution for a sum of two maximally monotone operators. Consequences on minimizing a sum of two convex functions will be discussed.

uk: вхід без авторизації

name: A parameterized Douglas-Rachford algorithm


uk: Математика, природничі науки

description: The possibilities of exploiting the special structure of d.c. programs, which consist of optimizing the difference of convex functions, are currently more or less limited to variants of the DCA proposed by Pham Dinh Tao and Le Thi Hoai An in 1997. These assume that either the convex or the concave part, or both, are evaluated by one of their subgradients. In this talk we propose an algorithm which allows the evaluation of both the concave and the convex part by their proximal points. Additionally, we allow a smooth part, which is evaluated via its gradient. In the spirit of primal-dual splitting algorithms, the concave part might be the composition of a concave function with a linear operator, which are, however, evaluated separately. For this algorithm we show that every cluster point is a solution of the optimization problem. Furthermore, we show the connection to the Toland dual problem and prove a descent property for the objective function values of a primal-dual formulation of the problem. Convergence of the iterates is shown if this objective function satisfies the Kurdyka - Lojasiewicz property. In the last part, we apply the algorithm to an image processing model.

uk: вхід без авторизації

name: A general double-proximal gradient algorithm for d.c. programming


uk: Математика, природничі науки

description: I will first give a short presentation of the objectives and scope of the CGAL open source project. I will the recently added algorithms that are relevant for geometric modeling and simulation: point set processing, polygon mesh processing and mesh generation. For each software component I will discuss the underlying design principles, show a live demo and explain how users can adapt and extend it to their specific needs. Finally, I will briefly review the existing projects for extending the mesh generation toolbox of CGAL.

uk: вхід без авторизації

name: The CGAL C++ library: survey and review of recent advances for geometric modeling.


uk: Математика, природничі науки

description: Use of computational topology for pattern analysis in turbulent flows Atmospheric science presents us with the problem of complex turbulent flows. We will present some techniques from computational topology which can be used in quantifying this spatio-temporal complexity, and for detecting minimal flow structures in direct numerical simulations.

uk: вхід без авторизації

name: Computational topology for pattern analysis in turbulent flow


uk: Математика, природничі науки

description: SOFA is an open-source framework for multi-physics simulation. SOFA aims at interactive and real-time applications, with an emphasis on medical simulation. SOFA benefits today from large, active and international community, including international universities, startups and companies. For more flexibility, SOFA is made up of a stable open-source core and many optional plugins (>100 plugins), providing innovative numerical methods and state-of-the-art algorithms. The SOFA core has a LGPL license (permissive and non-contaminating) fostering development of prototypes and products under any commercial license.

uk: вхід без авторизації

name: SOFA, an open-source framework for physics simulation : a tool for research, collaboration and innovation


uk: Математика, природничі науки

description: The collocation method with weighted extended B-splines (WEB-splines) represents a recently published approach for the spline approximation of the solution of stationary partial differential equations. In contrast to standard finite element methods, WEB-collocation requires no mesh generation and numerical integration, which leads to considerably faster computation times and an easier implementation. In this talk, the basics of WEB-collocation for general boundary value problems with mixed boundary conditions are described and the advantages over finite element methods are illustrated for Poisson's equation as typical model problem. On this basis, current research results from the application to singular and time-dependent problems are presented. The utilization of uniform spline spaces permits a straightforward generalization of the basic concept to hierarchical bases and the development of intuitive refinement strategies. The benefits of these adaptive WEB-collocation algorithms are shown in case of the model problem with a singular solution. Furthermore, considering the problem of simulating a tsunami, the combination of the WEB-collocation concept and a time-step iteration is presented to demonstrate a novel approximation scheme for time-dependent equations.

uk: вхід без авторизації

name: WEB-Collocation for Singular and Time-Dependent Problems


The flow limit of reflect-reflect-relax

uk: Математика, природничі науки

description: In the reflect-reflect-average (RRA) iteration scheme a point is averaged with the product of its reflection in two constraint sets to produce the next point. It was noticed in a recent study of the bit retrieval problem that the relaxed variant of RRA, called RRR, has superior performance in the limit of zero relaxation parameter. In this limit RRR defines a continuous flow (a system of ODEs), where the flow field is formed by the difference of a pair of points on the constraint sets. The flow limit can be implemented very efficiently (event-chain dynamics) when the flow field is piecewise constant, as it is in some NP-hard feasibility problems. I hope to have results for the Latin square completion problem in time for the workshop.

uk: вхід без авторизації


uk: Математика, природничі науки

description: Recently, several local and global linear convergence rate results for Douglas–Rachford splitting have appeared in the literature. Many of these are derived under strong monotonicity, Lipschitz continuity, and/or cocoercivity assumptions, and most focus on the convex optimization setting in the dual Douglas-Rachford algorithm, i.e., the alternating direction method of multipliers. In the monotone inclusion setting for Douglas-Rachford splitting, Lions and Mercier showed a linear convergence rate bound under the assumption that one of the two operators is strongly monotone and Lipschitz continuous. In this talk, we show that this rate is not sharp, and present a sharp contraction factor for the Douglas-Rachford operator under the stated assumptions. We also present, for many cases sharp, contraction factors for the Douglas-Rachford operator under different combinations of strong monotonicity, Lipschitz continuity, and cocoercivity assumptions. If these stronger properties are split between the operators, our results rely on the introduced notions of negatively averaged operators and negatively linearly regular operators, meaning that the negative operator is averaged and linearly regular respectively.

uk: вхід без авторизації

name: Sharp Contraction Factors for the Douglas-Rachford Operator


uk: Математика, природничі науки

description: We introduce and study the convergence properties of a projection-type algorithm for solving the variational inequality problem for point-to-set operators. No monotonicity assumption is used in our analysis. The operator defining the problem is only assumed to be continuous in the point-to-set sense, i.e., inner- and outer-semicontinuous. Additionally, we assume non-emptiness of the so-called dual solution set. We prove that the whole sequence of iterates converges to a solution of the variational inequality. Moreover, we provide numerical experiments illustrating the behavior of our iterates. Through several examples, we provide a comparison with a recent similar algorithm.

uk: вхід без авторизації

name: A projection algorithm for non-monotone variational inequalities


uk: Математика, природничі науки

description: We consider optimal control problems where the aim is to minimize the total variation in the control variables in addition to a general objective functional, resulting in a multi-objective optimal control problem. We study an illustrative convex problem, which appears in a simple form, and obtain asymptotic results for the challenging case when only the total variation is to be minimized. We also discuss problems which are in a more general form.

uk: вхід без авторизації

name: Optimal Control with Minimum Total Variation


uk: Математика, природничі науки

description: Minimizing a simple nonsmooth outer function composed with a smooth inner map offers a versatile framework for structured optimization. A unifying algorithmic idea solves easy subproblems involving the linearized inner map and a proximal penalty on the step. I sketch a typical such algorithm - ProxDescent - illustrating computational results and representative basic convergence theory. Although such simple methods may be slow (without second-order acceleration), eventual linear convergence is common. An intuitive explanation is a generic quadratic growth property - a condition equivalent to an "error bound" involving the algorithm's stepsize. The stepsize is therefore a natural termination criterion, an idea that extends to more general Taylor-like optimization models.

uk: вхід без авторизації

name: Error bounds and convergence of proximal methods for composite minimization


uk: Математика, природничі науки

description: In this talk we present a new iterative projection method for finding the closest point in the intersection of convex sets to any arbitrary point in a Hilbert space. This method, termed AAMR for averaged alternating modified reflections, can be viewed as an adequate modification of the Douglas-Rachford method that yields a solution to the best approximation problem. Under a constraint qualification at the point of interest, we show strong convergence of the method. In fact, the so-called strong CHIP fully characterizes the convergence of AAMR for every point in the space. We show some promising numerical experiments where we compare the performance of AAMR against other projection methods for finding the closest point in the intersection of pairs of finite dimensional subspaces.

uk: вхід без авторизації

name: Modifying the Douglas-Rachford algorithm to solve best approximation problems


Modeling through self-assembly

uk: Математика, природничі науки

description: Usually shapes are modeled through a top down approach. Think of smooth surfaces defined by a few control points. In this talk I will propose an opposite paradigm. This approach creates shapes from a bottom up approach. Think of the "Game of life." Shapes emerge from small scale interactions through local interactions. This approach is inspired by micro-biology. Key to this approach is to use a dynamics solver.

uk: вхід без авторизації


uk: Математика, природничі науки

description: We solve a fractional-time diffusion equation by a collocation-Galerkin method that uses the refinable spaces generated by the fractional B-splines as approximating spaces. The main advantage in using the fractional B-splines is in that their derivatives of both integer and fractional order can be expressed in a closed form that involves just the fractional difference operator. We analyze the performance of the method by solving some test problems.

uk: вхід без авторизації

name: On a fractional spline collocation-Galerkin method for a fractional diffusion equatio


uk: Математика, природничі науки

description: Since the introduction of Smoothness-Increasing Accuracy-Conserving (SIAC) Filtering for DG approximation of univariate hyperbolic equations by Cockburn et al., many generalizations of SIAC filtering have been proposed. Recently, new advancements in connecting the spline theory and SIAC filtering have paved the way for a more geometric view of this filtering technique. Based on which, various generalizations of the SIAC kernel have been proposed to make the filtering viable for more realistic applications. Examples include the introduction of SIAC line integral with applications for streamlining and flow visualization, hexagonal SIAC using nonseparable splines, and position dependent SIAC with nonuniform knot sequences. In this talk, I will introduce the basic concept of the SIAC filtering, its connection with well-established concepts from approximation theory, and discuss the recent advances in SIAC filtering.

uk: вхід без авторизації

name: Smoothness-Increasing Accuracy-Conserving (SIAC) Filtering and Its Application

name: Banff International Research Station (BIRS) for Mathematical Innovation and Discovery


uk: Математика, природничі науки

description: The classical inverse/implicit function theorems revolves around solving an equation in terms of a parameter and tell us when the solution mapping associated with this equation is a (differentiable) function. Already in 1927 Hildebrandt and Graves observed that one can put aside differentiability obtaining that the solution mapping is just Lipschitz continuous. The idea has evolved in subsequent extensions most known of which are various reincarnations of the Lyusternik-Graves theorem. In the last several decades it has been widely accepted that in order to derive estimates for the solution mapping and put them in use for proving convergence of algorithms, it is sufficient to differentiate what you can and leave the rest as is, hoping that the resulting problem is easier to handle. More sophisticated results have been obtained by employing various forms of metric regularity of mappings acting in metric spaces, aiming at applications to numerical analysis.

uk: вхід без авторизації

name: The Inverse Function Theorems of Lawrence M. Graves


uk: Математика, природничі науки

description: The talk presents some concepts and results from systems and control theory, focusing on convergence to and stability of a continuum of equilibria in a dynamical system. The well-studied and understood asymptotic stability of a compact set requires Lyapunov stability of the set: solutions that start close remain close to the set, and that every solution converge to the set, in terms of distance. Pointwise asymptotic stability of a set of equilibria requires Lyapunov stability of each equilibrium, and that every solution converge to one of the equilibria. This property is present, for example, in continuous-time steepest descent and convergent saddle-point dynamics, in optimization algorithms generating convergent Fejer monotone sequences, etc., and also in many consensus algorithms for multi-agent systems. The talk will present some background on asymptotic stability and then discuss necessary and sufficient conditions for pointwise asymptotic stability in terms of set-valued Lyapunov functions; robustness of this property to perturbations; and how the property can be achieved in a control system by optimal control.

uk: вхід без авторизації

name: Pointwise Asymptotic Stability of a Continuum of Equilibria


uk: Математика, природничі науки

description: We use recent zero duality gap results arising from Monotropic Programming problem for analyzing consistency of the convex feasibility problem in Hilbert spaces. We characterize consistency in terms of the lower semicontinuity of the infimal convolution of the associated support functions. Based on joint work with Victoria Martín-Márquez.

uk: вхід без авторизації

name: Convex Feasibility via Monotropic Programming


uk: Математика, природничі науки

description: Any casual observer of preprint servers like arXiv wil have noticed the weekly and sometimes daily appearance of new first-oder methods for structured optimization and (convex )relaxations for popular problems. Absent from this surge in new algorithmic strategies is any standard test-bed of real-world problems for trustworthy comparisons. The phase retrieval problem is one example of this sometimes absurd disconnect between proposed algorithms and the problems these routines are meant to solve. The ProxToolbox started as an effort to establish reproducibility of my numerical experiments, not only for students and other researchers, but for myself twenty years hence. It has since developed into what I hope to convince you is a platform for testing and comparing prox-based methods on data sets, both simulated and experimental, from some of the more popular applications of first order methods.

uk: вхід без авторизації

name: ProxToolbox: A Labortatory for Quantitative Convergence Analysis of Picard Iterations


uk: Математика, природничі науки

uk: вхід без авторизації

name: MODULI OF REGULARITY AND RATES OF CONVERGENCE FOR FEJER MONOTONE SEQUENCES


uk: Математика, природничі науки

description: In this talk, we discuss pointwise and ergodic convergence rates for a variable metric proximal alternating direction method of multiplicas (VM-PADMM) for solving linearly constrained convex optimization problems. The VM-PADMM can be seen as a class of ADMM variants, allowing the use of degenerate metrics (defined by noninvertible linear operators). We first propose and study nonasymptotic convergence rates of a variable metric hybrid proximal extragradient (VM-HPE) framework for solving monotone inclusions. Then, the convergence rates for the VM-PADMM are obtained essentially by showing that it falls within the latter framework.

uk: вхід без авторизації

name: Pointwise and ergodic convergence rates of a variable metric proximal ADMM


uk: Математика, природничі науки

uk: вхід без авторизації

name: From Brain to Hand to Statistics with Dynamic Smoothing


Similarity maps and splines on surfaces

uk: Математика, природничі науки

description: A variety of techniques were proposed to model smooth surfaces of arbitrary topology based on tensor product splines (e.g. subdivision surfaces, free-form splines, T-splines). Conversion of an input surface into such a representation is commonly achieved by constructing a global seamless parametrization, possibly aligned to a guiding cross-field and using this parametrization as a domain to construct the spline-based surface. (Informally, seamless parametrizations can be thought of as paramezations of surfaces cut to disks, with isoparametric line directions and spacing on the surface matching perfectly across the cuts). One major fundamental difficulty in designing robust algorithms for this task is the fact that for common types, e.g. subdivision surfaces (requiring a conforming domain mesh) or T-spline surfaces reliably obtaining a suitable parametrization that has the same topological structure (matching singularities and more generally rotations of parametric line directions along loops matching that of the cross-field) as the guiding field poses a major challenge. Even worse, not all fields do admit suitable parametrizations, and no concise conditions are known as to which fields do. I will discuss our recent work that addresses the problem by introducing two new concepts: (1) seamless similarity maps -- a relaxation of the seamless parametrization idea, allowing scale jumps across cuts (2) splines with half-edge knots, that relax the global knot interval consistency requirements on surfaces with nontrivial genus. It turns out, that for any given guiding field structure, a compatible parametrization of this kind exists and can be computed by a relatively simple algorithm; at the same time, for any such parametrization, a smooth piecewise rational surface with exactly the same structure as the input field can be constructed from it. This leads to fully automatic construction of high-order approximations of arbitrary surfaces, even with hiighly complex topology, potentially enabling, e.g., robust automatic conversion of surfaces to isogeometric form.

uk: вхід без авторизації


uk: Математика, природничі науки

uk: вхід без авторизації

name: Approximation with Ambient B-Splines and Intrinsic PDEs on Manifolds


Estimating Time-Varying Networks

uk: Математика, природничі науки

description: The problem of modeling the dynamical regulation process within a gene network has been of great interest for a long time. We propose to model this dynamical system with a large number of nonlinear ordinary differential equations (ODEs), in which the regulation function is estimated directly from data without any parametric assumption. Most current research assumes the gene regulation network is static, but in reality, the connection and regulation function of the network may change with time or environment. This change is reflected in our dynamical model by allowing the regulation function varying with the gene expression and forcing this regulation function to be zero if no regulation happens. We introduce a statistical method called functional SCAD to estimate a time-varying sparse and directed gene regulation network, and, simultaneously, to provide a smooth estimation of the regulation function and identify the interval in which no regulation effect exists. The finite sample performance of the proposed method is investigated in a Monte Carlo simulation study. Our method is demonstrated by estimating a time-varying directed gene regulation network of 20 genes involved in muscle development during the embryonic stage of Drosophila melanogaster.

uk: вхід без авторизації


uk: Математика, природничі науки

description: We propose a robust estimator for functional autoregressive models. This estimator, the Depth-based Least Squares (DLS) estimator, down-weights the influence of outliers by using the functional outlyingness as a centrality measure. The DLS estimator consists of two steps: identifying the outliers with a functional boxplot based on a defined depth, then down-weighting the outliers using the functional outlyingness. We prove that the influence function of the DLS estimator is bounded. Through a Monte Carlo study, we show that the DLS estimator performs better than the PCA and robust PCA estimators, which are the most commonly used.

uk: вхід без авторизації

name: Robust Depth-based Estimation of the Functional Autoregressive Model


Geometric Functional Data Analysis

uk: Математика, природничі науки

uk: вхід без авторизації


uk: Математика, природничі науки

description: We propose a new perspective on functional regression with a predictor process via the concept of manifold that is intrinsically finite-dimensional and embedded in an infinite-dimensional functional space, where the predictor is contaminated with discrete/noisy measurements. By a method of functional local linear manifold smoothing, we achieve a polynomial rate of convergence that adapts to the intrinsic manifold dimension and the level of noise/sampling contamination with a phase transition phenomenon depending on their interplay. This is in contrast to the logarithmic convergence rate in the literature of functional nonparametric regression. We demonstrate that the proposed method enjoys favourable finite sample performance relative to commonly used methods via simulated and real data examples.

uk: вхід без авторизації

name: Functional Regression on Manifold with Contamination


Do True Metabolic Pathway Models Exist?

uk: Математика, природничі науки

description: Metabolic systems are among of the oldest applications of mathematical modeling. Spanning a time period of over one hundred years, the repertoire of options for structuring metabolic models and for formulating reactions has been growing constantly, and yet, it is still unclear whether or to what degree some models are better than others and how the modeler is to choose among them. This situation begs the question whether there are representations of metabolic processes that are true over reasonably wide ranges, yet mathematically tractable. A glimpse into such representations is provided by Dynamic Flux Estimation which, under ideal conditions, reveals the actual shapes of functions representing metabolic processes, although not their mathematical formats. While intriguing, DFE is only directly applicable if a pathway system contains as many dependent variables as fluxes. Because most actual systems contain more fluxes than metabolite pools, this requirement is seldom satisfied. Some auxiliary methods have been proposed to alleviate this issue, but they were quite ad hoc. Here I demonstrate a generic strategy that renders DFE applicable to moderately underdetermined pathway systems. A second challenge with DFE is the need to identify explicit functional formats that have shapes as close as possible to those inferred. Clearly, even if this inference is feasible, the result is necessarily biased. As an alternative, I demonstrate that good time series data allow us to circumvent this step and to develop nonlinear dynamic models in an entirely nonparametric fashion. The resulting nonparametric models offer a surprisingly wide range of analytic tools, including stability and sensitivity analyses. I will finish with some comments on dynamical model reduction, using power-law models. Goel, G., I-C. Chou, and E.O. Voit: System estimation from metabolic time series data. Bioinformatics 24, 2505-2511, 2008. Dolatshahi, S., and E.O. Voit: Identification of metabolic pathway systems. Frontiers in Genetics 7:6, 2016. Faraji, M. and E.O. Voit: Nonparametric dynamic modeling. Math. Biosc. 287, 130-146, 2017. Faraji, M. and E.O. Voit: Stepwise inference of likely dynamic flux distributions from metabolic time series data. Bioinformatics, 33 (14): 2165-2172; 2017. Voit, E.O.: The best models of metabolism. WIREs Systems Biology and Medicine. (in press)

uk: вхід без авторизації


uk: Математика, природничі науки

description: The direction of outlyingness is crucial to describing the centrality of multivariate functional data. Motivated by this idea, we generalize classical depth to directional outlyingness for functional data. We investigate theoretical properties of functional directional outlyingness and find that it naturally decomposes functional outlyingness into two parts: magnitude outlyingness and shape outlyingness which represent the centrality of a curve for magnitude and shape, respectively. Using this decomposition, we provide a visualization tool for the centrality of curves. Furthermore, we design an outlier detection procedure based on functional directional outlyingness. This criterion applies to both univariate and multivariate curves and simulation studies show that it outperforms competing methods. Weather and electrocardiogram data demonstrate the practical application of our proposed framework. We further discuss an outlyingness matrix for multivariate functional data classification as well as plots for multivariate functional data visualization and outlier detection. The talk is based on joint work with Wenlin Dai.

uk: вхід без авторизації

name: Directional Outlyingness for Multivariate Functional Data


uk: Математика, природничі науки

description: Our goal is to develop an efficient framework for fitting stochastic continuous-time models to experimental data in cognitive psychology. As a simple test problem, we consider data from an eye-tracking study of attention in learning. For each subject, the data for each trial consists of the sequence of stimulus features that the subject fixates on, together with the duration of each fixation. We fit a stochastic differential equation model to this data, using the Approximate Bayesian Computation framework. For an individual subject we infer posterior distributions for the unknown parameters in the model.

uk: вхід без авторизації

name: Fitting a Stochastic Model to Eye Movement Time Series in a Categorization Task


uk: Математика, природничі науки

description: Emboldened by a string of insights gleaned from time series using likelihood-based inference and stochastic dynamical systems models, we undertook to exploit age-specific disease incidence data using an age-structured stochastic transmission model. In this talk, I explain the study's motivation, in questions surrounding the current resurgence of pertussis in countries with high vaccine coverage, and describe the model we formulated to address these questions. I point out the interesting features of the model implementation and the critical aspects of the inference methodology, with special attention to the challenges associated with this high-dimensional context. I highlight the surprises among the scientific conclusions we drew and conclude by speculating on the unreasonable effectiveness of stochastic models in population biology.

uk: вхід без авторизації

name: Looking for the limits to particle-filter based inference


uk: Математика, природничі науки

description: High speed fluorescence microscopy has created incredible opportunities to tag nano-/micro-scale biological agents (proteins, organelles, vesicles, etc) and track them dynamically. With opportunities comes challenges. With most of these microscopy methods, other biological entities in the environment (membranes, microtubules, actin networks) cannot be visualized simultaneously. Can we use the dynamic observations from the agents of interest to infer the structure of these other biological entities or at least not corrupt our inference of the dynamics of those original agents? In this talk, I will discuss two such biological challenges along with particular data sets: 1) Molecular motor/cargo complexes (agents) and their interaction with un-visualized microtubules. 2) Tracer particles interacting with membranes (cellular, nuclear, etc ).

uk: вхід без авторизації

name: Diffusing Particles and Surfaces Interacting in Cells


Bivariate Splines for Data Forecasting

uk: Математика, природничі науки

uk: вхід без авторизації


uk: Математика, природничі науки

description: We propose a new method for the analysis of functional data defined over spatio-temporal domains. These data can be interpreted as time evolving surfaces or spatially dependent curves. The proposed method is based on regression with differential regularization. We are in particular interested to the case when prior knowledge on the phenomenon under study is available. The prior knowledge is described in terms of a time-dependent Partial Differential Equation (PDE) that jointly models the spatial and temporal variation of the phenomenon. We consider various samplings designs, including geo-statistical and areal data. We show that the corresponding estimation problem are well posed and can be discretized in space by means of the Finite Element method, and in time by means of the Finite Difference method. The model can handle data distributed over spatial domains having complex shapes, such as domains with strong concavities and holes. Moreover, various types of boundary conditions can be considered. The proposed method is compared to existing techniques for the analysis of spatio-temporal models, including space-time kriging and methods based on thin plate splines and soap film smoothing. As a motivating example, we study the blood flow velocity field in the common carotid artery, using data from Echo-Color Doppler.

uk: вхід без авторизації

name: A time-dependent PDE regularization to model functional data defined over spatio-temporal domains


uk: Математика, природничі науки

description: Dimension reduction techniques play a very important role in analyzing a set of functional data that possess temporal or spatial dependence. Of these dimension reduction techniques, functional principal components (FPCs) analysis remains as a popular approach that extracts a set of latent components by maximizing variance in a set of dependent functional data. However, this technique may fail to adequately capture temporal or spatial autocorrelation in a functional data set. Functional maximum autocorrelation factors (FMAFs) are proposed for modelling and forecasting a temporal/spatially dependent functional data. FMAFs find linear combinations of original functional data that have maximum autocorrelation and are decreasingly predictable functions of time. We show that FMAFs can be obtained by searching for the rotated components that have smallest integrated first derivatives. Through a basis function expansion, a set of scores are obtained by multiplying extracted FMAFs with original functional data. Then, these scores are forecast using a vector autoregressive model under stationarity. Conditional on fixed FMAFs and observed functional data, the point forecasts are obtained by multiplying forecast scores with FMAFs. Interval forecasts can also be obtained by forecasting bootstrapped FMAF scores. Through a set of Monte Carlo simulation results, we study the finite-sample properties of the proposed FMAFs. Wherever possible, we compare the performance between the FMAFs and FPCs.

uk: вхід без авторизації

name: Maximum autocorrelation factors for function-valued spatial/temporal data


uk: Математика, природничі науки

description: In oceanic research, it is challenging to understand the patterns of winds and waves due to the complicated spatio-temporal dynamics. In this work, we propose new spectra-based methods for clustering hourly data of wind speed and wave height observed in the entire Red Sea. By clustering time series observed from different locations together, we identify spatial regions that share similar wind and wave directional spectra. We show that it is necessary to consider directional spectra for winds and waves, and that the clustering results may be very different, ignoring the direction.

uk: вхід без авторизації

name: Spectra-based clustering methods for visualizing spatio-temporal patterns of winds and waves in the Red Sea

name: Banff International Research Station (BIRS) for Mathematical Innovation and Discovery


uk: Математика, природничі науки

description: The analysis of complex data distributed over large or highly textured regions poses new challenges for spatial statistics. Available methods usually rely on global assumptions about the stationarity of the field generating the data and are unsuitable for large, textured or convoluted spatial domains, with holes or barriers. We here propose a novel approach for spatial prediction which cope with the data and the domain complexities through iterative random domain decompositions. The method is general and apt to the analysis of different types object data. A case study on the analysis and spatial prediction of density data relevant to the study of dissolved oxygen depletion in the Chesapeake Bay (US) will illustrate the potential of the novel approach. This is a joint work with Alessandra Menafoglio and Giorgia Gaetani, at MOX-Department of Mathematics, Politecnico di Milano.

uk: вхід без авторизації

name: Random domain decomposition for kriging non stationary object data


uk: Математика, природничі науки

description: In the talk, we will present and discuss a general framework for multi-aspect local non-parametric null-hypothesis testing for functional data defined on a common domain (Pini and Vantini, 2017). In detail: “multi-aspect” pertains to the fact the procedure allows the simultaneous investigation of many different data aspects like means, variances, quantiles of functional data and their associated differential and/or integral quantities; “local” pertains instead to the fact the procedure can impute the rejection to aspect-specific regions of the domain; finally, “non-parametric” refers to the fact that the specific implementation of the procedure is permutation-based and thus finite-sample exact and consistent independently on data Gaussianity. For ease of clarity, the focus will be on functional two-population tests and functional one-way ANOVA with an application on the statistical comparison of ultrasound tongue profiles pertaining to different allophones pronounced by the same speaker which can be modelled as functions varying on a spatio-temporal domain (Pini et al. 2017a). Finally, we will quickly show how to extend the approach to deal with more complex testing problems like functional two-way ANOVA and functional-on-scalar linear regression with applications to the analysis of spectral data (Pini et al. 2017b) and human movement data (Pini et al. 2015), respectively. Hébert-Losier, K., Pini, A., Vantini, S., Strandberg, J., Abramowicz, K., Schelin, L., Häger, C. K. (2015): "One-leg hop kinematics 20 years following anterior cruciate ligament rupture: Data revisited using functional data analysis", Clinical Biomechanics, Vol. 30(10), pp. 1153-1161. Pini, A., Vantini, S. (2017): “Interval-Wise Testing for Functional Data”, Journal of Nonparametric Statistics. 29 (2), pp. 407-424. Pini, A., Spreafico, L., Vantini, S., Vietti, A. (2017): Multi-aspect local inference for functional data: analysis of ultrasound tongue profiles. Tech. Rep. MOX 28/2017, Dept. of Mathematics, Politecnico di Milano. Pini, A., Vantini, S., Colosimo, B. M., Grasso, M. (2017): “Domain-Selective Functional Analysis of Variance for Supervised Statistical Profile Monitoring of Signal Data”, Journal of the Royal Statistical Society – Series C (to appear).

uk: вхід без авторизації

name: Non-parametric multi-aspect local null hypothesis testing for functional data


uk: Математика, природничі науки

uk: вхід без авторизації

name: Coding demonstration for triangular finite element method and PDE-smoothing


keywords:
Mathematics
Combinatorics
Sequences, series, summability
Discrete mathematics

uk: Математика, природничі науки

description: A D-finite power series satisfies a system of linear partial differential equations with polynomial coefficients of special type. This class of power series has been systematically investigated by Stanley in his book Enumerative Combinatorics (Volume II). We prove that a multivariate D-finite power series with coefficients from a finite set is rational. This generalizes a rationality theorem of van der Poorten and Shparlinski in 1996. As an application, we will show how this result can be used to study the nonnegative integer points on algebraic varieties. This is a joint work with Jason P. Bell.

uk: вхід без авторизації

name: Power Series with Coefficients from a Finite Set


S-units and D-finite power series

uk: Математика, природничі науки

description: Let K be a field of characteristic zero and let G be a finitely generated subgroup of K∗. Given a P-recursive sequence f(n) taking values in K, we study the problem of when f(n) takes values in G. We show that this problem can be interpreted purely dynamically and, when one does so, one can prove a much more general result about algebraic dynamical systems. Using this framework, we can then show that the set of n for which f(n)∈G is a finite union of infinite arithmetic progressions along with a set of zero Banach density, which simultaneously generalizes a result of Methfessel and a separate result due to Bezivin.

uk: вхід без авторизації


uk: Математика, природничі науки

description: The field of computer algebra can be divided into several mutually related subfields. Some of these are more relevant to combinatorialists than others, but we believe that there are some which should be known better. Therefore, for this overview talk, we have decided not only to discuss the most natural topics forming the subfield sometimes called symbolic combinatorics (featuring algorithms for recurrences and differential equations) but also discuss some of the techniques belonging to two other subfields that may be less known: exact arithmetic (with fast multiplication and working with homomorphic images) and Groebner basis (with techniques for reasoning about polynomial ideals). These topics are at the heart of computer algebra, and we believe that it will be handy for a computationally oriented combinatorialist to know about them.

uk: вхід без авторизації

name: Course #1: Some Lessons on Computer Algebra


uk: Математика, природничі науки

description: The kernel method has become one of the standard tools for solving lattice path enumeration problems. I will start by introducing the method in the context of constrained 1-dimensional random walks and simple polymer models. When we move up to 2 dimensions, the generating functions of random walk models can display a very broad range of analytic properties. In this context the kernel method becomes much richer and I will demonstrate a few of the ways in which it may be applied. Along the way I will give a few "warm-up" exercises and finish with a few open problems concerning more interacting boundary conditions.

uk: вхід без авторизації

name: Course #2: An introduction to the kernel method


uk: Математика, природничі науки

description: When studying special functions of the complex variable, one would like to determine whether a function is algebraic over the field of rational functions or not. To refine this classification, one could also be interested in the differential dependence of the function. Starting from the holonomic or D-finite functions, that is the ones that satisfy a linear differential equation, we consider a new class of complexity, called differential transcendence, that corresponds to the functions that do not satisfy a polynomial relation with their derivatives. A celebrated example is the Gamma function, which is differentially transcendental by a result of Hölder. When the special function is given by a linear functional equation, the difference Galois theory provides powerful and systematic tools that allows to determine such kind of relations. In this talk, I will try to give an overview of this Galois theory by focusing on examples and applications.

uk: вхід без авторизації

name: Course #3: An overview of difference Galois theory


uk: Математика, природничі науки

description: In this lecture we will focus on techniques coming from probability theory and analysis to study models of walks confined to multidimensional cones, with arbitrary big steps and possibly with weights. To give an example of the fruitful interaction between the above domains, we will restrict our attention to the computation of critical exponents which appear in the asymptotic behavior of confined walks. For instance, what is the asymptotic behavior of the number of excursions, i.e., of the number of walks starting and ending at given points, remaining in a fixed cone, as the number of steps goes to infinity ? In a first part, using an approximation of random walks by Brownian motion, we will present the seminal work of Denisov and Wachtel providing a solution to the above problem in the case of excursions. We shall also present partial results and conjectures related to the total number of walks confined to a cone. We will show new results concerning the non-D-finiteness of some series counting walks in the quarter plane. In a second part we shall be interested in discrete harmonic functions in cones. The generating function of these harmonic functions satisfies a functional equation, which happens to be closely related to the well-known functional equation that appears in the context of enumeration of confined walks. We shall explain the link between these harmonic functions and a one-parameter family of conformal mappings. These harmonic functions provide a second way to compute the critical exponents. We will present several conjectures.

uk: вхід без авторизації

name: Course #4: Probabilistic Tools for Lattice Path Enumeration


uk: Математика, природничі науки

uk: вхід без авторизації

name: Algorithmic Tools for the Asymptotics of Diagonals


uk: Математика, природничі науки

description: Given a sequence represented by a linear recurrence with polynomial coefficients and sufficiently many initial terms, a natural question is whether the transcendence of its generating function can be decided algorithmically. The question is non trivial even for sequences satisfying a recurrence of first order. An algorithm due to Michael Singer is sufficient, in principle, to answer the general case. However, this algorithm suffers from too high a complexity to be effective in practice. We will present a recent method that we have used to treat a non-trivial combinatorial example. It reduces the question of transcendence to a (structured) linear algebra problem.

uk: вхід без авторизації

name: How to prove algorithmically the transcendence of D-finite power series


uk: Математика, природничі науки

description: Creative telescoping is a powerful technique to tackle summation and integration problems symbolically, but it can be computationally very costly. Many existing algorithms compute two objects, called telescoper and certificate, but in many applications only the first one is of interest, while typically the second one is larger in size. In the past few years a new direction of research was initiated, namely to develop creative telescoping algorithms that are based on Hermite-type reductions, which avoid the computation of the certificate and therefore can be more efficient in practice. In our 2016 ISSAC paper, we have developed an algorithm for constructing minimal-order telescopers for algebraic functions, based on Trager's reduction and on a so-called polynomial reduction. Later we have extended this algorithm to fuchsian D-finite functions.

uk: вхід без авторизації

name: Reduction-Based Creative Telescoping for D-Finite Functions


uk: Математика, природничі науки

description: This talk focusses on the interaction between the kernel method, a powerful collection of techniques used extensively in the enumeration of lattice walks in restricted regions, and the relatively new field of analytic combinatorics in several variables (ACSV). In particular, the kernel method often allows one to write the generating function for the number of lattice walks restricted to certain regions as the diagonal of an explicit multivariate rational function, which can then be analyzed using the methods of ACSV. This pairing is powerful and flexible, allowing for results which can be generalized to high (or even arbitrary) dimensions, weighted step sets, and the enumeration of walks returning to certain boundary regions of the domains under consideration.

uk: вхід без авторизації

name: Lattice Path Enumeration and Analytic Combinatorics in Several Variables


uk: Математика, природничі науки

description: A very rich interplay between arithmetic, geometry, transcendence and combinatorics arises in the study of homogeneous linear differential equations and especially of those that “come from geometry” and the related study of Siegel G-functions. A remarkable result is that, by adding variables, we can see many transcendental G-functions (and thus many generating series) as arising in a natural way from much more elementary function, namely rational functions. This process, called diagonalization, can be thought of as a formal integration. I will discuss some properties enjoy by diagonals of rational functions and connect them with Lucas'congruences for binomial coefficients and algebraic independence of power series. This corresponds to some joint works with Jason Bell and Eric Delaygue.

uk: вхід без авторизації

name: Diagonals, congruences, and algebraic independence


uk: Математика, природничі науки

description: We find a formula for the asymptotics of the coefficients of a generating function of the form, H(z1,z1,...,zd)−β, as the indices approach infinity in a fixed ratio. Then, we look at how this formula can be applied to generating functions that enumerate the possible structures into which RNA sequences can fold. This work relies on the techniques in multivariate analytic combinatorics developed by Pemantle and Wilson. We combine the multivariate Cauchy integral formula with explicit contour deformations to compute the asymptotic formula. A challenge of using the formula is correctly identifying the points which contribute to asymptotics.

uk: вхід без авторизації

name: Multivariate Algebraic Generating Functions: Asymptotics and Examples


Winding angles of simple walks on Z^2

uk: Математика, природничі науки

description: A method will be described to determine generating functions for certain classes of simple walks on the square lattice, while keeping track of their winding angle around the origin. Together with a reflection principle the method can be used to count certain simple walks in wedges of various opening angles, and this is shown to lead in particular to a new proof of the counting of Gessel excursions. If time permits, I'll discuss a connection with the enumeration of planar maps and O(n) loop models.

uk: вхід без авторизації


Lattice walks on the half line

uk: Математика, природничі науки

description: I will discuss proofs of the algebraicity of generating functions for walks on the half-line with an arbitrary set of integer steps, with emphasis on Paul Monsky's approach that reduces the problem to the case of Motzkin walks with noncommuting weights.

uk: вхід без авторизації


uk: Математика, природничі науки

description: Several combinatorial objects (including several types of random walks) have a recursive combinatorial description that leads to a (system of) functional equation(s) for the corresponding counting generating function, where the right hand side of the equation has non-negative coefficients; sometimes there also appears a catalytic variable, for example for random walks restricted to some region or for the enumeration of planar maps. The purpose of this talk to show that the positivity condition leads to universal asymptotic properties of the underlying counting problem.

uk: вхід без авторизації

name: POSITIVE CATALYTIC AND NON-CATALYTIC POLYNOMIAL SYSTEMS OF EQUATIONS


uk: Математика, природничі науки

description: In the recent years, the nature of the generating series of the walks in the quarter plane has attracted the attention of many authors. The main questions are: are they algebraic, holonomic (solutions of linear differential equations) or at least hyperalgebraic (solutions of algebraic differential equations)?

uk: вхід без авторизації

name: Walks, Difference Equations and Elliptic Curves


uk: Математика, природничі науки

description: I will describe an algorithm (implemented in Maple) that produces a system of generators of the Lie algebra of an absolutely irreducible linear differential equation over the rational functions with complex coefficients. This is a joint work with M. Barkatou, T. Cuzeau et J.-A. Weil.

uk: вхід без авторизації

name: Direct problem in differential Galois theory

name: Banff International Research Station (BIRS) for Mathematical Innovation and Discovery


uk: Математика, природничі науки

description: We study the enumeration of three friendly directed walks on the square lattice. We show how one can count different types of shared sites (interactions) by solving the associated functional equations using the obstinate kernel method. However, we also highlight that the introduction of finer counting problems that introduce asymmetry can lead to as yet unsolvable functional equations. We survey related results over the past few years counting extra features in multiple path problems.

uk: вхід без авторизації

name: Counting shared sites of three friendly directed lattice paths and related problems


Counting Eulerian orientations

uk: Математика, природничі науки

description: Inspired by the paper of Bonichon, Bousquet-M\'elou, Dorbec and Pennarun, we give a system of functional equations for the ogfs for the number of planar Eulerian orientations counted by the edges, U(x), and the number of 4-valent planar Eulerian orientations counted by the number of vertices, A(x). The latter problem is equivalent to the 6-vertex problem on a random lattice, widely studied in mathematical physics. While unable to solve these functional equations, they immediately provide polynomial-time algorithms for the coefficients of the generating function. From these algorithms we have obtained 100 terms for U(x) and 90 terms for A(x). Analysis of these series suggests that they both behave as ⋅(1−μx)2/log(1−μx), where we make the confident conjectures that μ=4π for Eulerian orientations counted by edges and μ=43–√π for 4-valent Eulerian orientations counted by vertices.

uk: вхід без авторизації


uk: Математика, природничі науки

description: I will give a broad review of classes of integer sequences arising in combinatorics. Beside the more traditional asymptotic results, I will also emphasize the complexity aspects. Many examples will be presented. Some recent results will be mentioned as well.

uk: вхід без авторизації

name: The combinatorics and complexity of integer sequences


uk: Математика, природничі науки

description: I shall present a curious factorisation theorem for the number of rhombus tilings of a hexagon with vertical and horizontal symmetry axis, with triangular holes along the latter axis. Its proof is essentially based on (non-intersecting) lattice paths.

uk: вхід без авторизації

name: A factorisation theorem for the number of rhombus tilings of a hexagon with triangular holes


uk: Математика, природничі науки

description: We introduce a deformed version of Dyck paths (DDP), where additional to the steps allowed for Dyck paths, ‘jumps’ orthogonal to the preferred direction of the path are permitted. We consider the generating function of DDP, weighted with respect to their half length, area and number of jumps. This represents the first example of a hierarchy of exactly solvable two-dimensional lattice vesicle models showing higher-order multi-critical points with scaling functions expressible via generalized Airy functions, as conjectured by John Cardy.

uk: вхід без авторизації

name: Higher-order multicritical points in two-dimensional lattice polygon models


uk: Математика, природничі науки

description: We will survey a series of intriguing conjectures about log-concavity and t-stack sortable permutations, as well as some recent enumeration results. Connections to lattice paths and labeled plan trees will also be discussed.

uk: вхід без авторизації

name: t-stack sortable permutations and log-concavity


keywords:
Boeckle, Gebhard
Mathematics
Algebraic geometry
Number theory

uk: Математика, природничі науки

description: Let K be a finitely generated infinite field over the finite prime field Fp with separable closure Ks and let X be a smooth projective variety over K. By Deligne the cohomology groups Hie´t(XKs,Qℓ) for varying primes ℓ≠p form a (Q-rational) compatible system of Galois representations of Gal(Ks/K) and its restriction to the geometric Galois group GgeoK=Gal(Ks/KFsp) is semisimple. Using mainly algebraic geometry, representation theory and Bruhat-Tits theory, Cadoret, Hui and Tamagawa showed recently that also the family of reductions Hie´t(XKs,Fℓ) is semisimple as a representation of GgeoK for almost all ℓ, the key case being that of a global function field K. This has important consequence for the image of GgeoK for its action on the adelic module~Hie´t(XKs,AQ). In joint work with W. Gajda and S. Petersen, using automorphic methods as a main tool, we prove the analog of the above result for any E-rational compatible system of Galois representations of a global function field. In the talk I shall explain the context, indicate the applications and sketch how automorphic methods come to bear on the problem.

uk: вхід без авторизації

name: Images of compatible systems of Galois representations of global function fields


keywords:
Mathematics
Algebraic geometry
Number theory

uk: Математика, природничі науки

description: In the first part of this talk we are going to define certain integral structures, depending on a congruence level, for the sheaves of differential operators on a formal scheme which is a blow-up of a formal scheme which itself is formally smooth over a complete discrete valuation ring of mixed characteristic. When one takes the projective limit over all blow-ups, one obtains the sheaf of differential operators on the associated rigid space, introduced independently by K. Ardakov and S. Wadsley. In the second part we will explain what it means that formal models of flag varieties are D-affine (this concept is analogous to that of Beilinson-Bernstein and Brylinski-Kashiwara in the algebraic context). If time permits, we will explain an example which illustrates that methods and results from rigid cohomology can be used in connection with those sheaves to analyze locally analytic representations of p-adic groups. This is joint work with C. Huyghe, D. Patel, and T. Schmidt.

uk: вхід без авторизації

name: Arithmetic structures in sheaves of differential operators on formal schemes and D-affinity


Rigid systems and integrality

uk: Математика, природничі науки

description: We prove that the monodromy of a cohomologically rigid integrable connection (E,∇) on a smooth complex projective variety X is integral. This answers positively a special case of a conjecture by Carlos Simpson. To this aim, we prove that the mod p reduction of a rigid integrable connection (E,∇) has the structure of an isocrystal with Frobenius structure. We also prove that rigid integrable connections with vanishing p- curvatures are unitary. This allows one to prove new cases of Grothendieck’s p-curvature conjecture. Joint with Michael Groechenig.

uk: вхід без авторизації


uk: Математика, природничі науки

description: I will talk on a recent result and possible applications concerning the comparison between integral Monsky-Washnitzer cohomology and overconvergent de Rham-Witt cohomology for smooth and affine schemes. This extends work of Davis and Zureick-Brown. Joint with Johannes Sprang.

uk: вхід без авторизації

name: Integral Monsky-Washnitzer and overconvergent de Rham-Witt cohomology


Higher algebra and arithmetic

uk: Математика, природничі науки

description: This talk concerns a twenty-thousand-year old mistake: the natural numbers record only the result of counting and not the process of counting. As algebra is rooted in the natural numbers, the higher algebra of Joyal and Lurie is rooted in a more basic notion of number which also records the process of counting. Long advocated by Waldhausen, the arithmetic of these more basic numbers should eliminate denominators. Notable manifestations of this vision include the Bökstedt-Hsiang-Madsen topological cyclic homology, which receives a denominator-free Chern character, and the related Bhatt-Morrow-Scholze integral p-adic Hodge theory, which makes it possible to exploit torsion cohomology classes in arithmetic geometry. Moreover, for schemes smooth and proper over a finite field, the analogue of de Rham cohomology in this setting naturally gives rise to the cohomological interpretation of the Hasse-Weil zeta function by regularized determinants envisioned by Deninger.

uk: вхід без авторизації


uk: Математика, природничі науки

description: I will describe an extension of Berthelot's theory of arithmetic differential operators to a class of morphisms of adic formal schemes that are not necessarily of finite type, or even adic. If time permits I will explain how one can use this to give generalizations of the theory of convergent and overconvergent isocrystals.

uk: вхід без авторизації

name: Rings of arithmetic differential operators on tubes


uk: Математика, природничі науки

description: I will talk about a recent proof, joint with M. Gröchenig and D. Wyss, of a conjecture of Hausel and Thaddeus which predicts the equality of suitably defined Hodge numbers of moduli spaces of Higgs bundles with SL(n)- and PGL(n)-structure. The proof, inspired by an argument of Batyrev, proceeds by comparing the number of points of these moduli spaces over finite fields via p-adic integration.

uk: вхід без авторизації

name: Mirror symmetry for moduli spaces of Higgs bundles via p-adic integration


Rational motivic path spaces

uk: Математика, природничі науки

description: A central ingredient in Kim's work on integral points of hyperbolic curves is the "unipotent Kummer map" which goes from integral points to certain torsors for the prounipotent completion of the fundamental group, and which, roughly speaking, sends an integral point to the torsor of homotopy classes of paths connecting it to a fixed base-point. In joint work with Tomer Schlank, we introduce a space Ω of rational motivic loops, and we construct a double factorization of the unipotent Kummer map which may be summarized schematically as points → rational motivic points → Ω-torsors → π1-torsors. Our "connectedness theorem" says that any two motivic points are connected by a non-empty torsor. Our "concentration theorem" says that for an affine curve, Ω is actually equal to π1.

uk: вхід без авторизації


uk: Математика, природничі науки

description: We report on joint work-in-progress with Ambrus Pal on the following conjecture. Conjecture: Let X be a smooth variety over a finite field k and let E be an overconvergent F-isocrystal on X that has rank 2 and is absolutely irreducible. Suppose further that E has "infinite monodromy at a divisor at infinity." Then E comes from a family of abelian varieties.

uk: вхід без авторизації

name: Rank 2 F-isocrystals and abelian varieties


uk: Математика, природничі науки

description: I will first describe two problems in the arithmetic of elliptic curves over function fields, then my work in progress on these conjectures using tools from $p$-adic cohomology. The first ingredient is a pro-representability theorem of what could be called arithmetic deformations of crystals and Dieudonne crystals. The second ingredient is the application of the Taylor-Wiles method in this setting.

uk: вхід без авторизації

name: Formal deformations of crystals and arithmetic applications


Update on the companion problem

uk: Математика, природничі науки

description: Let k be a finite field of characteristic p, and let l be a prime not equal to p. An old conjecture of Deligne states that on any smooth variety over k, any lisse l-adic sheaf which is irreducible of finite determinant admits a "crystalline companion" with matching Frobenius characteristic polynomials at all closed points. We report on progress towards constructing crystalline companions in the category of overconvergent F-isocrystals, building on work of Drinfeld, Deligne, and Abe-Esnault.

uk: вхід без авторизації


A quantum Simpson correspondence

uk: Математика, природничі науки

description: A Simpson correspondence establishes an equivalence between a category of modules equipped with a connexion and a category of Higgs bundles. In a joint work with Michel Gros and Adolfo Quirós, we describe such an equivalence in the case of modules equipped with a q-connexion when q is a p-th root of unity. This is modeled on the characteristic p case and relies on the notion of quantum divided powers that we also discuss.

uk: вхід без авторизації


uk: Математика, природничі науки

description: I will report on my work in progress done jointly with Frits Beukers. In 1990s V. Batyrev used Dwork modules to describe the mixed Hodge structure on the middle cohomology of affine hypersurfaces in algebraic tori. Dwork modules are crystals where the Frobenius map and connection can be described explicitly. We use these crystals to show several p-adic congruences for the coefficients of powers of a Laurent polynomial. The congruence mentioned in the title involves the L-function of the toric exponential sums and yields p-adic unit-root formulas.

uk: вхід без авторизації

name: Atkin and Swinnerton-Dyer congruences for toric hypersurfaces


uk: Математика, природничі науки

description: We study the formal neighbourhood of a point in µ-ordinary locus of an integral model of a Hodge type Shimura variety, and prove the existence of a structure analogous to the Serre-Tate structure on the deformation space of an ordinary abelian variety.This is joint work with Rong Zhou.

uk: вхід без авторизації

name: Serre-Tate theory for Shimura varieties of Hodge type


uk: Математика, природничі науки

description: We introduce a notion of F-isocrystals with logarithmic decay and give a conjecture relating this notion to slope filtrations. When the unit-root subcrystal has rank one we prove this conjecture. Combining this with a monodromy theorem we give a new proof of the Drinfeld-Kedlaya theorem. We also prove a generalized version of Wan's conjecture on genus stability for towers of curves coming from geometry.

uk: вхід без авторизації

name: Slope filtrations of F-isocrystals, log decay, and genus stability for towers of curves


uk: Математика, природничі науки

description: In this talk, I will present an algorithm to identify sparse dependence structure in continuous and non-Gaussian probability distributions, given a corresponding set of data. The conditional independence structure of an arbitrary distribution can be represented as an undirected graph (or Markov random field), but most algorithms for learning this structure are restricted to the discrete or Gaussian cases. Our new approach allows for more realistic and accurate descriptions of the distribution in question, and in turn better estimates of its sparse Markov structure. The algorithm relies on exploiting the connection between the sparsity of the graph and the sparsity of transport maps, which deterministically couple one probability measure to another.

uk: вхід без авторизації

name: Beyond normality: Learning sparse probabilistic graphical models in the non-Gaussian setting


uk: Математика, природничі науки

description: Approximation of multivariate functions is a difficult task when the number of input parameters is large. Identifying the directions where the function doesn't significantly vary is a key step for complexity reduction. Among other dimension reduction techniques, the Active Subspace method uses gradients of a scalar-valued function to reduce the parameter space. In this talk, we extend this methodology for vector-valued functions, e.g. functions with multiple scalar outputs or functions taking values in function spaces. Numerical examples reveal the importance of the choice of the metric to measure errors.

uk: вхід без авторизації

name: Dimension reduction of the input parameter space of vector-valued functions

name: Banff International Research Station (BIRS) for Mathematical Innovation and Discovery


uk: Математика, природничі науки

description: The distinguished properties of stochastic systems in civil structures subjected to dynamic disastrous actions include: (1) Randomness involved in both structural properties, which are essentially random fields, and external dynamic excitations such as strong earthquakes and wind, which are essentially stochastic processes; (2) Strong nonlinearity of restoring force, including strength degradation and stiffness degradation, which could not be described by polynomials and should be captured by the elastoplastic damage mechanics; and (3) Large degrees of freedom in the order of magnitude of millions or larger. The coupling of randomness and nonlinearity in large degrees of freedom systems leads to great difficulty in uncertainty quantification and global reliability of such real-world civil structures, and hinders effective design trading off safety and economical efficiency. In this presentation, the probability density evolution method (PDEM) will be outlined. In this method, by combining the principle of preservation of probability and the underlying physical mechanism, a state variables decoupled generalized density evolution equation (GDEE) could be derived. This equation reveals that the change of probabilistic information of the response is determined by the change of the underlying physical state. The technically most appealing property of this partial differential equation is that the dimension of this equation depends only on the number of quantity of interest, rather than the dimension of the embedded system. Consequently, combining the embedded deterministic analyses, from which the mechanism of propagation of uncertainty is captured, and the solution of GDEE, will lead to instantaneous probability density function of the quantity of interest. The applications to the seismic response and global reliability of real-world civil structures will be exemplified. Challenging problems to be further resolved, as well as most recent advancement, will be discussed.

uk: вхід без авторизації

name: The probability density evolution method for uncertainty quantification and global reliability of complex civil structures


uk: Математика, природничі науки

description: Among the large variety of complex non-periodic structures, quasicrystals and quasiperiodic distributions play a special role. These structures have some of their physical properties (e.g. dielectric constant, potential, etc.) modulated according to a deterministic non-periodic pattern such as a substitution rules set or a cut & project construction. Such architectures have long been recognized to yield a pronounced long-range order manifesting as an infinite set of crystallographic Bragg peaks and a highly lacunar singular-continuous energy spectrum, with an infinite set of gaps arranged in a multifractal hierarchy. The possibility that such structures also possess distinct topological features has been discussed both in mathematics and physics literature including some descriptions of the spectrum through topological invariants. These topological numbers, emerging from the structural building rules, are known to label the dense set of spectral gaps. We present the topological properties of a finite quasiperiodic chains studied using the scattering and also the diffraction of waves. We show that the topological invariants may be measured from the winding of a chiral scattering phase as a function of a phason structural degree of freedom. Using a Fabry-Perot point of view, this chiral phase is also shown to drive the spectral traverse of conveniently emulated edge states. Furthermore, we present a method to obtain all available topological numbers from the diffraction pattern of a quasicrystal, a method which may be termed topological quasicrystallography. Existing experimental realizations will be addressed, as well as the possible generalizations.

uk: вхід без авторизації

name: Probing Topological Properties of Quasicrystals with Waves + Tight-binding Approximations (TbA) for Longitudinally Driven Lieb and Kagome Lattices


uk: Математика, природничі науки

description: In this talk I will describe our recent ideas of how to engineer nanostructures that generate topological transport of vibrations. These include situations with explicit time-reversal symmetry breaking (via an optical field with optical vorticity) as well as with time-reversal symmetry intact. In the latter case, I will show how the snowflake phononic crystal, first invented for the purposes of optomechanics, provides an ideal platform in which to implement both pseudomagnetic fields for vibrations as well as a topological insulator. Pseudomagnetic fields for sound at the nanoscale Christian Brendel, Vittorio Peano, Oskar Painter, and Florian Marquardt, Proceedings of the National Academy of Sciences (PNAS) 114, E3390–E3395 (2017) Snowflake Topological Insulator for Sound Waves Christian Brendel, Vittorio Peano, Oskar Painter, and Florian Marquardt, arXiv:1701.06330 (2017) Topological Phases of Sound and Light Vittorio Peano, Christian Brendel, Michael Schmidt, and Florian Marquardt, Phys. Rev. X 5, 031011 (2015)

uk: вхід без авторизації

name: Topology of Light and Sound at the Nanoscale


uk: Математика, природничі науки

description: Geometry, topology and broken symmetry often play a powerful role in determining the organization and properties of materials. A recent example is the discovery that the excitation spectra of materials -- be they electronic, optical, or mechanical -- may be topologically non-trivial. I will explore the use of `spinning tops' to explore this physics. In particular I will discuss an experimental and theoretical study of a simple kind of active meta-material – coupled gyroscopes – that naturally encodes non-trivial topology in its vibrational spectrum. These materials have topologically protected edge modes which we observe in experiment. Crucially, the geometry of the underlying lattice controls the presence of time reversal symmetry that is essential to the non-trivial topology of the spectrum. We exploit this to control the chirality of the edge modes by simply deforming the lattice. Moving beyond ordered lattices we show that amorphous gyroscopic networks are naturally topological. We construct them from arbitrary point sets -- including hyperuniform, jammed, quasi-crystalline and uniformly random -- and control their topology through simple, local decorations.

uk: вхід без авторизації

name: Spinning top-ology: Order, disorder and topology in mechanical gyro-materials and fluids


uk: Математика, природничі науки

description: Predictive computational science is an emerging discipline concerned with assessing the predictability of mathematical and computational tools, particularly in the presence of inevitable uncertainty and limited information. In this talk, I will present a new comprehensive predictive methodology embedded in a new hybrid fuzzy-stochastic framework to predict physical events described by partial differential equations (PDEs) and subject to both random (aleatoric) and non-random (epistemic) uncertainty. In the new framework the uncertain parameters will be characterized by random fields with fuzzy moments. This will result in a new class of PDEs with hybrid fuzzy-stochastic parameters, coined fuzzy-stochastic PDEs, for which forward and inverse problems need to be solved. I will demonstrate the importance and feasibility of the new methodology by applying it to a complex problem: prediction of the response of materials with hierarchical microstructure to external forces. This model problem will serve as an illustrative example, one that cannot be tackled by today’s UQ methodologies.

uk: вхід без авторизації

name: Hybrid fuzzy-stochastic predictive modeling and computation


uk: Математика, природничі науки

description: We consider uncertainty quantification for high frequency waves. A crucial assumption for stochastic collocation methods to converge rapidly is the stochastic regularity of the QoI. In the high frequency regime, the derivatives in the stochastic variable should preferably be bounded independently of the wavelength. We show that, despite the highly oscillatory character of the waves, QoIs defined as local averages of the squared modulus of the wave solution approximated by Gaussian beams, indeed have this property.

uk: вхід без авторизації

name: Stochastic regularity of a quadratic observable of high frequency waves


uk: Математика, природничі науки

description: In this talk we will study the generalized polynomial chaos-stochastic Galerkin (gPC-SG) approach to kinetic equations with uncertain coefficients/inputs, and multiple time or space scales, and show that they can be made asymptotic-preserving, in the sense that the gPC-SG scheme preserves various asymptotic limits in the discrete space. This allows the implementation of the gPC methods for these problems without numerically resolving (spatially, temporally or by gPC modes) the small scales. Rigorous analysis, based on hypocoercivity of the collision operator, will be provided for both linear transport and nonlinear Vlasov-Poisson-Fokker-Planck system to study the regularity and long-time behavior (sensitivity analysis) of the solution in the random space, and to prove that these schemes are stochastically asymptotic preserving.

uk: вхід без авторизації

name: Uncertainty quantification for multiscale kinetic equations with uncertain coefficients


uk: Математика, природничі науки

description: When a small quantum system is subject to multiple periodic drives, it may realize multidimensional topological phases. In my talk, I will explain how to make such constructions, and show how a spin-1/2 particle driven by two elliptically-polarized light beams could realize the Bernevig-Hughes-Zhang model of 2 topological insulators. The observable consequence of such construction is quantized pumping of energy between the two drive sources.

uk: вхід без авторизації

name: Topological frequency conversion in strongly driven quantum systems


keywords:
Mathematics
Combinatorics
Computer science
Discrete mathematics

uk: Математика, природничі науки

description: Consider the following two ways to colour a graph where the requirement that adjacent vertices get distinct colours is relaxed. A colouring has DEFECT $d$ if each monochromatic component has maximum degree at most $d$. A colouring has CLUSTERING $c$ if each monochromatic component has at most $c$ vertices. This talk surveys recent progress on these types of colourings for the following graph classes: planar graphs, graphs embeddable in surfaces, graphs with given maximum degree, graphs with linear crossing number, linklessly or knotlessly embeddable graphs, graphs with given Colin de Verdiere parameter, graphs with given thickness, graphs with given book-thickness, graphs excluding $K_t$ as a minor, graphs excluding $K_{s,t}$ as a minor, and graphs excluding an arbitrary graph $H$ as a minor. Several open problems are discussed.

uk: вхід без авторизації

name: Tutorial on defective and clustered graph colouring


uk: Математика, природничі науки

description: We continue the survey of defective and clustered colorings of minor-closed classes of graphs of graphs, focusing on the techniques and conjectures. The discussed techniques include: islands, separators, decompositons of graphs in minor closed classes into graphs of bounded treewidth, and linear decompositions of graphs of bounded treewidth. Based on joint work with Dvorak, and with Scott, Seymour and Wood.

uk: вхід без авторизації

name: Tutorial on defective colouring continued


Coloring graphs with forbidden minors

uk: Математика, природничі науки

description: As pointed out by Seymour in his recent survey on Hadwiger's conjecture, proving that graphs with no \(K_7\) minor are 6-colorable is the first case of Hadwiger's conjecture that is still open. It is not known yet whether graphs with no \(K_7\) minor are 7-colorable. Using a Kempe-chain argument along with the fact that an induced path on three vertices is dominating in a graph with independence number two, we first give a very short and computer-free proof of a recent result of Albar and Goncalves, and generalize it to the next step by showing that every graph with no \(K_t\) minor is \((2t-6)\)-colorable, where \(t\in\{7,8,9\}\). We then prove that graphs with no \(K_8^-\) minor are 9-colorable, and graphs with no \(K_8^=\) minor are 8-colorable. Finally we prove that if Mader's bound for the extremal function for \(K_t\) minors is true, then every graph with no \(K_t\) minor is \((2t-6)\)-colorable for all \(t\ge6\). This implies our first result.

uk: вхід без авторизації


uk: Математика, природничі науки

description: A class of graphs is \(\chi\)-bounded if there exists a function \(f:\mathbb{N}→\mathbb{N}\) such that for every graph \(G\) in the class and every induced subgraph \(H\) of \(G\), if \(H\) has no clique of size \(q+1\), then the chromatic number of \(H\) is less than or equal to \(f(q)\). We denote by \(W_n\) the wheel graph on \(n+1\) vertices. We show that the class of graphs having no vertex-minor isomorphic to \(W_n\) is \(\chi\)-bounded. This generalizes several previous results; \(\chi\)-boundedness for circle graphs, for graphs having no \(W_5\) vertex-minors, and for graphs having no fan vertex-minors. This is joint work with Hojin Choi, O-joung Kwon, and Paul Wollan.

uk: вхід без авторизації

name: Chi-boundedness of graph classes excluding wheel vertex-minors


uk: Математика, природничі науки

description: In this talk I will discuss the use of a Domain Decomposition method to reduced the computational complexity of classical problems arising in Uncertainty Quantification and stochastic Partial Differential equations. The first problem concerns the determination of the Karhunen-Loeve decomposition of a stochastic process given its covariance function. We propose to solve independently the decomposition problem over a set of subdomains, each with low complexity cost, and subsequently assemble a reduced problem to determined the global problem solution. We propose error estimates to control the resulting approximation error. Second, these ideas are extended to construct an efficient sampling approach for elliptic problems with stochastic coefficients expanded in a KL form. Here, we rely on the resolution of low complexity local stochastic elliptic problems to exhibit contributions to the condensed stochastic problem for the unknown boundary values at the internal subdomain boundaries. By relying intensively on local resolutions, that can be performed independently, the proposed approaches are naturally suited to parallel implementation and we will provide scalability results.

uk: вхід без авторизації

name: A domain decomposition method for stochastic elliptic differential equations


keywords: Probability theory and stochastic processes

keywords: Industrial mathematics / modelling and simulation

uk: Математика, природничі науки

description: For half a century computational scientists have been numerically simulating complex systems. Uncertainty is recently becoming a requisite consideration in complex applications which have been classically treated deterministically. This has led to an increasing interest in recent years in uncertainty quantification (UQ). Another recent trend is the explosion of available data. Bayesian inference provides a principled and well-defined approach to the integration of data into an a priori known distribution. The posterior distribution, however, is known only point-wise (possibly with an intractable likelihood) and up to a normalizing constant. Monte Carlo methods have been designed to sample such distributions, such as Markov chain Monte Carlo (MCMC) and sequential Monte Carlo (SMC) samplers. Recently, the multilevel Monte Carlo (MLMC) framework has been extended to some of these cases, so that numerical approximation error can be optimally balanced with statistical sampling error, and ultimately the Bayesian inverse problem can be solved for the same asymptotic cost as solving the deterministic forward problem. This talk will concern the recent development of various MLMC algorithms for Bayesian inference problems.

uk: вхід без авторизації

name: Multilevel Monte Carlo methods for Bayesian inference


uk: Математика, природничі науки

description: Scientists and engineers use computer simulations to study relationships between a physical model's input parameters and its output predictions. However, thorough parameter studies---e.g., constructing response surfaces, optimizing, or averaging---are challenging, if not impossible, when the simulation is expensive and the model has several inputs. To enable parameter studies in these cases, the engineer may attempt to reduce the dimension of the model's input parameter space. I will (i) describe computational methods for discovering low-dimensional structures in the parameter-to-quantity-of-interest map, (ii) propose strategies for exploiting the low-dimensional structures to enable otherwise infeasible parameter studies, and (iii) review results from several science and engineering applications.

uk: вхід без авторизації

name: Parameter space dimension reduction for forward and inverse uncertainty quantification


uk: Математика, природничі науки

description: In this work we propose an Uncertainty Quantification methodology for the evolution of sedimentary basins undergoing mechanical and geochemical compaction processes, which we model as a coupled, time-dependent, non-linear, monodimensional (depth-only) system of PDEs with uncertain parameters. Specifically, we consider multi-layered basins, in which each layer is characterized by a different material. The multi-layered structure gives rise to discontinuities in the dependence of the state variables on the uncertain parameters. Because of these discontinuites, an appropriate treatment is needed for surrogate modeling techniques such as sparse grids to be effective. To this end, we propose a two-steps methodology which relies on a change of coordinate system to align the discontinuities of the target function within the random parameter space. Once this alignement has been computed, a standard sparse grid approximation of the state variables can be performed. The effectiveness of this procedure is due to the fact that the physical locations of the interfaces among layers feature a smooth dependence on the random parameters and are therefore amenable to sparse grid polynomial approximations. We showcase the capabilities of our numerical methodologies through some synthetic test cases. References: Ivo Colombo, Fabio Nobile, Giovanni Porta, Anna Scotti, Lorenzo Tamellini, Uncertainty Quantification of geochemical and mechanical compaction in layered sedimentary basins, Computer Methods in Applied Mechanics and Engineering.

uk: вхід без авторизації

name: Uncertainty quantification of geochemical and mechanical compaction in layered sedimentary basins


uk: Математика, природничі науки

description: Let G be a graph with large chromatic number. What induced subgraphs must it contain? It may contain a large complete subgraph, but what can we say if this is not the case? We will survey recent work on this topic, concentrating on the question of finding induced cycles. In particular, we will discuss recent results with Paul Seymour, Maria Chudnovsky and Sophie Spirkl.

uk: вхід без авторизації

name: Holes in graphs of large chromatic number


uk: Математика, природничі науки

description: We will report on recent significant progress towards a structural description of large enough $k$-crossing-critical graphs, for every fixed k>1.

uk: вхід без авторизації

name: Towards a structure theorem for crossing-critical graphs

name: Banff International Research Station (BIRS) for Mathematical Innovation and Discovery


Crossing numbers

uk: Математика, природничі науки

description: One of the most useful tools in topological graph theory is the so-called Crossing Lemma of Ajtai, Chvatal, Newborn, Szemeredi (1982) and Leighton (1983). It states, roughly speaking, that if a graph drawn in the plane has much more edges than vertices, then the number of crossings between its edges is much larger than the number of edges. We extend this result to simple topological multigraphs, that is, for multigraphs drawn in the plane such that (1) any two independent edges meet in at most one point, (2) no two edges that share an endpoint have any interior point in common, and (3) both lenses enclosed by two edges that have the same endpoint contain at least one vertex in their interiors.

uk: вхід без авторизації


uk: Математика, природничі науки

description: A drawing of a graph in the plane is pseudolinear if every edge can be extended to an open arc that separates the plane into two regions such that any two such arcs have exactly one point in common and that common point is a crossing. We characterize pseudolinear drawings in terms of forbidden subconfigurations. A polynomial time algorithm for finding either the extensions or a forbidden subconfiguration is also given.

uk: вхід без авторизації

name: Characterizing pseudolinear drawings of graphs in the plane


uk: Математика, природничі науки

description: We find a graph of genus 5 and its drawing on the orientable surface of genus 4 with every pair of independent edges crossing an even number of times. This shows that the strong Hanani-Tutte theorem cannot be generalized to the orientable surface of genus 4. As a base step in the construction we use a counterexample to the unified Hanani-Tutte theorem on the torus. The result was obtained together with Radoslav Fulek.

uk: вхід без авторизації

name: Counterexample to the Hanani-Tutte theorem on the surface of genus 4


uk: Математика, природничі науки

description: We resolve in the affirmative conjectures of A. Skopenkov and Repovs (1998), and M. Skopenkov (2003) generalizing the classical Hanani--Tutte theorem to the setting of approximating maps of graphs in the plane by embeddings. Our proof of this result is constructive, and implies the existence of a polynomial-time algorithm for the following problem. An instance of the problem consists of (i) a graph $G$ whose vertices are partitioned into clusters and whose inter-cluster edges are partitioned into bundles, and (ii) a 2-dimensional surface $S$ given as the union of a set of pairwise disjoint discs corresponding to the clusters and a set of pairwise non-intersecting strips, ``pipes'', corresponding to the bundles, connecting certain pairs of these discs. We are to decide whether $G$ can be embedded inside $S$ so that the vertices in every cluster are drawn in the corresponding disc, the edges in every bundle pass only through its corresponding pipe, and every edge crosses the boundary of each disc at most once.

uk: вхід без авторизації

name: Hanani-Tutte for approximating maps of graphs


uk: Математика, природничі науки

description: We consider optimal control problems governed by PDEs with uncertain parameter fields, and in particular those with objective functions given by the mean and variance of the control objective. To make the problem tractable, we invoke a quadratic Taylor approximation of the objective with respect to the uncertain parameter field. This enables deriving explicit expressions for the mean and variance of the control objective in terms of its gradient and Hessian with respect to the uncertain parameter. The stochastic optimal control problem is then formulated as a PDE-constrained optimization problem with constraints given by the forward and adjoint PDEs defining these gradients and Hessians. The expressions for the mean and variance of the control objective under the quadratic approximation involve the trace of the (preconditioned) Hessian, and are thus prohibitive to evaluate for (discretized) infinite-dimensional parameter fields. To overcome this difficulty, we employ a randomized eigensolver to extract the dominant eigenvalues of the decaying spectrum. The resulting objective functional can now be readily differentiated using adjoint methods along with eigenvalue sensitivity analysis to obtain its gradient with respect to the controls. Along with the quadratic approximation and truncated spectral decomposition, this ensures that the cost of computing the objective and its gradient with respect to the control--measured in the number of PDE solves--is independent of the (discretized) parameter and control dimensions, leading to an efficient quasi-Newton method for solving the optimal control problem. Finally, the quadratic approximation can be employed as a control variate for accurate evaluation of the objective at greatly reduced cost relative to sampling the original objective. Several applications with high-dimensional uncertain parameter spaces will be presented.

uk: вхід без авторизації

name: Efficient and scalable methods for large-scale stochastic PDE-constrained optimal control


uk: Математика, природничі науки

description: This talk concerns the long-time evolution of stochastic ODE and PDE with random coefficients and white noise forcing by the method of PDEs.

uk: вхід без авторизації

name: Long time propagation of stochasticity by dynamical PCEs