PH-Jitter

uk: Математика, природничі науки

description: Recently, the statistics of short bars in persistence diagrams started to draw a lot of attention (especially from material sciences) as a descriptor of functions or spaces. I will talk about reformulation of some known results in terms of the PH point processes. New results deal with the generic PH-dimension for Lipschitz or Hoelder functions on manifolds, and statistics of short bars (and their chiralities) for Brownian motions.

uk: вхід без авторизації


Nicolas Berkouk

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: Multi-graded betti numbers, one of the only invariant we have to study multi-modules, are highly not stable with respect to the interleaving distance. However, they arise from the existence of free minimal resolutions and we propose to show how we can equip the derived category of persistent modules (the one in which resolutions « naturally » live) with a derived distance to make those resolutions stable. As a byproduct, we get the non-existence of a right (or left) exact functor from the category of modules to the one that decomposes in thin summands.

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: Stable resolutions of multi-persistent modules

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


Generalized Persistence Diagrams

uk: Математика, природничі науки

description: I will interpret the persistence diagram of Cohen-Steiner, Edelsbrunner, and Harer as the Möbius inversion of the rank function. I will then show how to generalize the persistence diagram to the setting of constructible persistence modules valued in any small symmetric monoidal category.

uk: вхід без авторизації


uk: Математика, природничі науки

description: The lines or grids of vector spaces appearing in the study of persistence can be seen as special instances of representations of posets (= presheaves on posets). In the finite case, representations of posets (or arbitrary quivers with relations) have independently been studied in the context of finite dimensional algebras. In my talk I will review the setup and some classical results of representation theory of finite dimensional algebras which one might hope to apply in the study of persistence.

uk: вхід без авторизації

name: Some background from representation theory


The persistent homotopy type distance

uk: Математика, природничі науки

description: I'll present the persistent homotopy type distance d_HT to compare two real valued functions defi ned on possibly different but homotopy equivalent topological spaces. The underlying idea in the defi nition of d_HT is to measure the minimal shift that is necessary to apply to one of the two functions in order that the sublevel sets of the two functions become homotopically equivalent. This distance is interesting in connection with persistent homology. Indeed, dHT still provides an upper bound for the bottleneck distance between the persistence diagrams of the intervening functions. Moreover, because homotopy equivalences are weaker than homeomorphisms, this implies a lifting of the standard stability results provided by the supremum distance and by the natural pseudo-distance d_NP. From a different standpoint, d_HT extends the supremum distance and d_NP in two ways. First, appropriately restricting the category of objects to which d_HT applies, it can be made to coincide with the other two distances. Secondly, dHT has an interpretation in terms of interleavings that naturally places it in the family of distances used in persistence theory.

uk: вхід без авторизації


Floer and Khovanov, 20 years after

uk: Математика, природничі науки

description: In his seminal paper on his "Jones polynomial homology," Khovanov identifies Floer homology as one of the underlying motivations for his work. In the 20 years since then, the connections between Khovanov homology (broadly interpreted) and Floer homology have been a fruitful topic for research, but there is still much that remains mysterious. I'll describe some of our successes at understanding the relations between the two theories over the last 20 years, and speculate on where things may go in the next 20.

uk: вхід без авторизації


uk: Математика, природничі науки

description: Involutive Heegaard Floer homology is a variant on the 3-manifold invariant Heegaard Floer homology which incorporates the data of the conjugation symmetry on the Heegaard Floer complexes, and is in principle meant to correspond to $\mathbb{Z}_4$-Seiberg Witten Floer homology. It can be used to obtain two new invariants of homology cobordism and two new concordance invariants of knots, one of which (unlike other invariants arising from Heegaard Floer homology) detects non-sliceness of the figure-eight knot. We introduce involutive Heegaard Floer homology and its associated invariants and use it to give a new criterion for an element in the integer homology cobordism group to have infinite order, similar but not identical to a recent criterion given by Lin-Ruberman-Saviliev. Much of this talk is joint work with C. Manolescu; other parts are variously also joint with I. Zemke or with J. Hom and T. Lidman.

uk: вхід без авторизації

name: Involutive Heegaard Floer homology and the homology cobordism group


uk: Математика, природничі науки

description: I will begin with a survey of concordance invariants constructed from Heegaard Floer homology. I then will describe a sequence of concordance invariants νn constructed from truncated Heegaard Floer homology that generalize the Ozsvath-Szabo ν invariant and the Hom-Wu ν+ invariant.

uk: вхід без авторизації

name: Truncated Heegaard Floer homology and concordance invariants


uk: Математика, природничі науки

description: Local weak convergence is a powerful framework for study of sparse graph limits and has been successfully applied in obtaining exact expectation asymptotics in probabilistic combinatorial optimization, statistical physics and random graph theory. In particular, it can be used to show that sum of lifetime sum of $H_0$-persistent diagram on a mean field model (complete graph with i.i.d. weights) converges to $\zeta(3)$, where $\zeta$ is the Riemann-zeta function. Further, using this framework the minimum cost function on the complete bipartite graph with i.i.d. weights was shown to converge to $\zeta(2)$. In this talk, we shall look at some underlying ideas behind such results and wonder about the possibility of extensions to random topology. As is to be expected, when we move from random graphs to random complexes, there will be fewer answers and more questions.

uk: вхід без авторизації

name: Local weak convergence, Zeta limits and random topology


uk: Математика, природничі науки

description: In their paper "The theory of multidimensional persistence", Carlsson and Zomorodian write "Our study of multigraded objects shows that no complete discrete invariant exists for multidimensional persistence. We still desire a discriminating invariant that captures persistent information, that is, homology classes with large persistence." In this talk I will discuss how tools from commutative algebra give computable invariants able to capture homology classes with large persistence. Specifically, multigraded associated primes provide a stratification of the region where a multigraded module does not vanish, while multigraded Hilbert series and local cohomology give a measure of the size of components of the module supported on different strata. These invariants generalize in a suitable sense the invariant for the one-parameter case.

uk: вхід без авторизації

name: Stratifying multi-parameter persistent homology


uk: Математика, природничі науки

description: The goal of this talk is to present some on-going work on proving stability for certain subcategories of multipersistence modules, specifically the ones that are decomposable into direct sums of free, rectangle, or interval modules. This is joint work with Magnus Botnan and Jérémy Cochoy.

uk: вхід без авторизації

name: Stability for certain subcategories of multipersistence modules


Fibered simple knots

uk: Математика, природничі науки

description: I will describe a characterization of which simple knots in lens spaces fiber. This is joint work with John Luecke.

uk: вхід без авторизації


uk: Математика, природничі науки

description: We will describe a new interpretation of bordered Heegaard Floer invariants in the case of a manifold M with torus boundary. In our setting, these invariants, originally defined as homotopy classes of (differential) modules over a particular algebra, take the form of homotopy classes of immersed curves in the boundary of M decorated with local systems. Moreover, pairing two bordered Floer invariants corresponds to taking the Floer homology of two sets of decorated immersed curves. In most cases this simply counts the minimal intersection number, which leads to a number of applications. This is joint work with Liam Watson and Jake Rasmussen.

uk: вхід без авторизації

name: Bordered Floer homology via immersed curves: properties and applications


uk: Математика, природничі науки

description: This talk will describe some of the key components of theorem stated in the previous talk, namely, the passage from differential modules over the torus algebra to immersed curves. We give a description of type D structures in terms of train tracks in the (marked) torus, and develop a yoga for simplifying these train tracks to immersed curves with local systems. This is joint work with Jonathan Hanselman and Jake Rasmussen.

uk: вхід без авторизації

name: Bordered Floer homology via immersed curves: the structure theorem


uk: Математика, природничі науки

description: I will explain the construction of a new homology theory for three-manifolds, defined using perverse sheaves on the SL(2,C) character variety. Our invariant is a model for an SL(2,C) version of Floer’s instanton homology. I will present a few explicit computations for Brieskorn spheres, and discuss the connection to the Kapustin-Witten equations and Khovanov homology. This is joint work with Mohammed Abouzaid.

uk: вхід без авторизації

name: A sheaf-theoretic model for SL(2,C) Floer homology


uk: Математика, природничі науки

description: Classical Morse theory connects the topology of a manifold with critical points of a Morse function. Stratified Morse theory considers the topology of a stratified space with critical points of a stratified Morse function. Discrete Morse theory is considered as a combinatorial adaptation of Classical Morse theory, and is applied to simplicial or cell complexes. It seems natural to ask whether there could be a combinatorial adaptation of Stratified Morse theory. This talk will serve as a conversation starter, where I would like to discuss a (possibly) discrete version of stratified Morse theory and its (potential) application in data analysis.

uk: вхід без авторизації

name: A Discrete Version of Stratified Morse Theory: A "Peachy Temptation"


uk: Математика, природничі науки

description: I will present a simple-minded view of interleaving: we want two diagrams to fit together inside a bigger diagram. I will show how this idea can be used to draw pictures that prove a number of persistence results. Surprisingly (to us!) carefully working out the theory underlying these ideas led us to discover that interleaving distance and Gromov-Hausdorff distance are both special cases of a richer theory. Joint work with Vin de Silva and Jonathan Scott.

uk: вхід без авторизації

name: A pictorial approach to persistent homology


Skeleta of Weinstein manifolds

uk: Математика, природничі науки

description: The Floer theory of a cotangent fiber in a symplectic cotangent bundle T*M can be understood via the topology of the manifold M. More generally, a Weinstein manifold has a core isotropic skeleton and we can try to understand the symplectic topology of the Weinstein manifold in terms of the topology of the skeleton. Unfortunately, generically the skeleton has singularities which make its topology difficult to understand and which lead to loss of information. We will discuss a nice minimal set of singularities which we can understand combinatorially, and try to show that all Weinstein manifolds can be deformed to have a skeleton with only these nice singularities. These singularities coincide with Nadler’s “arboreal singularities” where Floer theoretic calculations can locally be done combinatorially.

uk: вхід без авторизації


Equivariant Floer homology

uk: Математика, природничі науки

description: Given a Lie group G acting on a symplectic manifold preserving a pair of Lagrangians setwise, I will describe a construction of G-equivariant Lagrangian Floer homology. This does not require G-equivariant transversality, which allows the construction to be flexible. Time permitting, I will talk about applying this for the O(2)-action on Seidel-Smith's symplectic Khovanov homology. This is joint with Kristen Hendricks and Robert Lipshitz.

uk: вхід без авторизації


A new concordance group of links

uk: Математика, природничі науки

description: The knot concordance group has been the subject of much study since its introduction by Ralph Fox and John Milnor in 1966. One might hope to generalize the notion of a concordance group to links; however, the immediate generalization to the set of links up to concordance does not form a group since connected sum of links is not well-defined. In 1988, Jean Yves Le Dimet defined the string link concordance group, where a link is based by a disk and represented by embedded arcs in D^2 × I. In 2012, Andrew Donald and Brendan Owens defined groups of links up to a notion of concordance based on Euler characteristic. However, both cases expand the set of links modulo concordance to larger sets and each link has many representatives in these larger groups. In this talk, I will present joint work with Matthew Hedden where we define a link concordance group based on the the “knotification” construction of Peter Ozsv´ath and Zoltan Szab´o, giving a definition of a link concordance group where each link has a unique group representative. I will also present invariants for studying this group using both group theory and Heegaard-Floer Homology.

uk: вхід без авторизації


TDA for Spiral Defect Chaos

uk: Математика, природничі науки

description: A look at some research directions inspired by studying spiral defect chaos in Rayleigh-Benard convection via topological methods.

uk: вхід без авторизації


uk: Математика, природничі науки

description: I will report on work in progress, joint with U. Bauer, M. Botnan and S. Oppermann. One of the main motivations for this project was to try to determine the complexity of 2-parameter clustering. Although this eventually led to purely representation theoretic results, the focus of this talk is what one of these results tells us in the realm of clustering. To be more precise, we show that the category of 2-parameter persistence modules with epimorphisms either horizontally, vertically, or both, is nearly as bad as the full category of 2-parameter persistence modules.

uk: вхід без авторизації

name: A representation theoretic take on clustering


Persistence and noncommutative geometry

uk: Математика, природничі науки

description: Motivated by the attempt to build on the top of reasonable data ( e.g. weighted networks) so-called BF-theories (a kind of topological quantum field theories), we will extend the usual approach of multidimensional persistent (co-)homology by considering diagrams of incidence algebras of finite posets. By using results of Gerstenhaber et al. we will show what is the link between diagram cohomology, Hochschild cohomology, and multidimensional persistence connecting seemingly uncorrelated research areas as topological data analysis and topological quantum field theory.

uk: вхід без авторизації


uk: Математика, природничі науки

description: Dynamic computer experiments are those with time series outputs. Research on such experiments has been gaining momentum recently. In this talk, we consider two problems of dynamic computer experiments. We propose a computationally efficient modeling approach to build emulators for large-scale dynamic computer experiments. This approach sequentially finds a set of local design points based on a new criterion specifically designed for emulating dynamic computer simulators. Singular value decomposition based Gaussian process models are built with the sequentially chosen local data. To update the models efficiently, an empirical Bayesian approach is introduced. When a target observation is available, estimating the inputs of the computer simulator that produce the matching response as close as possible is known as inverse problem. We propose a new criterion-based estimation method to address the inverse problem of dynamic computer experiments.

uk: вхід без авторизації

name: Computer Experiments III: Recent developments in dynamic computer experiments


uk: Математика, природничі науки

description: Most “learning” in big data is driven by the data alone. Some people may believe this is sufficient because of the sheer data size. If the physical world is involved, this approach is often insufficient. In this talk I will give a recent study to illustrate how physics and data are used jointly to learn about the “truth” of the physical world. In the quest for advanced propulsion systems, a new design methodology is needed which combines engineering physics, computer simulations and statistical modeling. There are two key challenges: the simulation of high-fidelity spatial-temporal flows (using the Navier-Stokes equations) is computationally expensive, and the analysis and modeling of this data requires physical insights and statistical tools. First, a surrogate model is presented for efficient flow prediction in swirl injectors with varying geometries, devices commonly used in many engineering applications. The novelty lies in incorporating properties of the fluid flow as simplifying model assumptions, which allows for quick emulation in practical turnaround times, and also reveals interesting flow physics which can guide further investigations. Next, a flame transfer function framework is proposed for modeling unsteady heat release in a rocket injector. Such a model is useful not only for analyzing the stability of an injector design, but also identifies key physics which contribute to combustion instability.

uk: вхід без авторизації

name: Computer Experiments II: Spatial-temporal Kriging, Navier-Stokes, and Combustion Instability


uk: Математика, природничі науки

description: Besides optimization and parameter estimation, discrimination between rival models has always been a prime objective of experimental design. A good review of the early developments is given in Hill (1978). A big leap from these rather ad-hoc approaches was Atkinson & Fedorov (1975), who introduced $T$-optimality derived from the likelihood ratio test under the assumption that one model is true and its parameters are fixed at nominal values. Maximization of the noncentrality parameter is equivalent to maximizing the power of the respective likelihood ratio test. When the models are nested, $T$-optimality can be shown to be equivalent to $D_s$-optimality for the parameters that embody the deviations from the smaller model (see eg. Fedorov and Khabarov 1986). For this setting the optimal design questions are essentially solved and everything hinges on the asymmetric nature of the NP-lemma. However, the design problem itself is inherently symmetric as it is usually the purpose of the experiment to determine which of two different models is true, and nestedness of the models is a less common situation. The purpose of the talk is to propose a new criterion, termed $\Delta$-optimality, to solve the discrimination design problem for non-nested non-linear regression models, without having to resort to asymmetry as in the references above. We will suppose that we do not have a prior probability distribution on the unknown parameters of the models, which rules out Bayesian approaches such as Felsenstein (1992). Nevertheless, we will assume a specific kind of prior knowledge about the unknown parameters, extending the approach of local optimality. We will demonstrate methodological and computational advantages of the proposed criterion and illustrate its use in a practical setting from pharmacology.

uk: вхід без авторизації

name: A design criterion for symmetric model discrimination


uk: Математика, природничі науки

description: We will present new constructions of small exotic symplectic 4-manifolds via rational blow-down surgery. This is a joint work with Anar Akhmedov.

uk: вхід без авторизації

name: New construction of small exotic 4-manifolds


uk: Математика, природничі науки

description: The Euler characteristic is multiplicative in fiber bundles. On the other hand, the signature is not. Atiyah and, independently, Kodaira showed it by giving surface bundles over surfaces with non-zero signatures. Since then, many examples with non-zero signatures have been constructed. The signature of a surface bundle over a surface has some restrictions, for examples, it is dividable by 4 and vanishes if the base genus is 0 or 1. Bryan and Donagi constructed examples over a genus-2 surface with non-zero signatures. The signatures and the genera of their examples are sporadic. In this talk, for any positive integer n, we give a surface bundle of fiber genus g over a surface of genus 2 with signature 4n and a section of self-intersection 0 if g is greater than or equal to 39n. Such example are constructed using mapping class group arguments.

uk: вхід без авторизації

name: Signatures of surface bundles over surfaces

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


The 4-Dimensional Light Bulb Theorem

uk: Математика, природничі науки

description: We generalize the classical light bulb theorem to four dimensions. I.e. a smooth 2-sphere in S2×S2 that intersects S2×0 once and is homologous to 0×S2 is smoothly isotopically standard.

uk: вхід без авторизації


uk: Математика, природничі науки

description: Automated experimental systems, involving minimal human intervention, are becoming more popular and common, providing economical and fast data collection. We discuss some statistical issues around the design of experiments and data modelling for such systems. Our application is to “closed-loop” optimisation of chemical processes, where automation of reaction synthesis, chemical analysis and statistical design and modelling increases lab efficiency and allows 24/7 use of equipment. Our approach uses nonparametric regression modelling, specifically Gaussian process regression, to allow flexible and robust modelling of potentially complex relationships between reaction conditions and measured responses. A Bayesian approach is adopted to uncertainty quantification, facilitated through computationally efficient Sequential Monte Carlo algorithms for the approximation of the posterior predictive distribution. We propose a new criterion, Expected Gain in Utility (EGU), for optimisation of a noisy response via fully-sequential design of experiments, and we compare the performance of EGU to extensions of the Expected Improvement criterion, which is popular for optimisation of deterministic functions. We also show how the modelling and design can be adapted to identify, and then down-weight, potentially outlying observations to obtain a more robust analysis.

uk: вхід без авторизації

name: Robustness III: Closed-loop automatic experimentation for optimisation


uk: Математика, природничі науки

uk: вхід без авторизації

name: A substitute for square lattice designs for 36 treatments


Distributionally Robust Optimal Designs

uk: Математика, природничі науки

description: The optimal design of experiments for nonlinear (or generalized-linear) models can be formulated as the problem of finding a design $\xi$ maximizing a criterion $\Phi(\xi,\theta)$, where $\theta$ is the unknown quantity of interest that we want to determine. Several strategies have been proposed to deal with the dependency of the optimal design on the unknown parameter $\theta$. Whenever possible, a sequential approach can be applied. Otherwise, Bayesian and Maximin approaches have been proposed. The robust maximin designs maximizes the worst-case of the criterion $\Phi(\xi,\theta)$, when $\theta$ varies in a set $\Theta$. In many cases however, such a design performs well only in a very small subset of the region $\Theta$, so a maximin design might be far away from the optimal design for the true value of the unknown parameter. On the other hand, it has been proposed to assume that a prior for $\theta$ is available, and to minimize the expected value of the criterion with respect to this prior. One objection to this approach is that when a sequential approach is not possible, we rarely have precise distributional information on the unkown parameter $\theta$. In the literature on optimization under uncertainty, the Bayesian and maximin approaches are known as "stochastic programming" and "robust optimization", respectively. A third way, somehow in between the two other paradigms, has received a lot of attention recently. The distributionally robust approach can be seen as a robust counterpart of the Bayesian approach, in which we optimize against the worst-case of all priors belonging to a family of probability distributions. In this talk, we will give equivalence theorems to characterize distributionally-robust optimal (DRO) designs. We will show that DRO-designs can be computed numerically by using semidefinite programming (SDP) or second-order cone programming (SOCP), and we will compare DRO-designs to Bayesian and maximin-optimal designs in simple cases.

uk: вхід без авторизації


uk: Математика, природничі науки

description: A necessary and sufficient condition for a smooth, compact 4-manifold to admit the structure of a Stein domain was given many years ago by Eliashberg, in terms of the existence of a certain kind of handle decomposition. In practice it is not always clear whether such a handle decomposition exists, even on a 4-manifold that is topologically very simple. We describe examples of 4-dimensional 2-handlebodies with a single 2-handle that do not obviously satisfy Eliashberg’s criteria yet still admit a Stein structure, and also examples of contractible 4-manifolds that do not admit any Stein structure despite satisfying various necessary conditions. The latter examples have bearing on a conjecture of Gompf asserting that no Brieskorn homology sphere admits a pseudoconvex embedding in complex 2-space.

uk: вхід без авторизації

name: Existence and non-existence of Stein structures on simple 4-manifolds


Graded roots and contact structures

uk: Математика, природничі науки

description: To every negative definite plumbing, Nemethi associates a computable combinatorial object called graded root. In the case the plumbing is the resolution of an almost rational singularity, the corresponding graded root captures the Heegaard Floer homology of the boundary 3-manifold. In this talk, I'll demonstrate how to detect the contact Ozsvath-Szabo invariant inside a graded root when the contact structure is compatible with an almost rational plumbing. As an application, we obstruct the existence of a Stein cobordism between the canonical contact structures on links of singularities. This is a joint work with F. Ozturk.

uk: вхід без авторизації


uk: Математика, природничі науки

description: A cork is a contractible smooth 4-manifold with an involution on its boundary that does not extend to a diffeomorphism of the entire manifold. Corks can be used to detect exotic structures; in fact, any two smooth structures on a closed simply-connected 4-manifold are related by a cork twist. Recently, Auckly-Kim-Melvin-Ruberman showed that for any finite subgroup G of SO(4) there exists a contractible 4-manifold with an effective G-action on its boundary so that the twists associated to the non-trivial elements of G do not extend to diffeomorphisms of the entire manifold. In this talk, we will use Heegaard Floer techniques originating in work of Akbulut-Karakurt to give a different proof of this phenomenon.

uk: вхід без авторизації

name: Equivariant corks and Heegaard Floer homology


uk: Математика, природничі науки

description: We consider the construction of robust sampling designs for the estimation of threshold probabilities in spatial studies. A threshold probability is a probability that the value of a stochastic process at a particular location exceeds a given threshold. We propose designs which estimate a threshold probability efficiently, and also deal with two possible model uncertainties: misspecified regression responses and misspecified variance/covariance structures. The designs minimize a loss function based on the relative mean squared error of the predicted values (i.e., relative to the true values). To this end an asymptotic approximation of the loss function is derived. To address the uncertainty of the variance/covariance structures of this process, we average this loss over all such structures in a neighbourhood of the experimenter's nominal choice. We then maximize this averaged loss over a neighbourhood of the experimenter's fitted model. Finally the maximum is minimized, to obtain a minimax design.

uk: вхід без авторизації

name: New Researchers I: Robust design for the estimation of a threshold probability


uk: Математика, природничі науки

description: A common problem in Phase II clinical trials is the comparison of dose response curves corresponding to different treatment groups. If the effect of the dose level is described by parametric regression models and the treatments differ in the administration frequency (but not in the sort of drug) a reasonable assumption is that the regression models for the different treatments share common parameters. During the talk we develop optimal design theory for the comparison of different regression models with common parameters. We derive upper bounds on the number of support points of admissible designs, and explicit expressions for D-optimal designs for frequently used dose response models with a common location parameter. If the location and scale parameter in the different models coincide, the problem becomes much harder and therefore we determine minimally supported designs and sufficient conditions for their optimality in the class of all designs.

uk: вхід без авторизації

name: New Researchers II: Optimal designs for dose response curves with common parameters


uk: Математика, природничі науки

description: Finding Bayesian optimal designs for a nonlinear model is generally a difficult task, especially when there are several factors in the study. Such optimal designs are often analytically intractable and require expensive effort to compute them. A main problem is the unknown number of support points required for the optimal design. The often used Monte Carlo approximations require very large samples from the parameter space to have reasonable accuracy and are thus unrealistic for moderate to high dimensional models. In this talk, we propose an effective and assumptions free approach for finding numerical Bayesian optimal designs with a few real applications to longitudinal models in HIV studies. Our algorithm hybridizes particle swarm optimization and sparse grids methodology (PSOSG) and we demonstrate its potential for finding Bayesian optimal designs for a variety of models using user-specified prior distributions. The optimality of all our generated designs is verified using an equivalence theorem and if time permits, we also discuss applications of PSOSG to find other types of optimal designs.

uk: вхід без авторизації

name: New Researchers IV: Sparse grid hybridized PSO for finding Bayesian optimal designs


uk: Математика, природничі науки

description: Random coefficient regression (RCR) models are popular in many fields of statistical application; especially in biosciences and medical research. In these models observational units (individuals) are assumed to come from the same population with an unknown population mean and differ from each other by individual random parameters. Besides the estimation of the population mean parameter, the prediction of the individual response is often of prior interest. In the particular case of multiple group RCR models individuals in different groups get different kinds of treatment. If group sizes are fixed and the unknown mean parameters may differ from group to group, statistical analysis can be performed in each group separately (see Prus, M.: Optimal Designs for the Prediction in Hierarchical Random Coefficient Regression Models. Ph.D. thesis, Otto-von-Guericke University Magdeburg (2015).). This talk presents analytical results for optimal group sizes for the prediction of the individual parameters in multi group RCR models with a common population mean for all individuals across all groups.

uk: вхід без авторизації

name: New Researchers III: Optimal designs for individual prediction in multiple group random coefficient regression models


uk: Математика, природничі науки

description: Particle swarm optimization (PSO) is a population-based global optimization and evolves a group of solutions stochastically, which was motivated by the behavior of a flock of birds or school of fish in nature. PSO is used to solve a wide array of different optimization problems because of its attractive advantages, such as the ease of implementation and its gradient free stochastic algorithm. It has been proved to be an efficient method for many global optimization problems, and not suffering from the difficulties encountered by other evolutionary computation techniques. In this talk, we will review PSO and its variants and then discuss several applications for complicated statistical models. In particular, we will discuss on computational issues of PSO in practice and show how to use it for pharmacokinetics/pharmacodynamics (PK/PD) modeling, including our recent applications to adaptive Phase II clinical trial designs.

uk: вхід без авторизації

name: Statistical modeling and applications of particle swarm optimization


uk: Математика, природничі науки

description: We shall describe the SO(3)-monopole cobordism approach to proving two results concerning gauge-theoretic invariants of closed, four-dimensional, smooth manifolds. First, we shall explain how the SO(3)-monopole cobordism are used to prove that all four-manifolds with Seiberg-Witten simple type satisfy the superconformal simple type condition defined by Marino, Moore, and Peradze (1999). This result implies a lower bound, conjectured by Fintushel and Stern (2001), on the number of Seiberg-Witten basic classes in terms of topological data. Second, we shall explain how the SO(3)-monopole cobordism and the superconformal simple type property are used to prove Witten's Conjecture (1994) relating the Donaldson and Seiberg-Witten invariants. Our presentation is primarily based on our articles arXiv:1408.5307 and arXiv:1408.5085 and book arXiv:math/0203047 (to appear in Memoirs of the American Mathematical Society).

uk: вхід без авторизації

name: SO(3) monopoles and relations between Donaldson and Seiberg-Witten invariants


uk: Математика, природничі науки

description: Using the knot filtration on the Heegaard Floer chain complex, Ozsváth and Szabó defined an invariant of knots in the three sphere called τ(K), which they also showed is a lower bound for the 4-ball genus. Generalizing their construction, I will show that for a (not necessarily null-homologous) knot, K, in a rational homology sphere, Y, we obtain a collection of τ-invariants, one for each spin-c structure. In addition, these invariants can be used to obtain a lower bound on the genus of a surface with boundary K properly embedded in a negative definite 4-manifold with boundary Y.

uk: вхід без авторизації

name: τ-invariants for knots in rational homology spheres


uk: Математика, природничі науки

description: The presence of missing response values complicates statistical analyses. However, incomplete data are particularly problematic when constructing optimal designs, as it is not known at the design stage which values will be missing. When data are missing at random (MAR) it is possible to incorporate this information into the optimality criterion that is used to find designs. However, when data are not missing at random (NMAR) such a framework can lead to inefficient designs. We first investigate an issue common to all missing data mechanisms: The covariance matrix of the estimators does not exist, so it is not clear how well the inverse of the information matrix will approximate the observed covariance matrix. To this end, we propose and study a new approximation to the observed covariance matrix for situations where the missing data mechanisms is MAR. We then address the specific challenges that NMAR values present when finding optimal designs for linear regression models. We show that the optimality criteria will depend on model parameters that traditionally do not affect the design, such as regression coefficients and the residual variance. We also develop a framework that improves efficiency of designs over those found assuming values are MAR.

uk: вхід без авторизації

name: Optimal design when outcome values may be missing


uk: Математика, природничі науки

description: To design longitudinal studies with nonlinear mixed effect models (NLMEM), optimal design based on the expected Fisher information matrix (FIM) can be used. A new method evaluating the FIM based on Monte-Carlo Hamiltonian Monte-Carlo (MC/HMC) was developed and implemented in the R package MIXFIM using Stan for HMC sampling. This approach requires a priori knowledge on models and parameters, leading to locally optimal designs. The objective of this work was to extend this MC/HMC-based method to evaluate the FIM in NLMEM accounting for uncertainty in parameters and in models. We showed an illustration of this approach to optimize robust designs for repeated count data. When introducing uncertainty on the population parameters, we evaluated the robust FIM as the expectation of the FIM computed by MC/HMC on these parameters. Then, the compound D-optimality criterion was used to find a common CD-optimal design for several candidate models. A compound DE-criterion combining the determinants of the robust FIMs was also calculated to find the CDE-optimal design which was robust with respect to both model and parameters. These methods were applied in a longitudinal Poisson count model which event rate parameter.

uk: вхід без авторизації

name: Using Hamiltonian Monte-Carlo to design longitudinal count studies accounting for parameter and model uncertainties


uk: Математика, природничі науки

description: We construct optimal designs for group testing experiments where the goal is to estimate the prevalence of a trait using a test with uncertain sensitivity and specificity. Using optimal design theory for approximate designs, we show that the most efficient design for simultaneously estimating the prevalence, sensitivity, and specificity requires three different group sizes with equal frequencies. However, if estimating prevalence as accurately as possible is the only focus, the optimal strategy is to have three group sizes with unequal frequencies. Based on a Chlamydia study in the United States, we compare performances of competing designs and provide insights into how the unknown sensitivity and specificity of the test affect the performance of the prevalence estimator. We demonstrate that the proposed locally D- and Ds-optimal designs have high efficiencies even when the prespecified values of the parameters are moderately misspecified. Extensions on budget-constrained optimal group testing designs will also be discussed, where both subjects and tests incur costs, and assays have uncertain sensitivity and specificity that may be linked to the group sizes.

uk: вхід без авторизації

name: Optimal group testing designs for estimating prevalence with uncertain testing errors


uk: Математика, природничі науки

description: We consider the problem of construction of optimal experimental designs for linear regression models with correlated observations. In the first part of the talk, we assume that the ordinary least square estimator is used. The approach of Bickel-Herzberg is reviewed and several generalization of this approach are discussed. Then we consider the problem of construction of BLUE and some of its discrete approximations. In this case, the problem of construction of asymptotically optimal experimental designs is much more evolved. The celebrated Sacks-Ylvisaker approach is reviewed and several approximate methods of construction of optimal estimators and designs are discussed.

uk: вхід без авторизації

name: Optimal design in regression models with correlated errors


uk: Математика, природничі науки

description: Fractional polynomials (FP) have been shown to be much more flexible than polynomials for fitting continuous outcomes in the biological and health sciences. Despite their increasing popularity, design issues for FP models have never been addressed. D- and I-optimal experimental designs will be computed for prediction using FP models. Their properties will be evaluated and a catalogue of design points useful for FP models will be provided. As applications, we consider linear mixed effects models for longitudinal studies. To provide greater flexibility in modeling the shape of the response, we use fractional polynomials and not polynomials to approximate the mean response. An example using gene expression data will be considered comparing the designs used in practice. An additional an interesting problem is finding designs for effective model discrimination for FP models. This will be explored from the KL-optimality point of view.

uk: вхід без авторизації

name: Optimal designs for longitudinal studies with fractional polynomial models


uk: Математика, природничі науки

description: Fractional polynomial models are potentially useful for response surface investigations. With the availability of routines for fitting nonlinear models in statistical packages they are increasingly being used. However as in all experiments the design should be chosen such that the model parameters are estimated as efficiently as possible. The design choice for such models involves the usual difficulties of nonlinear models design. We find Bayesian optimal exact designs for several fractional factorial models. The optimum designs are compared to various standard designs in response surface problems. Some unusual problems in the choices of prior and optimization method will be noted.

uk: вхід без авторизації

name: Bayesian optimal designs for fitting fractional polynomial response surface models

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


On Data Reduction of Big Data

uk: Математика, природничі науки

description: Extraordinary amounts of data are being produced in many branches of science. Proven statistical methods are no longer applicable with extraordinary large data sets due to computational limitations. A critical step in Big Data analysis is data reduction. In this presentation, I will review some existing approaches in data reduction and introduce a new strategy called information-based optimal subdata selection (IBOSS). Under linear and nonlinear models set up, theoretical results and extensive simulations demonstrate that the IBOSS approach is superior to other approaches in term of parameter estimation and predictive performance. The tradeoff between accuracy and computation cost is also investigated. When models are mis-specified, the performance of different data reduction methods are compared through simulation studies. Some ongoing research work as well as some open questions will also be discussed.

uk: вхід без авторизації


uk: Математика, природничі науки

description: Skype presentation: Using a mathematical programming lens, we will look at a parametric family of optimal design problems in depth. Two popular problems, the approximate D- and A- optimal design problems, will be highlighted as two special members and some of the results will be customised for them. First, we will present mathematical formulations, derive their duals and obtain optimality conditions. Second, we will survey different types of algorithms that are in use and discuss their convergence properties. Third, we will provide a branch and bound framework together with techniques to generate lower and upper bounds to obtain exact designs. Along the way, we will be identifying immediate research questions, as well as theoretical and computational challenges awaiting to be tackled.

uk: вхід без авторизації

name: A mathematical programming look at optimal experimental design problems


uk: Математика, природничі науки

description: We construct an invariant of closed spin^c 4-manifolds. This invariant is defined using families of Seiberg-Witten equations and formulated as a cohomology class on a certain abstract simplicial complex. We also give examples of 4-manifolds which admit positive scalar curvature metrics and for which this invariant does not vanish. This non-vanishing result of our invariant provides a new class of adjunction-type genus constraints on configurations of embedded surfaces in a 4-manifold whose Seiberg-Witten invariant vanishes.

uk: вхід без авторизації

name: A cohomological Seiberg-Witten invariant emerging from the adjunction inequality


uk: Математика, природничі науки

description: A good experimental design in a non-parametric framework, such as Gaussian process modelling in computer experiments, should have satisfactory space-filling properties. Minimax-distance designs minimize the maximum distance between a point of the region of interest and its closest design point, and thus have attractive properties in this context. However, their construction is difficult, even in moderate dimension, and one should in general be satisfied with a design that is not too strongly suboptimal. Several methods based on a discretization of the experimental region will be considered, such as the determination of Chebyshev-centroidal Voronoi tessellations obtained from fixed-point iterations of Lloyds' method, and the construction of any-time (nested) suboptimal solutions by greedy algorithms applied to submodular surrogates of the minimax-distance criterion. The construction of design measures that minimize a regularized version of the criterion will also be investigated.

uk: вхід без авторизації

name: On the construction of minimax-distance (sub-)optimal designs


uk: Математика, природничі науки

description: I will construct infinite families of Rohlin invariant one Brieskorn homology spheres bounding rational balls (a joint work with Kyle Larson). I will then switch over discussing homology spheres bounding contractible manifolds. A contractible manifold W can fail to be a cork, either if there is no exotic involution on its boundary, or W fails to be Stein. We will give examples of Stein non-corks, and non-Stein non-loose corks. It is still an open question whether there are loose-corks that can not be corks (this part is a joint work with Danny Ruberman).

uk: вхід без авторизації

name: Homology spheres bounding rational balls


uk: Математика, природничі науки

description: We introduce the Kodaira dimension of contact 3-manifolds and show that contact 3-manifolds with distinct Kodaria dimensions behave differently when it comes to the geography of various kinds of fillings. We also prove that, given any contact 3-manifold, there is a lower bound of $2\ chi+ 3\ sigma $ for all its minimal symplectic fillings. This generalizes the similar bound of Stipsicz for Stein fillings. This talk is based on joint works with Cheuk Yu Mak, and partly with Koichi Yasui.

uk: вхід без авторизації

name: Geography of symplectic fillings in dimension 4


uk: Математика, природничі науки

description: The Wall's stable h-cobordism theorem states that homotopy equivalent, smooth simply-connected 4-manifolds become diffeomorphic after stabilizing, i.e. connected summing with some finite number of a S^2-bundle over S^2. And, in fact, all known examples need only one stabilization to be diffeomorphic. In this talk, we will talk about the analogous stabilization question for knotted surfaces in simply-connected 4-manifolds produced by all of the known constructions based on Fintushel-Stern knot surgery. And we will prove that any pair of these knotted surfaces that preserve the fundamental groups of their complements become all diffeomorphic after single stabilization.

uk: вхід без авторизації

name: Surfaces in 4-manifolds and 1-stable equivalence


uk: Математика, природничі науки

description:
Quantum control is an important prerequisite for quantum devices [1]. A major obstacle is the fact that a quantum system can never completely be isolated from its environment. The interaction of a quantum system with its environment causes decoherence. Optimal control theory is a tool that can be used to identify control strategies in the presence of decoherence. I will show how to adapt optimal control theory to quantum information tasks for open quantum systems [2].

 A key application of quantum control is to identify performance bounds, for tasks such as state preparation or quantum gate implementation, within a given architecture. One such bound is the quantum speed limit, which determines the shortest possible duration to carry out the task at hand. For open quantum systems, interaction with the environment may lead to a speed-up of the desired evolution. Here, I will show how initial correlations between system and environment may not only be exploited to speed up qubit reset but also to increase state preparation fidelities. Geometric control techniques provide an intuitive understanding of the underlying dynamics [3].

 Control tasks such as state preparation or gate implementation are typically optimized for known, fixed parameters of the system. Showcasing the full capabilities of quantum optimal control, I will discuss how recent advances in quantum control techniques allow for going even further. Using a fully numerical quantum optimal control approach, it is possible to map out the entire parameter landscape for superconducting transmon qubits. This allows to determine the global quantum speed limit for a universal set of gates with gate errors limited solely by the qubit lifetimes. It thus provides the optimal working points for a given architecture [4].

 While the interaction of qubits with their environment is typically regarded as detrimental, this does not need to be the case. I will show that the back-flow of amplitude and phase encountered in non-Markovian dynamics can be exploited to carry out quantum control tasks that could not be realized if the system was isolated [5]. The control is facilitated by a few strongly coupled, sufficiently isolated environmental modes. These can be found in a variety of solid-state devices including superconducting circuits.

 [1] S. Glaser et al.: Training Schrödinger's cat: quantum optimal control, Eur. Phys. J. D 69, 279 (2015)
 [2] C. P. Koch: Controlling open quantum systems: Tools, achievements, limitations, J. Phys. Cond. Mat. 28, 213001 (2016)
 [3] D. Basilewitsch et al.: Beating the limits with initial correlations, arXiv:1703.04483
 [4] M. H. Goerz et al.: Charting the circuit-QED Design Landscape Using Optimal Control Theory, arXiv:1606.08825
 [5] D. M. Reich, N. Katz & C. P. Koch: Exploiting Non-Markovianity for Quantum Control, Sci. Rep. 5, 12430 (2015)

uk: вхід без авторизації

name: Optimal control of open quantum systems: Theoretical foundations and applications to superconducting quantum devices


uk: Математика, природничі науки

description: We focus on quantum systems subject to external interactions (laser, magnetic fields) taken as controls, which are contaminated with non perturbative noise. The measured observables come thus in the form of probability laws; we ask the following question: is it possible, from the knowledge of these probability laws, to recover the free and interaction (dipole) Hamiltonians ? We see that the theoretical answer is positive (provided some assumptions on the controllability of the quantum system hold); then we explore numerical approaches which exploit particular metrics on the space of probability laws.

uk: вхід без авторизації

name: Identification of quantum Hamiltonians in presence of non-perturbative noisy data


uk: Математика, природничі науки

description: A numerical scheme that solves the time-dependent Dirac equation is presented in which the time evolution is performed by an operator-splitting decomposition technique combined with the method ofcharacteristics. On a classical computer, this numerical method has some nice features: it is very versatile and most notably, it can be parallellized efficiently. This makes for an interesting numerical tool for the simulation of quantum relativistic dynamical phenomena such as the electron dynamics in very high intensity lasers. Moreover, this numerical scheme can be implemented on a digital quantum computer due to its simple structure: the operator splitting is a sequence of streaming operators followed by rotations in spinor space. This structure is actually reminiscent of quantum walks, which can be implemented efficiently on quantum computers. We determine the resource requirements of the resulting quantum algorithm and show that under some conditions, it has an exponential speedup over the classical algorithm. Finally, an explicit decomposition of this algorithm into elementary gates from a universal set is carried out using the software Quipper. It is shown that a proof-of-principle calculation may be possible with actual quantum technologies.

uk: вхід без авторизації

name: Numerical scheme for the solution of the Dirac equation on classical and quantum computers


uk: Математика, природничі науки

description: The minimum mode following method for ﬁnding ﬁrst order saddle points on a potential energy surface is used, for example, in simulations of long time scale evolution of materials and surfaces of solids. Such simulations are increasingly being carried out in combination with computationally demanding electronic structure calculations of atomic interactions. Therefore, it becomes essential to reduce, as much as possible, the number of function evaluations needed to ﬁnd the relevant saddle points. Several improvements to the method are presented here and tested on a benchmark system involving rearrangements of a heptamer island on a closed packed crystal surface. Instead of using a uniform or Gaussian random initial displacement of the atoms, as has typically been done previously, the starting points are arranged evenly on the surface of a hypersphere and its radius is adjusted during the sampling of the saddle points. This increases the diversity of saddle points found and reduces the chances of converging again to previously located saddle points. The minimum mode is estimated using the Davidson method, and it is shown that signiﬁcant savings in the number of function evaluations can be obtained by assuming the minimum mode is unchanged until the atomic displacement exceeds a threshold value.

uk: вхід без авторизації

name: Numerical improvements in methods to ﬁnd ﬁrst order saddle points on potential energy surfaces


Pruning the MCTDH direct product basis

uk: Математика, природничі науки

description: I shall present two pruned, nondirect product multi-conﬁguration time dependent Hartree (MCTDH) methods for solving the Schr¨odinger equation. Both use a basis of products of natural orbitals. Standard MCTDH uses optimized 1D basis functions, called single particle functions, but the size of the basis scales exponentially with D, the number of coordinates.By replacing t → −iβ , β ∈ R>0 , we use the pruned methods to determine solutions of the time-independent Schroedinger equation. For a 12D Hamiltonian, we compare the pruned approach to standard MCTDH calculations for basis sizessmall enough that the latter are possible and demonstrate that pruning the basis reduces the CPU cost of computing vibrational energy levels of acetonitrile by more than two orders of magnitude. One of the pruned MCTDH methods uses an algebraic pruning constraint. The other uses a ﬂexible basis that expands as the calculation proceeds. Results obtained with the expanded basis are compared to those obtained with the established multi-layer MCTDH (ML-MCTDH) scheme. Although ML-MCTDH is somewhat more eﬃcient when low or intermediate accuracy is desired, pruned MCTDH is more eﬃcient when high accuracy is required.

uk: вхід без авторизації


uk: Математика, природничі науки

description: Controllable devices provide novel ways for the simulation of complex quantum open systems. In this talk, we will present different experimental platforms, developed in our group, where the dynamics of Born-Markov open quantum systems can be successfully simulated. In particular, we will discuss the observation of the so-called environment-assisted quantum transport in electrical oscillator networks, the survival of quantum coherence between indistinguishable particles propagating in quantum networks affected by noise, and the implementation of the first noise-enabled optical ratchet system.

uk: вхід без авторизації

name: Simulation of Born-Markov Open Quantum Systems in Electronic and Photonic Systems


uk: Математика, природничі науки

description: The nuclear dynamics of molecules following photoexcitation are fundamentally quantum dynamical processes owing to the breakdown of the Born-Oppenheimer approximation in regions close to the intersection of potential energy surfaces. To describe such processes, the solution of the time-dependent Schrödinger equation is necessitated. If large amplitude nuclear motion is involved, then this is a particularly challenging task for conventional gridbased quantum dynamics methods, owing to the need to construct model Hamiltonians that accurately describe multiple potential energy surfaces and the couplings between them over a large subvolume of nuclear conﬁguration space. A powerful alternative is to expand the nuclear wavefunction in terms of localised time-dependent parameterised basis functions, and to exploit this locality to calculate Hamiltonian matrix elements using information at a small number of nuclear geometries. Such methods not only promise to break the curse of exponential scaling suffered by conventional grid-based methods, but also allow for quantum dynamics calculations to be performed ‘on-the-ﬂy’ using information from ab initio electronic structure calculations calculated as and when it is needed. In this talk, I will discuss one such method: the ab initio multiple spawning (AIMS) method. In the AIMS method, the solution of the time-dependent Schrödinger equation is achieved via the expansion of the nuclear wavefunction in an adaptive set of Gaussian basis functions, which is increased in size when needed in order to efﬁciently describe the transfer of population between electronic states. In the ﬁrst part of the talk, the theoretical foundations and details of the AIMS method will be discussed. In the second part, representative examples of the study of excited state molecular dynamics will be given, illustrating the power and success of the AIMS method in studying photochemical processes.

uk: вхід без авторизації

name: Studying Photochemical Processes Using the Ab Initio Multiple Spawning Method


uk: Математика, природничі науки

description: Use of elliptically-polarized light opens the possibility of investigating effects that are not accessible with linearly-polarized pulses. This talk presents new physical effects that are predicted for ionization of the helium atom by few-cycle, elliptically-polarized attosecond pulses. For double ionization of He by an intense elliptically-polarized attosecond pulse, we predict a nonlinear dichroic effect (i.e., the difference of the two-electron angular distributions in the polarization plane for opposite helicities of the ionizing pulse) that is sensitive to the carrier-envelope phase, ellipticity, peak intensity I, and temporal duration of the pulse [1]. For single [2,3] and double ionization [4] of He by two oppositely circularly-polarized, time-delayed attosecond pulses, we predict that the photoelectron momentum distributions in the polarization plane have helical vortex structures that are exquisitely sensitive to the time-delay between the pulses, their relative phase, and their handedness [2-4]. These effects manifest the ability to control the angular distributions of the ionized electrons by means of the attosecond pulse parameters. Our predictions are obtained numerically by solving the two-electron time-dependent Schrödinger equation for the six-dimensional case of elliptically-polarized attosecond pulses. They are interpreted analytically by means of perturbation theory analyses of the two ionization processes. *This work is supported in part by the U.S. Department of Energy (DOE), Office of Science, Basic Energy Sciences (BES), Award No. DE-FG03-96ER14646. [1] J.M. Ngoko Djiokap, N.L. Manakov, A.V. Meremianin, S.X. Hu, L.B. Madsen, and A.F. Starace, “Nonlinear dichroism in back-to-back double ionization of He by an intense elliptically-polarized few-cycle XUV pulse,” Phys. Rev. Lett. 113, 223002 (2014). [2] J.M. Ngoko Djiokap, S.X. Hu, L.B. Madsen, N.L. Manakov, A.V. Meremianin, and A.F. Starace, “Electron Vortices in Photoionization by Circularly Polarized Attosecond Pulses,” Phys. Rev. Lett. 115, 113004 (2015). <br/> [3] J.M. Ngoko Djiokap, A.V. Meremianin, N.L. Manakov, S.X. Hu, L.B. Madsen, and A.F. Starace, “Multistart Spiral Electron Vortices in Ionization by Circularly Polarized UV Pulses,” Phys. Rev. A 94, 013408 (2016). [4] J.M. Ngoko Djiokap, A.V. Meremianin, N.L. Manakov, S.X. Hu, L.B. Madsen, and A.F. Starace, “Kinematical Vortices in Double Photoionization of Helium by Attosecond Pulses,” Phys. Rev. A (accepted 19 June 2017, in press).

uk: вхід без авторизації

name: Time-Dependent Quantum Mechanics: Ionized Electron Vortices Produced by Elliptically Polarized Attosecond Pulses


uk: Математика, природничі науки

description:
A large part of strong-field physics relies on trajectory-based semiclassical methods for the description of ionization and the subsequent dynamics, both for intuitive understanding and as quantitative models, including the workhorse Strong-Field Approximation and its various extensions. In this work I examine the underpinnings of this trajectory-based language in the saddle-point analysis of temporal integrals deformed over complex contours, and how this formalism can be extended to include interactions with the ion. I show that a first-principles approach requires the use of complex-valued positions as well as times, and that the evaluation of the ionic Coulomb potential at these complex-valued positions imposes new constraints on the allowed temporal integration contours. I show how the navigation of these constraints is infused with physical content and a rich geometry, and how the correct traversal of the resulting landscape has clear consequences on the photoelectron spectrum.

 References:
 1. E. Pisanty and M. Ivanov. Phys. Rev. A 93, 043408 (2016).
 2. E. Pisanty and M. Ivanov. J. Phys. B: At. Mol. Opt. Phys. 49, 105601 (2016).

uk: вхід без авторизації

name: Slalom in complex time: semiclassical trajectories in strong-field ionization and their analytical continuations


uk: Математика, природничі науки

description: In most of the past studies of processes involving interaction of lasers with atoms and molecules the tiny photon momentum has not been taken into account nor the issue of momentum sharing between a photoelectron and an ion has not been addressed despite the fact than when intense lasers are used a huge amount of infrared photons are absorbed. This situation has been related to the fact that in most theoretical investigations the dipole approximation has been used for description of the photoionization processes. In this talk I emphasize the importance of using the non-dipole approaches in description of the interaction of intense lasers with atoms and molecules. I will review some surprising results obtained by us using numerical solutions of the time-dependent Schroedinger equation in [1-3] and present new results related to the photon-momentum effect using counter-propagating pulses and the specific non-dipole effects in diatomic molecules. [1] S. Chelkowski, A.D. Bandrauk, and P.B. Corkum, Phys.Rev.Let. 113, 263005 (2014). [2] S. Chelkowski, A.D. Bandrauk, and P.B. Corkum, Phys.Rev. A 92, 051401 (R) (2015). [3] S. Chelkowski, A.D. Bandrauk, and P.B. Corkum, Phys.Rev. A 95, 053402 (2017).

uk: вхід без авторизації

name: Beyond-dipole approximation effects in photoionization: importance of the photon momentum


uk: Математика, природничі науки

description: The well-know pseudo-spectral second-order splitting method has been the work-horse algorithm for solving the time-dependent Schrodinger equation for decades. However, it has been only ~30 years that one learns how to generalize this algorithm to fourth and higher orders. During that period, one also learned about the crucial distinction between solving time-reversible and time-irreversible quantum evolution equations. In this work, I show that there is only one class of fourth-order, forward time step algorithm that can solve both time-reversible and time-irreversible equations with equal efficiency. Other higher order algorithms based on multi-product expansion and complex coefficient will also be mentioned.

uk: вхід без авторизації

name: Higher order forward time-step algorithms for solving diverse evolution equations


uk: Математика, природничі науки

description: It will be shown that the Rabi model and the ion laser interaction Hamiltonians may be related via a simmilarity transformation which allows much faster regimes than the ones reached with the rotating wave approximation. This is because the speed of the interaction is dictated the Rabi (intensity of the laser) and our approach does not impose a condition on it. <br/> It will also be shown that the unitary transformation may be done in the time dependent case, and the Rabi model Hamiltonian parameters will depend also on time.

uk: вхід без авторизації

name: Ion-Laser Interactions and the Rabi Model

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


uk: Математика, природничі науки

description: The electronic system is driven far from its ground state in many applications today: attosecond control and manipulation of electron dynamics and the consequent ion dynamics, photovoltaic design, photoinduced processes in general. Time-dependent density functional theory is a good candidate by which to computationally study such problems. Although it has had much success in the linear response regime for calculations of excitation spectra and response, its reliability in the fully non-perturbative regime is less clear even though it is increasingly used. In the first part of the talk, I will show some of our recent work exploring exact features of the time-dependent exchange-correlation potential that are necessary to yield accurate dynamics and new approaches to develop functionals going beyond the adiabatic approximation. In the second part of the talk, I broaden the focus to the description of coupled electron-ion motion. When the coupling to quantum nuclear dynamics is accounted for, we find additional terms in the potential acting on the electronic subsystem, that fully account for electron-nuclear correlation, and that can yield significant differences to the traditional potentials used when computing coupled electron-ion dynamics.

uk: вхід без авторизації

name: Capturing Electron-Electron Electron-Ion Correlations in Strong Fields


uk: Математика, природничі науки

description: Ending up with a lot of dynamics data ( be it trajectories or wavefunctions) is a common situation we face whenever we are working with real (and sometimes, evenmodel) systems. How can we make sense of these massive data in terms of concepts we understand? In the past two decades I and my team have been using modern dynamical systems theory to uncover simple structures buried under massive trajectory calculations. The power of this tool is due to a very simple property: It allows you to focus on collective behavior of families of trajectories rather than individual ones, thereby helping you to isolate generic behavior. I will give you specific examples from our research which illustrate the use of this powerful tool.

uk: вхід без авторизації

name: Using Modern Dynamical Systems Theory to Interpret Your Data


Quantum walking in curved spacetime

uk: Математика, природничі науки

description: A discrete-time Quantum Walk (QW) is essentially a unitary operator driving the evolution of a single particle on the lattice. Some QWs admit a continuum limit, leading to familiar PDEs (e.g. the Dirac equation). In this paper, we study the continuum limit of a wide class of QWs, and show that it leads to an entire class of PDEs, encompassing the Hamiltonian form of the massive Dirac equation in (1+1) curved spacetime. Therefore a certain QW, which we make explicit, provides us with a unitary discrete toy model of a test particle in curved spacetime, in spite of the fixed background lattice. Mathematically we have introduced two novel ingredients for taking the continuum limit of a QW, but which apply to any quantum cellular automata: encoding and grouping.

uk: вхід без авторизації


uk: Математика, природничі науки

description: We numerically study the interaction of a terahertz pulse with monolayer graphene. We use a numerical solution of the two-dimensional Dirac equation in Fourier space with time-evolution based on split-operator method to describe the dynamics of electron-hole pair creation in graphene. We notice that the electron momentum density is affected by the carrier-envelope phase (CEP) of the few-cycle terahertz laser pulse that induces the electron dynamics. Two main features are observed: (1) interference pattern for any values of the CEP and (2) asymmetry, for non-zero values of the CEP. We explain the origin of the quantum interferences and the asymmetry within the adiabatic-impulse model by finding conditions to reach minimal adiabatic gap between the valence band and the conduction band in graphene. The quantum interferences emanate from successive non-adiabatic transitions at this minimal gap. We discuss how these conditions and the interference pattern are modified by the CEP. This opens the door to control fundamental time-dependent electron dynamics in the tunneling regime in Dirac materials. Also, this suggests a way to measure the CEP of a terahertz laser pulse when it interacts with condensed matter systems.

uk: вхід без авторизації

name: Non-adiabatic dynamics in graphene controlled by the carrier-envelope phase of a few-cycle laser pulse


uk: Математика, природничі науки

description:
To solve the equation of motion for a quantum-mechanical wavefunction $\psi$ the Schrödinger equation, for a many-particle system is a major problem in many branches of physics. Based on the idea that the joint probability density $|\psi|^2$ can be written as the product of a marginal probability density (that depends only on some of the particle variables) and a conditional probability density, it can be shown that there exists a marginal wavefunction which also obeys a Schrödinger equation, but with an eﬀective potential that encodes the interaction with all parti- cles. In this talk, I present an application of this idea to a many-electron system: By taking only the variables of one electron as the marginal coordinates, an exact single-electron picture that describes the correlated electron dynamics of many electrons is obtained. All many-electron interactions are then encoded in the structure and time-dependence of an eﬀective single-electron potential. This approach is applied to the description of strong ﬁeld phenomena, because they are often interpreted in a single-electron picture, while at the same time the understanding and measurement of many-electronic interactions is a main topic in this ﬁeld. First results for 2- and 3-electron model systems in strong laser ﬁelds are presented and used to illustrate how an exact single electron picture of ionization or high-harmonic generation looks like. Additionally, I show ﬁrst steps towards a feasible method for the calculation of the many-electron dynamics in complex molecules.

 [1] Axel Schild, E.K.U. Gross, Phys. Rev. Lett. 118, 163202 (2017).

uk: вхід без авторизації

name: An Exact Single-Electron Picture of Many-Electron Processes and its Application to the Dynamics of Molecules in Strong Laser Fields


uk: Математика, природничі науки

description: Quantum feedback control is challenging to implement as a measurement on a quantum state only reveals partial information of the state. A feedback procedure can be developed based on a trusted model of the system dynamic, which is typically not available in practical applications. We aim to devise tractable methods to generate effective feedback procedures that do not depend on trusted models. As an application, we construct a reinforcement-learning algorithm to generate adaptive for quantum-enhanced phase estimation in the presence of arbitrary phase noise. Our algorithm exploits noise-resistant differential evolution and introducesan accept-reject criterion. Our robust method shows a path forward to realizing adaptive quantum metrology with unknown noise properties.

uk: вхід без авторизації

name: Reinforcement Learning for Robust Adaptive Quantum-Enhanced Metrology


uk: Математика, природничі науки

description: Dirac equation, proposed by Paul Dirac in 1928, is a relativistic version of the Schroedinger equation for quantum mechanics. It describes the evolution of spin-1/2 massive particles, e.g. electrons. Due to its applications in graphene and 2D materials, Dirac equations has drawn considerable interests recently. We are concerned with the numerical methods for solving the Dirac equation in the non-relativistic limit regime, involving a small parameter inversely proportional to the speed of light. We begin with commonly used numerical methods in literature, including finite difference time domain and time splitting spectral, which need very small time steps to solve the Dirac equation in the non-relativistic limit regime. We then propose and analyze a multi-scale time integrator pseudospectral method for the Dirac equation, and prove its uniform convergence in the non-relativistic limit regime. We will extend the study to the nonlinear Dirac equation case.

uk: вхід без авторизації

name: Numerical methods for the Dirac equation in the non-relativistic limit regime


uk: Математика, природничі науки

description: In this talk we give the notion of complete colorings in graphs, achromatic number, Knesser graphs and Steiner triple systems . Also, we explain how the Steiner triple systems solve the problem about the existence of complete colorations on Knesser graphs that attain the upper bound of the achromatic number, where the achromatic number of a graph G is the maximum integer value for the number of chromatic classes in a complete and proper coloring of G.

uk: вхід без авторизації

name: The achromatic number of Knesser graphs and their relationship with Steiner triple systems


uk: Математика, природничі науки

description: The Upper and Lower bound Theorems in convex geometry deal, respectively, with the maximum and minimum number of facets that a convex d-dimensional polytope with n vertices can have. In this talk we will offer bounds for such numbers for what we refer to as the projective case of this problem.

uk: вхід без авторизації

name: Towards a projective Upper/Lower bound theorem


Symmetric Cubic Graphs as Cayley Graphs

uk: Математика, природничі науки

description: A graph X is {\em symmetric} if its automorphism group acts transitively on the arcs of X, and {\em s-arc-transitive} if its automorphism group acts transitively on the set of s-arcs of X. Furthermore, if the latter action is sharply-transitive on s-arcs, then X is {\em s-arc-regular.} It was shown by Tutte (1947, 1959) that every finite symmetric cubic graph is s-arc-regular for some s≤5. Djokovi\v c and Miller (1980) took this further by showing that there are seven types of arc-transitive group action on finite cubic graphs, characterised by the stabilisers of a vertex and an edge. The latter classification was refined by Conder and Nedela (2009), in terms of what types of arc-transitive subgroup can occur in the automorphism group of X. In this talk we consider the question of when a finite symmetric cubic graph can be a Cayley graph. We show that in five of the 17 Conder-Nedela classes, there is no Cayley graph, while in two others, every graph is a Cayley graph. In eight of the remaining ten classes, we give necessary conditions on the order of the graph for it to be Cayley; there is no such condition in the other two. Also we use covers (and the `Macbeath trick') to show that in each of those last ten classes, there are infinitely many Cayley graphs, and infinitely many non-Cayley graphs. This research grew out of some recent discussions with Klavdija Kutnar and Dragan Maru{\v s}i{\v c}.

uk: вхід без авторизації


Edge-transitive maps

uk: Математика, природничі науки

description: From the minute 10:05 of video does not have audio. In 1997 Graver and Watkins partitioned edge-transitive maps into 14 classes, distinguished by the quotient of each map by its full automorphism group. These classes include those of regular, chiral and just-edge-transitive maps. I shall describe the possible topological and combinatorial properties of the maps in these classes, including maps with boundary, together with the possibilities for their automorphism groups. In the case of finite simple groups, the latter builds on results of Nuzhin and others relevant to regular maps (with a small but important correction), and more recently of Leemans and Liebeck for chiral maps.

uk: вхід без авторизації


uk: Математика, природничі науки

description: In a series of three papers, Monson and Schulte showed that, subject to some mild constraints, certain families of real reflection groups can be reduced modulo odd primes to yield finite string C-subgroups of orthogonal groups. Further, polytopes of any desired rank can be constructed this way. In this talk I will show that the latter is also true for orthogonal groups defined over any non-prime field of characteristic 2. Of course, the modular reduction method cannot be used for such groups, so the polytopes are constructed from scratch using analogues of reflections.

uk: вхід без авторизації

name: Orthogonal groups in characteristic 2 acting on polytopes of high rank


uk: Математика, природничі науки

description: An abstract polytope is flat if every facet is incident on every vertex. In this talk, we prove that no chiral polytope has flat finite regular facets and finite regular vertex- figures. We then determine the three smallest non-flat regular polytopes in each rank, and use this to show that for n≥8, a chiral n-polytope has at least 48(n−2)(n−2)! flags.

uk: вхід без авторизації

name: Non-flat regular polytopes and restrictions on chiral polytopes


Cellular automata and finite groups

uk: Математика, природничі науки

description: Let G be a finite group and A a finite set. A cellular automaton is a transformation of the configuration space A^G that commutes with the natural action of G. We shall present some algebraic results on the monoid of cellular automata over A^G, and discuss some of the difficulties of extending these to a more general setting.

uk: вхід без авторизації


Embeddings of biplanes

uk: Математика, природничі науки

description: Given a combinatorial design D and its incidence graph ID, the embedding of D on a surface S is defined by a transformation on the embedding of the graph ID on S. In this talk will be introduce this concepts to describe the embedding of the possible bipalnes known so far.

uk: вхід без авторизації


uk: Математика, природничі науки

description: We describe how the symmetries in configuration spaces for mass partition problems are the key to solving such problems. In particular, we will focus on the following problem: Given positive integers r and d, and d smooth finite measures in R^d, there is a partition of R^d into r convex sets that simultaneously splits each measure evenly.

uk: вхід без авторизації

name: Symmetries in configuration spaces for mass partition


uk: Математика, природничі науки

description: We present several novel methods for solving general (pseudo-) monotone variational inequalities. The first method uses fixed stepsize and is similar to the proximal reflected gradient method: it also requires only one value of operator and one prox-operator per iteration. However, its extension — the dynamic version — has a notable distinction. In every iteration it defines a stepsize, based on a local information about operator, without running any linesearch procedure. Thus, the iteration costs of this method is almost the same as in the first one with a fixed stepsize, but it converges without the Lipschitz assumption on the operator. We further discuss possible generalizations of the methods, in particular for solving large-scale nonlinear saddle point problems. Some numerical experiments are reported.

uk: вхід без авторизації

name: Golden Ratio Algorithms for Variational Inequalities


uk: Математика, природничі науки

description: We propose a unifying scheme for several algorithms from the literature dedicated to the solving of monotone inclusion problems involving compositions with linear continuous operators in infinite dimensional Hilbert spaces. We show that a number of primal-dual algorithms for monotone inclusions and also the classical ADMM numerical scheme for convex optimization problems, along with some of its variants, can be embedded in this unifying scheme. While in the first part of the talk convergence results for the iterates are reported, the second part is devoted to the derivation of convergence rates obtained by combining variable metric techniques with strategies based on a suitable choice of dynamical step sizes.

uk: вхід без авторизації

name: ADMM for monotone operators: convergence analysis and rates


uk: Математика, природничі науки

description: The Douglas-Rachford method has been employed successfully to solve a variety of non-convex feasibility problems. In particular, it shows surprising stability when applied to finding the intersections of hypersurfaces. We prove local convergence in the generalization of a case prototypical of the phase retrieval problem. In so doing, we also discover phenomena which may inhibit convergence. Finally we illustrate an application to solving boundary valued ordinary differential equations. This talk includes discoveries from three closely related works: 1. With Brailey Sims, Matthew Skerritt. ''Computing Intersections of Implicitly Specified Plane Curves.'' To appear in Journal of Nonlinear and Convex Analysis. 2. With Jonathan M. Borwein, Brailey Sims, Anna Schneider, Matthew Skerritt. ''Dynamics of the Douglas-Rachford Method for Ellipses and p-Spheres.'' Submitted to Set Valued and Variational Analysis. 3. With Bishnu Lamichhane and Brailey Sims. ''Application of Projection Algorithms to Differential Equations: Boundary Value Problems,'' in preparation with plans to submit to ANZIAM Journal.

uk: вхід без авторизації

name: The Douglas-Rachford method for finding intersections of hypersurfaces


uk: Математика, природничі науки

uk: вхід без авторизації

name: Nesterov accelerated gradient method in the subcritical case α≤ 3


uk: Математика, природничі науки

description: In this talk, we present a new framework for establishing error bounds for a class of structured convex optimization problems, in which the objective function is the sum of a smooth convex function and a general closed proper convex function. Such a class encapsulates fairly general constrained minimization problems, as well as various regularized loss minimization formulations in machine learning, signal processing, and statistics. Our framework not only allows us to recover a number of existing error bound results in a unified and transparent manner but also leads to new error bounds for problems whose objective functions do not necessarily have a polyhedral epigraph. As an illustration, we show that a class of nuclear-norm regularized loss minimization problems admits an error bound under a strict complementarity-type regularity condition. We then complement this result by constructing an example to show that the said error bound could fail to hold without the regularity condition. If time permits, we discuss the algorithmic implications of our results, particularly on the convergence rate analysis of a family of successive quadratic approximation methods for solving the aforementioned class of problems.

uk: вхід без авторизації

name: A Unified Approach to Error Bounds for Structured Convex Optimization Problems


uk: Математика, природничі науки

description: We propose new primal-dual decomposition algorithms for solving systems of inclusions involving sums of linearly composed maximally monotone operators. At each iteration, only a subset of the monotone operators needs to be processed, as opposed to all operators as in established methods. Deterministic strategies are used to select the blocks of operators activated at each iteration. In addition, asynchronous implementation is allowed. The first method provides weakly convergent primal and dual sequences under general conditions, while the second is a variant in which strong convergence is guaranteed without additional assumptions. The novelty of this class of algorithms will be discussed and comparisons with the state of the art will be performed.

uk: вхід без авторизації

name: Parallel, Block-Iterative, Primal-Dual Monotone Operator Splitting

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


uk: Математика, природничі науки

description: We give a simple proof and explicit formulae for the C1,1-convex extension problem studied by D. Azagra and C. Mudarra. As an application, we obtain an easy constructive proof for the Glaeser-Whitney problem of C1,1 extensions on a Hilbert space. This is a joint work with M. Haddou, O. Ley and E. Le Gruyer.

uk: вхід без авторизації

name: On the Glaeser-Whitney extension problem


uk: Математика, природничі науки

uk: вхід без авторизації

name: Minimization of quadratic functions on convex sets without asymptotes


uk: Математика, природничі науки

description: The Douglas–Rachford algorithm is a very popular splitting technique for finding a zero of the sum of two maximally monotone operators. The behaviour of the algorithm remains mysterious in the general inconsistent case, i.e., when the sum problem has no zeros. However, more than a decade ago, it was shown that in the (possibly inconsistent) convex feasibility setting, the shadow sequence remains bounded and its weak cluster points solve a best approximation problem. In this talk, we advance the understanding of the inconsistent case significantly by presenting a complete proof of the full weak convergence in the convex feasibility setting. We also provide linear rate of convergence and strong convergence in special cases.

uk: вхід без авторизації

name: On the Douglas–Rachford operator in the (possibly) inconsistent case and related progress


uk: Математика, природничі науки

description: We present an iterative proximal inertial forward-backward method with memory effects, based on recent advances in solving scalar convex optimization problems and monotone inclusions, for determining weakly efficient solutions to convex vector optimization problems consisting in vector-minimizing the sum of a differentiable vector function with a nonsmooth one, by making use of some adaptive linear scalarization techniques. During the talk, the difficulties encountered while formulating the algorithm and proving its convergence will be stressed, while the related (still unsolved) challenge of extending the celebrated FISTA method from scalar to vector optimization problems will be mentioned, too. The talk is based on joint work with Radu Ioan Boț.

uk: вхід без авторизації

name: An inertial forward-backward method for solving vector optimization problems


On Decomposing the Proximal Map

uk: Математика, природничі науки

description: The proximal map has played a significant role in convex analysis and various splitting algorithms. For "simple" functions this proximal map is available in closed-form while in general it needs to be computed by iterative numerical algorithms hence being inefficient. Motivated by applications in machine learning, we study when the proximal map of a sum of functions decomposes into the composition of the (simple) proximal maps of the individual summands. We present a simple sufficient condition and we demonstrate its surprising effectiveness by unifying and extending several results in seemingly unrelated fields. We end our discussion with a few open directions.

uk: вхід без авторизації


Of Coarse it’s Fine

keywords:
Mathematics
Numerical analysis
Mechanics of deformable solids
Applied computer science

uk: Математика, природничі науки

description: Numerical algorithms for simulating continua have long been based on the idea that high resolution and high accuracy go hand-in-hand. While engineering and computational mechanics have pushed resolution limits in pursuit of simulation fidelity, computer graphics has often gone the other way, sacrificing resolution and accuracy for performance. However, what if we want both high-performance and predictive accuracy ? In this talk I will discuss how to avoid this seeming fundamental trade-off using measurement-based and data-driven simulation techniques as well as some examples of these methods applied to both forward and inverse problems in graphics and 3D printing. I will conclude with some thoughts on where these methods might take us in the future.

uk: вхід без авторизації


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


uk: Математика, природничі науки

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.

uk: вхід без авторизації

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


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