ISU Analysis and Probability Seminar

Jan 17 (3:20pm) Carver 390 - Organizational meeting

Jan 26 (1:00pm) Carver 401 - Juraj Foldes (University of Virginia)

Title: Nonlinear Schrodinger equation with randomized initial conditions

Abstract: During the talk, we will discuss the local solutions of the super-critical cubic Schrödinger equation (NLS) on the whole space with general differential operator. Although such a problem is known to be ill-posed, we show that the random initial data yield almost sure local well-posedness. Using estimates in directional spaces, we improve and extend known results for the standard Schrödinger equation in various directions: higher dimensions, more general operators, weaker regularity assumptions on the initial conditions. In particular, we show that in 3D, the classical cubic NLS is stochastically, locally well-posed for any initial data with regularity in $H^\varepsilon$ for any $\varepsilon > 0$, compared to the known results $\varepsilon > 1/6$ . The proofs are based on precise estimates in frequency space using various tools from Harmonic analysis. This is a joint project with Jean-Baptise Casteras (Lisbon University) and Gennady Uraltsev (University of Virginia, University of Arkansas).

Feb 7 (3:20pm) Carver 401 - Chad Berner (ISU)

Title: Frame-like Fourier expansions for finite Borel measures on $\mathbb{R}$

Abstract: It is known that if a finite Borel measure $\mu$ on $[0,1)$ possesses a frame of exponential functions for $L^{2}(\mu)$, then $\mu$ is of pure type. In this talk, we discuss the existence of a class of finite Borel measures $\mu$ on $[0,1)$ that are not of pure type that possess frame-like Fourier expansions for $L^{2}(\mu)$. We also discuss some properties and classifications of certain measures possessing this type of Fourier expansion. Additionally, we establish a frame-like Fourier expansion for $L^{2}(\mu)$ where $\mu$ is a singular Borel probability measure on $\mathbb{R}$ using theory of slice singular measures and the Kaczmarz algorithm.

Feb 14 (3:20pm) Carver 401 - Yuming Zhang (Auburn University)

Title: Blow-ups and global solvability of a free boundary problem

Abstract: We study two McKean-Vlasov equations involving hitting times. The corresponding Fokker-Planck equations are super-cooled Stefan type problems (free boundary problems). The equations are used in financial networks, computational neuroscience and some fluid models. We discuss sufficient, simple conditions such that the McKean-Vlasov dynamics is well-defined for all time $t \geq 0$ or having blow-ups in finite time. We take the PDE approach - develop a new comparison principle for the first equation, while for the second equation, the argument is based on a relative entropy analysis. This is joint work with Erhan Bayraktar, Gaoyue Guo and Wenpin Tang.

Feb 28 (3:20pm) Carver 401 - Michael Salins (Boston University)

Title: Finite time explosion for stochastic partial differential equations

Abstract: I outline some recent results about the explosive behavior of stochastic partial differential equations with superlinear deterministic forcing and superlinear multiplicative noise terms.

Mar 6 (3:20pm) Carver 401 - Abhishek Halder (ISU)

Title: New Developments in Schrödinger Bridge, Stochastic Control and Stochastic Learning

Abstract: We will summarize several recent advances made in the theory and algorithms for solving problems at the intersection of generalized variants of the Schrödinger bridge, stochastic control and stochastic learning. We will provide the mathematical background, historical contexts, and the motivations behind the current resurgence of these ideas across different research communities. We will explain some new results -- both theoretical and computational -- and point to several directions that remain open.

Mar 13 Spring Break

Apr 3 (3:20pm) Carver 401 - Pêdra Andrade (University of Lisbon)

Title: Spectral partition problems with measure and inclusion constraints

Abstract: In this talk, we will discuss a class of spectral optimal partition problems with measure and inclusion constraints for partitions of a given bounded domain. We show the existence of an optimal open partition by establishing that the corresponding eigenfunctions are locally Lipschitz continuous. Our proofs are based on a weak formulation involving a minimization problem of a penalized functional, where the variables are functions rather than domains, suitable deformations, blowup techniques, and a monotonicity formula.

Apr 10 (3:20pm) Carver 401 - Carlos Rautenberg (George Mason University)

Title: Optimization Problems with Distributional and Weak Gradient Constraints

Abstract: We consider a class of models of growth of heterogeneous sand piles described by optimization problems with gradient (distributional or weak) constraints and mixed boundary conditions. These lead (via Fenchel duality) to variational problems involving partial differential operators on non-standard vector-valued measure spaces with divergences (represented by measures or by functions). Further, the latter include mixed boundary conditions that are established by means of a vanishing normal trace characterization of the vectorial measure. We determine existence of solutions, stability and strong duality results. We finalize the talk with numerical tests.

Apr 17 (3:20pm) Carver 401 - Dissertation Defense: Mitch Haeuser (ISU)

Title: Regularity estimates for elliptic equations involving fractional Neumann boundary conditions

Apr 24 (3:20pm) Carver 401 - Rocío Díaz Martín (Vanderbilt University)

Title: Applications of Optimal Transport Theory

Abstract: Optimal Transport (OT) theory has emerged as a powerful tool across various fields, including machine learning and signal processing analysis. We will start with an overview of OT theory, focusing on the Wasserstein distance for probability measures through Monge’s, Kantorovich’s, and dynamic formulations of the OT problem. We will emphasize why the Wasserstein distance is useful for discriminating distributions. Additionally, we will introduce an "embedding" or "transform" based on OT: the “Linear Optimal Transport Embedding” (LOT), which significantly reduces the computational cost of the Wasserstein distance. Furthermore, we will discuss its 1D version, known as the Cumulative Distribution Transform (CDT). If time permits, we will explore the balanced mass limitation in classical OT, thus delving into Partial Optimal Transport and Unbalanced Optimal Transport. 

The seminar will cover parts of joint works with A. Aldroubi, Y. Bai, S. Kolouri, I. Medri, S. Thareja, G. Rohde, and the MINT Lab at Vanderbilt University.

May 1 (3:20pm) Carver 401 and Zoom - Enrique Alvarado (UC Davis)

Title: The Maximum Distance Problem for Fractals

Abstract: The maximum distance problem is similar to the traveling salesman problem, which asks to find the shortest curve through a finite set E of points in Euclidean space. The important constraint here is that the curve must contain the points. The maximum distance problem allows for more room. It asks for the shortest curve whose r-neighborhood covers your points. Because of this extra room, E is allowed to contain an infinite number of points, and in fact, E only has to be bounded. There are a certain number of interesting questions that have to do with the asymptotic behavior of the solutions to the maximum distance problem. For a fixed E, what happens as r goes to 0? What happens if we approximate our set with points? Do our solutions converge in some way? In this talk we will investigate these types of questions for sets E that are of fractal nature.

Aug 30 (3:20pm) Carver 401 - Organizational meeting

Sep 6 (3:20pm) Carver 401 - Zhenhua Wang (ISU)

Title: Convergence of Policy Iteration for Entropy Regularized Stochastic Control Problems

Abstract: We study an entropy-regularized continuous time stochastic control problem. We first provide the policy improvement property, then with a sequel of estimates we show the compactness of the value sequence $(v^n)_n$ generated by the Policy Iteration Algorithm (PIA), and derive the convergence of the sequence to the optimal value. The existence of the solution to the HJB equation and regularity of the optimal value are achieved as by-products. Some related questions for future study will also be given at the end. This is a joint work with Yu-Jui Huang and Zhou Zhou.

Sep 13 (3:20pm) Carver 401 - Zhenhua Wang (ISU)

Title: Convergence of Policy Iteration for Entropy Regularized Stochastic Control Problems (Part 2)

Abstract: We study an entropy-regularized continuous time stochastic control problem. We first provide the policy improvement property, then with a sequel of estimates we show the compactness of the value sequence $(v^n)_n$ generated by the Policy Iteration Algorithm (PIA), and derive the convergence of the sequence to the optimal value. The existence of the solution to the HJB equation and regularity of the optimal value are achieved as by-products. Some related questions for future study will also be given at the end. This is a joint work with Yu-Jui Huang and Zhou Zhou.

Sep 20 (3:20pm) Carver 401 and Zoom - Bingyan Han (University of Michigan)

Title: Fitted value iteration methods for bicausal optimal transport 

Abstract: We develop a fitted value iteration (FVI) method to compute bicausal optimal transport (OT) where couplings have an adapted structure. Based on the dynamic programming formulation, FVI adopts a function class to approximate the value functions in bicausal OT. Under the concentrability condition and approximate completeness assumption, we prove the sample complexity using (local) Rademacher complexity. Furthermore, we demonstrate that multilayer neural networks with appropriate structures satisfy the crucial assumptions required in sample complexity proofs. Numerical experiments reveal that FVI outperforms linear programming and adapted Sinkhorn methods in scalability as the time horizon increases, while still maintaining acceptable accuracy. This is a joint work with Prof. Erhan Bayraktar. The preprint is available at https://arxiv.org/abs/2306.12658. 

Sep 27 (3:20pm) Carver 401 - Aidan Howells (University of Wisconsin-Madison)

Title: Stochastic reaction networks within interacting compartments

Abstract: Stochastic reaction networks, which are typically modeled as continuous-time Markov chains on $mathbb Z^d_{ge0}$, have proven to be a useful tool for the understanding of processes, chemical and otherwise, in homogeneous environments. There are multiple avenues for generalizing away from the assumption that the environment is homogeneous, with the proper modeling choice dependent upon the context of the problem being considered. One such generalization, introduced by Duso and Zechner in 2020, involves a varying number of interacting compartments, or cells, each of which contains an evolving copy of the stochastic reaction system. The novelty of the model is that these compartments also interact via the merging of two compartments (including their contents), the splitting of one compartment into two, and the appearance and destruction of compartments. We will discuss results pertaining to explosivity, transience, recurrence, and positive recurrence of the model, and explore a number of examples demonstrating some possible non-intuitive behaviors. The talk will begin with an introduction to stochastic reaction networks generally, and is intended to be accessible to non-specialists.

Oct 11 (3:20pm) Carver 401 - Dallas Albritton (University of Wisconsin-Madison)

Title: Non-uniqueness of Leray solutions to the forced Navier-Stokes equations

Abstract: In a seminal work, Leray demonstrated the existence of global weak solutions to the Navier-Stokes equations in three dimensions. Are Leray's solutions unique? This is a fundamental question in mathematical hydrodynamics, which we answer in the negative within the "forced" category, by exhibiting a one-parameter family of distinct Leray solutions with zero initial velocity and identical body force. This is joint work with Elia Brué and Maria Colombo.

Oct 18 (3:20pm) Carver 401 - Rohan Sarkar (University of Connecticut)

Title: Spectral theory of self-similar Markov processes by means of weak similarity relation

Abstract: Two operators are called weakly similar when they are intertwined via an injective (possibly non-invertible) operator. This talk will focus on the spectral theory of self-similar Markov semigroups on $L^2(R_+)$. These semigroups are not self-adjoint in general, which makes the spectral analysis challenging. Using the pseudo-differential operator representation of the generator, thanks to Lamperti’s characterization theorem, we prove under a very mild condition that (almost) all self-similar Markov semigroups are weakly similar to each other. As a by-product, we provide conditions under which these semigroups have point, continuous or residual spectrum, and the expressions of eigenfunctions/co-eigenfunctions when they exist. For some specific examples, these eigenfunctions can be expressed in terms of some special functions. 

This talk will be based on the paper “Weak similarity orbit of log self-similar Markov semigroups on the Euclidean space” which is a joint work with my Ph.D. advisor Pierre Patie (Cornell University).

Nov 1 (3:20pm) Carver 401 and Zoom - Eric Friedlander (St. Norbert College)

Title: Many-Server Asymptotics for Join-the-Shortest-Queue: Large Deviations and Rare Events

Abstract: This talk will focus on the Join-the-Shortest-Queue load balancing policy in an asymptotic regime where the number of processors, n, scales with the arrival rate. We establish a large deviation principle (LDP) for the system as, as n increases to infinity. Some of the technical obstacles in the proof will be discussed, as well as implications of the result. In particular, we address how the LDP can be used to estimate rates of decay for probabilities of rare-events, such as long-queues.

Nov 8 (3:20pm) Carver 401 and Zoom - Jing Wang (Purdue University)

Title: Spectral bounds for exit times of diffusions on metric measure spaces

Abstract: Click here for the abstract.

Nov 15 (3:20pm) Carver 401 - Tianxiang Gao (ISU)

Title: Neural Network and Gaussian Process Correspondence for Large Depth Neural Networks

Abstract: In recent years, wide neural networks have attracted significant attention due to their connection to Gaussian processes and kernel-based methods. This connection enables precise data fitting while preserving generalization abilities, a phenomenon known as benign overfitting. Previous research has primarily focused on shallow networks with finite depths, leaving a critical gap in our understanding of large-depth and even infinite-depth neural networks, such as neural ordinary differential equations (ODEs) and deep equilibrium models (DEQs).

Our work investigates Neural ODEs and DEQs as infinite-depth neural networks with shared weight matrices across layers. A key revelation from our analysis is that as the width of network layers approaches infinity, they converge to a Gaussian process, establishing the Neural Network and Gaussian Process (NNGP) correspondence. Notably, this convergence holds even when the limits of depth and width are interchanged. Additionally, we demonstrate the strictly positive definiteness of the associated Gaussian covariance function or NNGP kernel. As a result, we establish global convergence for a gradient-based method when training large-depth neural networks. In particular, our findings show that when Neural ODEs and DEQs are over-parameterized, gradient descent with random initialization converges to a global minimum at a linear rate with high probability.

This work provides valuable insights into the characteristics of wide neural networks and their relationship with Gaussian processes, significantly advancing our understanding of these powerful machine learning models and the underlying neural scaling laws.

Nov 29 (3:20pm) Carver 401 - Wonjun Lee (University of Minnesota)

Title: Monotone Generative Modeling via a Gromov-Monge Embedding

Abstract: Generative Adversarial Networks (GANs) are powerful tools for creating new content, but they face challenges such as sensitivity to starting conditions and mode collapse. To address these issues, we propose a deep generative model that utilizes the Gromov-Monge embedding (GME). It helps identify the low-dimensional structure of the underlying measure of the data and then map it, while preserving its geometry, into a measure in a low-dimensional latent space, which is then optimally transported to the reference measure. We guarantee the preservation of the underlying geometry by the GME and $c$-cyclical monotonicity of the generative map, where $c$ is an intrinsic embedding cost employed by the GME. The latter property is a first step in guaranteeing better robustness to initialization of parameters and mode collapse. Numerical experiments demonstrate the effectiveness of our approach in generating high-quality images, avoiding mode collapse, and exhibiting robustness to different starting conditions.

Jan 25 (3:15pm) Carver 401 and Zoom - Stefan Steinerberger (University of Washington, Seattle)

Title: The Flow of Roots of Polynomials under Differentiation 

Abstract: Suppose we have a real-valued polynomial of very high degree all of whose roots are on the real line and suppose that we start differentiating many times. One could wonder about the behavior of the roots of these derivatives: how do they behave? I will describe a nonlinear PDE that describes the result -- as it turns out, the same question has arisen in a very different framework in the study of Free Probability. This allows one to make progress on questions in Operator Algebra by studying polynomials! There will be many pictures and many open questions. 

Feb 1 (3:15pm) Carver 401 - Organizational meeting

Feb 8 (3:20pm) Carver 401 - Group discussion on qualifying exams and graduate courses (Analysis)

Mar 1 (3:20pm) Carver 401 - Ruoyu Wu (ISU)

Title: Large deviations, weak convergence, and calculus of variations for some interacting particle systems 

Abstract: In the first part of this talk, we will briefly introduce the theory of large deviations, some classic results, and the weak convergence approach developed in the last twenty years. Different from obtaining classic exponential probability estimates, such an approach combines techniques from stochastic control theory and weak convergence of probability measures. In the second part, we will show large deviation principles for several interacting particle systems arising from the study of random graphs and queueing systems. A particular emphasis will be given on the join-the-shortest-queue load-balancing queueing system. Calculus of variations problems associated with the large deviation rate functions will be analyzed.

Mar 22 (3:20pm) Carver 401 - Eric Weber (ISU)

Title: Vector-Valued de Branges Spaces

Abstract: The spectral theory of the backward shift operator on the Hardy space has numerous interconnecting components, each of which has been a source of interesting analysis throughout the 20th century. Nearly all of these interconnecting components are single-variable results, and trying to extend them to several variables has been either difficult or known to be impossible. But we are going to try anyway! I will give some of the historical context, and recent attempts (a few which have been successful!) to extend the single variable theory.

Mar 29 (3:20pm) Carver 401 - Pratima Hebbar (Grinnell College)

Title: Branching Diffusion in Periodic Media 

Abstract: We describe the behavior of branching diffusion processes in periodic media. For a super-critical branching process, we distinguish two types of behavior for the normalized number of particles in a bounded domain, depending on the distance of the domain from the region where the bulk of the particles is located. At distances that grow linearly in time, we observe intermittency (i.e., the k-th moment dominates the k-th power of the first moment for some k), while, at distances that grow sub-linearly in time, we show that all the moments converge. A key ingredient in our analysis is a sharp estimate of the transition kernel for the branching process, valid up to linear in time distances from the location of the initial particle.

Apr 5 (2:10pm) Carver 401 - Preliminary Exam: Mitch Haeuser (ISU)

Title: Nonlocal Equations on the Boundary

Apr 19 (3:20pm) Carver 401 - Sarah McCarty (ISU)

Title: Piecewise Linear Functions Representable with Infinite-Width Shallow Neural Networks

Abstract: A shallow infinite-width neural network represents a function as an integral with respect to a signed, finite measure on an appropriate parameter space of affine functions composed with the rectified linear unit (ReLU). Finite-width networks are represented by measures consisting of only finitely many atoms. The representational power of infinite-width neural networks is poorly characterized, especially compared to finite-width networks. We will show that every piecewise linear function expressible with an infinite-width shallow neural network is in fact expressible with a finite-width network. To this end, we associate the points in the parameter space with both hyperplanes in the domain of the function and with points in R^{n+1}. These associations are nearly bijective maps and induce more easily understood pushforward measures. With these mappings and a large non-co-hyperplanar set, we can use density arguments and Cramer-Wold Theorem to deduce the result.

May 3 (3:20pm) Carver 401 - David Herzog (ISU)

Title: Explicit rates of convergence to equilibrium for Langevin dynamics and other stochastic models

Abstract: The goal of this work is to understand how a stochastic differential equation converges to its stationary distribution as key parameters are varied in the system.  We begin by reviewing related results for underdamped Langevin dynamics as the friction parameter is either taken to zero or infinity.  Even in this simple setting, such results are relatively rare and lack key dynamical understanding.  We will then connect these known results with recent results of J. Bedrossian and K. Liss on finite-dimensional projections of 2d stochastic Navier-Stokes as the viscosity is taken to zero.  Again, in this setting, convergence bounds are deduced as they depend on small viscosity, but lack dynamical understanding.  The long-term goal is to find effective dynamics for all of these systems (as viscosity or the friction are taken to zero) that may shed light on the phenomenon of anomalous dissipation.

Aug 31 (3:15pm) Carver 401 - Organizational meeting

Sep 7 (3:15pm) Carver 401 and Zoom - Chad Berner (ISU)

Title: Fourier series from singular measures

Abstract: Any square integrable function on the torus is a norm limit of its Fourier series, but what if you change the measure from Lebesgue measure to a singular measure? It turns out you will lose orthogonality of the exponentials, but by the Kaczmarz algorithm, any function that is square integrable in this new measure space has a Fourier series converging in norm. We discuss further results in higher dimensions as well as analytic operators and their relation to the Hardy space and Fourier series.

Sep 21 (3:15pm) Carver 401 and Zoom - Pablo Raúl Stinga (ISU)

Title: Fractional derivatives: Fourier, memory effects and recent advances

Abstract: We consider the historically overlooked definition of fractional derivative by Fourier and apply our method of semigroups to show that it coincides with the fractional derivative of Marchaud. Applications to viscoelastic materials and recent advances in the theory of one-sided fractional Sobolev spaces are presented. This is based on joint works with Ana Bernardis, Francisco J. Martín-Reyes, José L. Torrea and Mary Vaughan.

Sep 28 (3:15pm) Carver 401 and Zoom - Fernando Charro (Wayne State University)

Title: Asymptotic Mean-Value Formulas for Nonlinear Equations

Abstract: In recent years there has been an increasing interest in whether a mean-value property, known to characterize harmonic functions, can be extended in some weak form to solutions of nonlinear equations. This question has been partially motivated by the surprising connection between Random Tug-of-War games and the normalized p-Laplacian discovered some years ago by Peres et al., where a nonlinear asymptotic mean-value property for solutions of a PDE is related to a dynamic programming principle for an appropriate game. In this talk we discuss asymptotic mean-value formulas for a class of nonlinear second-order equations that includes the classical Monge-Ampère and k-Hessian equations among other examples.

Oct 5 (3:15pm) Carver 401 and Zoom - Animesh Biswas (University of Nebraska-Lincoln )

Title: Nonlocal mean curvature with integrable kernel 

Abstract: Click here for the abstract.

Nov 2 (3:15pm) Carver 401 and Zoom - Monica Torres (Purdue University)

Title: Divergence-measure fields: Gauss-Green formulas and normal traces

Abstract: Click here for the abstract.

Nov 16 (3:15pm) Carver 401 and Zoom - Jiuyi Zhu (Louisiana State University) 

Title: Bounds of nodal sets of eigenfunctions 

Abstract: Motivated by Yau's conjecture, the study of the measure (sizes) of nodal sets (zero-level sets) of  eigenfunctions has been attracting much attention.  We investigate the nodal sets of Steklov eigenfunctions, Neumann eigenfunctions, and Dirichlet eigenfunctions in the domain and on the boundary of the domain. For the analytic domain, we show the sharp upper bounds of  interior nodal sets  for Steklov eigenfunctions, and the sharp upper bounds  of the intersections of nodal sets with the boundary for  Neumann and Dirichlet eigenfunctions. If time permits, we  will discuss the unified way to obtain the sharp upper bounds of nodal sets for  eigenfunctions of  bi-Laplace equations, and upper bounds of nodal sets in elliptic periodic homogenization. Part of the work in the talk is joint with Carlos Kenig, Fang-Hua Lin and Jinping Zhuge.

Mar 2 (4:10pm) Carver 401 and Zoom -  Jina Kim (ISU)

Title: From finite to infinite dimensions: functional inequalities for linear diffusions with degenerate noise

Abstract: In this talk, we introduce functional inequalities for specific classes of hypoelliptic stochastic differential equations. In particular, these inequalities include the reverse log-Sobolev inequality and Wang-type Harnack inequality for a large class of linear SDEs with degenerate noise. They are obtained from gradient bounds of the semigroup of the process and making use of the generalized version of the carré du champ operator. From the results we take steps further and also look for quasi-invariance of the same form of SDEs that take values in an infinite-dimensional Hilbert space. This is to gain some type of smoothness of the heat kernel measure of an infinite-dimensional process, and done by using the "projected" bound.

Mar 9 (4:10pm) Zoom -  Lee Przybylski (ISU)

Title: The Persistence Landscapes of Affine Fractals

Abstract: We develop a method for calculating the persistence landscapes of affine fractals using the parameters of the corresponding transformations.  Given an iterated function system of affine transformations that satisfies a certain compatibility condition, we prove that there exists an affine transformation acting on the space of persistence landscapes which intertwines the action of the iterated function system.  This latter affine transformation is a strict contraction and its unique fixed point is the persistence landscape of the affine fractal.  We present several examples of the theory as well as confirm the main results through simulations.

Mar 23 (4:10pm) Zoom -  Virginia Naibo (Kansas State University)

Title : The Neumann problem in graph Lipschitz domains in the plane. 

Abstract: New aspects of the solvability of the classical Neumann boundary value problem in a graph Lipschitz domain in the plane will be presented. When the domain is the upper half-plane and the boundary data is assumed to belong to weighted  Lebesgue or weighted Lorentz spaces, it will be shown that the solvability of the Neumann problem in these settings may be characterized in terms of Muckenhoupt weights and related weights, respectively. For a general graph Lipschitz domain $\Omega$, as proved in an unpublished work by E. Fabes and C. Kenig, there exists $\varepsilon>0$  such that the Neumann problem is solvable with data in $L^p(\partial\Omega)$ for $1<p<2+\varepsilon;$ it will be shown that the Neumann problem is solvable at the endpoint $2+\varepsilon$ with data in the  Lorentz space $L^{2+\varepsilon}(\partial\Omega).$ Examples of the results in Schwarz-Christoffel Lipschitz domains and related domains will be given. This is joint work with María Jesús Carro (Universidad Complutense de Madrid) and Carmen Ortiz-Caraballo (Universidad de Extremadura).

Apr 6 (4:10pm) Zoom - Guo-Jhen Wu (KTH Royal Institute of Technology) 

Title: Quasi-stationary distribution and ergodic control problems

Abstract: We introduce two ergodic control problems that can be used to analyze the quasi-stationary distributions (QSDs) associated with a diffusion process. The two problems are in some sense dual, with one defined in terms of the generator associated with the diffusion process and the other in terms of its adjoint. The first ergodic control problem can be used to characterize the Q-process associated with the QSD, and the cost potential of the second ergodic control problem to characterize the QSD itself. We briefly mention how the control problems can be used to construct numerical approximations of the QSD. This is joint work with Amarjit Budhiraja (University of North Carolina at Chapel Hill), Paul Dupuis (Brown University), and Pierre Nyquist (KTH).

Apr 13 (4:10pm) Zoom - Rodolfo H. Torres (University of California, Riverside)

Title: John-Nirenberg Inequalities and Weight Invariant BMO Spaces

Abstract: We will explore new deep connections between John-Nirenberg inequalities and Muckenhoupt weight invariance for a large class of spaces including or resembling versions of the classical space of functions of Bounded Mean Oscillation (BMO). The results are formulated in a very general framework in which $BMO$-type  spaces are constructed using a base of sets, employed also to define weights with respect to a non-negative measure (not necessarily doubling), and an appropriate oscillation functional. This includes as particular cases many different function spaces on geometric settings of interest. As a consequence the weight invariance of several $BMO$ spaces considered in the literature is proved. Most of the invariance results obtained under this unifying approach are new even in the most classical settings. This is joint work with Jarod Hart.

Apr 20 (4:10pm) Zoom - Ruimeng Hu (UCSB)

Title: Convergence of Empirical Measures, Mean-Field Games and Signatures 

Abstract: In this talk, we first propose a new class of metrics and show that under such metrics, the convergence of empirical measures in high dimensions is free of the curse of dimensionality, in contrast to Wasserstein distance. Proposed metrics originate from the maximum mean discrepancy, which we generalize by proposing criteria for test function spaces. Examples include RKHS, Barron space, and flow-induced function spaces. One application studies the construction of Nash equilibrium for the homogeneous n-player game by its mean-field limit (mean-field game). Then we discuss mean-field games with common noise and propose a deep learning algorithm based on fictitious play and signatures in rough path theory. The first part of the work collaborates with Jiequn Han and Jihao Long; the second part is the joint work with Ming Min.

Apr 27 (4:10pm) Zoom - Michael Perlmutter (UCLA)

Title: Deep Learning for Data with Geometric Structure 

Abstract: The field of Geometric Deep Learning aims to extend the success of Convolutional Neural Networks to data sets that lack a Euclidean, grid-like structure and are more naturally modeled as manifolds or (possibly directed). A major advance in this field has been the rise of graph convolutional networks which extend the success of CNNs to the graph domain by defining convolution either as a localized averaging operation or via the eigendecomposition of the graph Laplacian.

Despite the success of these networks, standard graph convolutional networks also have some limitations. They have difficulty (i) incorporating high-frequency information and long-range dependencies. (ii) handling directed graphs. In my talk, I will talk about how we can overcome these difficulties via (i) the graph and manifold scattering transforms which capture high-frequency information and long-range dependencies via wavelet filters, (ii) MagNet, a directed graph neural net that encodes directional information via a Hermitian matrix known as the magnetic Laplacian. 

May 4 (4:10pm) Zoom - Michael Conroy (University of Arizona)

Title: Renewal Theory for Maxima on Trees 

Abstract: This talk focuses on the all-time supremum W of the perturbed branching random walk, which is the so-called endogenous solution to the stochastic fixed-point equation known as the high-order Lindley equation. Under certain assumptions (in particular that the paths of each branch of the random walk have negative drift), W satisfies the tail asymptotic P(W > t) ~ He^{-at} for certain constants H, a > 0. Using tools from Markov renewal theory and spine changes of measure for branching processes, we establish the tail asymptotic by analyzing the forward iterations of the map defining the fixed-point equation. As a bonus to this approach, we obtain an unbiased, strongly-efficient, and easy-to-simulate estimator for the rare event probability P(W > t). This talk presents joint work with Bojan Basrak (University of Zagreb), Mariana Olvera-Cravioto (UNC Chapel Hill), and Zbigniew Palmowski (Wroclaw University of Science and Technology).  

Sep 8 (4:10pm) Carver 401 and Zoom -  David Herzog (ISU)

Title: A functional law of the iterated logarithm for weakly hypoelliptic diffusions 

Abstract: This is part two of three talks in the spring and fall. We discuss the method of stochastic characteristics to help motivate laws of the iterated logarithm. This method is often used to solve second-order linear boundary-valued PDEs, such as the Dirichlet or Poisson Problem in a bounded domain in R^n. If the operator L defining the equation is uniformly elliptic in a neighborhood of the domain and the boundary is smooth, then we can always use this method with ease to find an expression for the unique classical solution, provided the data is smooth enough. However, if we drop the ellipticity assumption, say L is only hypoelliptic or "weakly" hypoelliptic (i.e. the randomness is degenerate), then the candidate expression typically makes sense, but it may no longer satisfy the equation. The problem moving from the elliptic to hypoelliptic settings is that the natural elliptic approximation of L may not approximate the defining stochastic characteristics near the boundary, even though the processes defined by these operators are close in a very strong sense. In order to determine when it does approximate L near the boundary, we establish laws of the iterated logarithm for the diffusion at time zero.

Sep 15 (4:10pm) Carver 401 and Zoom -  David Herzog (ISU)

Title: A functional law of the iterated logarithm for weakly hypoelliptic diffusions

Abstract: This is part three of three talks in the spring and fall. We discuss the method of stochastic characteristics to help motivate laws of the iterated logarithm. This method is often used to solve second-order linear boundary-valued PDEs, such as the Dirichlet or Poisson Problem in a bounded domain in R^n. If the operator L defining the equation is uniformly elliptic in a neighborhood of the domain and the boundary is smooth, then we can always use this method with ease to find an expression for the unique classical solution, provided the data is smooth enough. However, if we drop the ellipticity assumption, say L is only hypoelliptic or "weakly" hypoelliptic (i.e. the randomness is degenerate), then the candidate expression typically makes sense, but it may no longer satisfy the equation. The problem moving from the elliptic to hypoelliptic settings is that the natural elliptic approximation of L may not approximate the defining stochastic characteristics near the boundary, even though the processes defined by these operators are close in a very strong sense. In order to determine when it does approximate L near the boundary, we establish laws of the iterated logarithm for the diffusion at time zero.

Oct 13 (4:10pm) Zoom - Debankur Mukherjee (Georgia Tech)

Title: Load Balancing Under Strict Compatibility Constraints

Abstract: We will talk about large-scale systems operating under the JSQ(d) policy in the presence of stringent task-server compatibility constraints. Consider a system with N identical single-server queues and M(N) task types, where each server is able to process only a small subset of possible task types. Each arriving task selects d≥2 random servers compatible to its type, and joins the shortest queue among them. The compatibility constraint is naturally captured by a fixed bipartite graph GN between the servers and the task types. When GN is complete bipartite, the mean-field approximation is proven to be accurate as N→∞. However, such dense compatibility graphs are infeasible due to their overwhelming implementation cost and prohibitive storage capacity requirement at the servers. We will characterize a class of sparse compatibility graphs for which the mean-field approximation remains valid asymptotically as N→∞. Both process-level limit and steady-state convergence results will be discussed. This is a joint work with Daan Rutten. 

Oct 20 (4:10pm) Zoom - Mikil Foss (University of Nebraska-Lincoln)

Title: Trace Theorems for Non-Differentiable Functions

Abstract: Given a uniformly continuous function on an open domain, there is a unique extension to the boundary that preserves the continuity. The trace operator provides a function that captures the boundary values for this extension. Gagliardo’s trace theorem extends this concept to the Sobolev spaces. There have been generalizations of Gagliardo’s theorem in many directions. Typically, trace theorems require some differentiability of the function in its domain and some regularity of the domain’s boundary. These assumptions ensure a there is a well-defined boundary value function. Moreover, this trace will, itself, possess some differentiability and a certain Lebesgue point property. I will present a trace theorem that provides a well-defined boundary-value function that exists in a fractional Sobolev space and has the Lebesgue point property yet requires no differentiability within the domain and allows very irregular boundaries. The result is motivated by boundary-value problems involving nonlocal operators that are defined for integrable but not necessarily differentiable functions. 

Oct 27 (4:25pm) Zoom - Farzad Sabzikar (ISU)

Title: How Does Tempering Affect the Local and Global Properties of Fractional Brownian Motion?

Abstract: In this talk, we discuss the effects of tempering the power-law kernel of the moving average representation of a fractional Brownian motion (fBm) on some local and global properties of this Gaussian stochastic process. Tempered fractional Brownian motion (TFBM) and tempered fractional Brownian motion of the second kind (TFBMII) are the processes that are considered in order to investigate the role of tempering. Tempering does not change the local properties of fBm including the sample paths and p-variation, but it has a strong impact on the Breuer–Major theorem, asymptotic behavior of the third and fourth cumulants of fBm, and the optimal fourth-moment theorem.

Nov 3 (4:10pm) Carver 401 and Zoom - Evan Camrud (ISU)

Title: Bedrossian, Blumenthal, and Punshon-Smith's bounds on Lyapunov exponents for SDEs 

Abstract: This will be a literature review talk on a paper that has been very interesting to my research. Bedrossian et al. obtain lower estimates for the maximal and sum Lyapunov exponents of the stochastic Lorenz 96 dynamical system. The positivity of the maximal exponent implies chaos in the dynamics while the negativity of the sum exponent implies Lebesgue volume contraction. 

Nov 17 (4:10pm) Zoom - Jaydeb Sarkar (Indian Statistical Institute)

Title: Schur functions - A brief survey 

Abstract: The Schur class, denoted by S(D), is the set of all functions analytic and bounded by one in modulus in the open unit disc D in the complex plane. Elements of S(D) are called Schur functions. A classical result going back to Issai Schur states: A function f is a Schur function if and only if f admits a linear fractional transformation (or transfer function realization). Linear fractional transformations are attached with colligation matrices or scattering matrices on Hilbert spaces. Schur's view of bounded analytic functions is one of the most used (and useful) tools in classical and modern complex analysis, function theory, operator theory, electrical network theory, signal processing, linear systems, operator algebras, and image processing (just to name a few).

In the first part of this talk, we will give a brief (but within the span of little more than a century) historic perspective and introduction to Schur theory and discuss its interactions with some classical problems in function theory and operator theory (like Nevanlinna-Pick interpolation). In the second part of the talk, we will review Schur's approach (ubiquity and its complications) to functions of several complex variables from a linear analysis point of view. 

Dec 1 (4:10pm) Zoom - Simon Bortz (University of Alabama)

Title: Recent Developments in Parabolic Uniform Rectifiability 

Abstract: In the 1980’s harmonic analysis was being rapidly adapted to `rough’ settings. Among the important early developments was the proof of the L^2 boundedness of the (principal value) Cauchy integral operator on a Lipschitz graph due to Coifman, McIntosh and Meyer. Soon after Coifman, David and Meyer extended this result to higher dimensions and more general singular integral operators. A natural question arose: Is there a characterization of sets for which all nice singular integral operators are L^2 bounded? David and Semmes found many characterizations of these sets and called them uniformly rectifiable. More recently these sets have played a significant role in potential theory.

In this talk, I will discuss the history of uniformly rectifiable sets and some of my recent work on their parabolic analogs. The parabolic theory presents some obstacles in the form of `non-local’ smoothness and, in fact, many of the characterizations of uniform rectifiability fail to have parabolic analogs.

Dec 8 (4:10pm) Zoom - Gideon Simpson (Drexel University)

Title: Nonlocal Diffusions with Additive Noise 

Abstract: In this talk I will discuss recent work on infinite dimensional nonlocal diffusions with additive noise. These may be obtained as the continuum limits of noise driven Kuramoto oscillator systems.  A well-posedness theory is developed for this infinite dimensional problem, and convergence results are obtained for both the associated semi-discrete and fully discrete problems.  This provides a basis for studying the associated metastability problems of the continuum limit Kuramoto system, which may also be viewed as an approximation of a high, but finite, dimensional problem.  Ongoing progress on studying the metastability, both numerically and analytically will be presented.  Novel challenges in the analysis are also highlighted.  This work is in collaboration with Georgi Medvedev (Drexel). 

Feb 10 (4:10pm) - Ananda Weerasinghe (ISU) 

Title: Diffusion approximations and optimal control of processing systems in heavy traffic

Abstract:  A running maximum control of a diffusion process is considered. To find an optimal control, we consider the associated two-dimensional degenerate Hamilton-Jacobi-Bellman(HJB) equation and construct a sufficiently smooth solution. This in turn leads to a bounded, feed-back type optimal control. 

This diffusion control problem(DCP) is motivated by the service rate control of an infinite-server processing system in heavy traffic where the control mechanism is enforced to limit the storage capacity. The DCP is obtained as the diffusion approximation of the processing system control problem.  An asymptotically feed-back type optimal control mechanism for the processing system is obtained using the solution to DCP.

Feb 17 (4:10pm) - Qi Feng (USC)

Title: Accelerating Convergence of Stochastic Gradient MCMC: algorithm and theory

Abstract: Stochastic Gradient Langevin Dynamics (SGLD) shows its advantages in multi-modal sampling and non-convex optimization, which implies broad application in machine learning, e.g. uncertainty quantification for AI safety problems, etc. The core issue in this field is about the acceleration of the SGLD algorithm and convergence rate of the continuous time Langevin diffusion process to its invariant distribution. In this talk, I will present the general idea of entropy dissipation and convergence rate analysis. In the first part, I will show stochastic gradient MCMC algorithms based on replica exchange Langevin dynamics, and empirical studies of our algorithm on optimization and uncertainty estimates for synthetic experiments and image data. In the second part, I will talk about the convergence rate analysis of reversible/non-reversible degenerate Langevin dynamics (i.e. variable coefficient underdamped Langevin dynamics). The talk is based on a series of joint works with W. Deng, L. Gao, G. Karagiannis, W. Li, F. Liang, and G. Lin.

Feb 24 (4:10pm) - Ananda Weerasinghe (ISU)

Title: A diffusion control problem with a path-dependent cost functional 

Abstract: We analyze and obtain the solution to a diffusion control problem (DCP) with a cost structure consists of two types of costs: a cost due to increase of the running maximum of the state process and a control cost where the control term effects the linear drift of the state process and it helps to slow down the growth of the running maximum. This DCP is obtained as the diffusion approximation of an infinite-server controlled stochastic processing system. The solution to this DCP enables us to find nearly optimal controls for the control problem associated with the processing system.

Mar 3 (4:10pm) - Nicole Buczkowski (University of Nebraska-Lincoln)

Title: Stability of solutions with respect to changes in data and parameters of nonlocal models

Abstract: Mathematical models that are physically relevant must guarantee existence and uniqueness of solutions, as well as continuity with respect to the data. Thus, small changes in data or parameters will lead to appropriate changes in the solution. These continuity properties are also relevant in numerical analysis, where small variations that are beyond computer precision should not be reflected in the numerical solutions produced. Nonlocal models, with their capability to handle discontinuities, record long range interactions through a kernel which gives additional flexibility. By varying data given by external forces, boundary conditions, the kernel, or the type of nonlinearity used in the model the solution changes accordingly. These changes are quantified through stability or continuity results, which we obtained in linear and nonlinear settings.

Mar 17 (4:10pm) - Kyle Liss (University of Maryland)

Title: Quantitative hypoellipticity and exponential convergence to equilibrium for a class of SDEs 

Abstract: Various physical systems (fluids, nonlinear waves, etc.) exhibit turbulent, generally chaotic behavior when subject to forcing and weak damping. In the study of such systems, two important problems once unique ergodicity has been established are to understand properties of the invariant measure and quantify the convergence of generic time averages to the unique stationary statistics. In this talk, we discuss recent work on the long-time behavior of a class of hypoelliptic, weakly damped SDEs that covers some prototypical chaotic/turbulent systems such as Lorenz-96 and Galerkin truncations of the stochastic Navier-Stokes equations. Our main results are an optimal, quantitative estimate on the exponential convergence to equilibrium in the limit of vanishing and balanced noise and dissipation, and uniform (in the small noise/dissipation parameter) pointwise estimates on the stationary density. Exponential convergence for the model under consideration has been known for some time, but our quantitative estimates are new. Our proof uses a scheme that combines a weak Poincaré inequality argument with quantitative hypoelliptic regularization for the associated time-dependent Kolmogorov equation. The uniform estimates on the stationary density are crucial to carrying out our approach, and are proven with hypoelliptic De Giorgi and Moser type iterations. This is joint work with Jacob Bedrossian (University of Maryland).

Mar 24 (4:15pm) - Xin Liu (Clemson University)

Title:  Allocation control for Bipartite Matching Queues

Abstract: Motivated by the study of organ transplant systems, we study a multi-class bipartite matching system with one side labeled “patients” and the other side labeled “organs”. We assume that patients may die/delist or move between classes due to changes of their health status. We introduce two stochastic queueing control problems (QCPs) with the same control process that governs the allocation of each arriving organ. For the first QCP, its objective is to maximize the expected life years of all population in the system during a finite time horizon. The second QCP is to minimize a long run average linear holding cost. For both QCPs, we first construct proper deterministic control problems, referred to as the fluid control problems (FCPs) and show that the FCPs provide the best achievable performance for the corresponding QCPs. We next develop an asymptotic framework, in which large-scale matching systems are considered, and show that the fluid scaled QCP attains the corresponding FCP optimal value asymptotically. We propose a simple priority type policy for the first QCP and a randomized allocation policy for the second QCP based on the optimal solutions of the corresponding FCPs. Finally, we establish the asymptotic optimality of the proposed policies through scaling limit theorems.

Mar 31 (4:10pm) - Hayley Olson (University of Nebraska-Lincoln)

Title: Tempered and Truncated Fractional Operators - Exploring Reduction of Computational Costs 

Abstract: Fractional calculus operators have been around nearly as long as the familiar integer-order derivatives. The introduction of an exponential tempering to the fractional operator has recently found applications in geophysics and finance. However, the infinite radius of interaction of the operator makes it computationally hefty. In this talk, we explore the use of a truncated fractional operator -- restricted to a bounded domain of integration -- in order to capture the same action of the tempered operator but with an easier computational load.

Apr 7 (4:10pm) - Cecilia Mondaini (Drexel University)

Title: Rates of convergence to statistical equilibrium: a general approach and applications 

Abstract: Randomness is an intrinsic part of many physical systems. For example, it might appear due to uncertainty in the initial data, or in the derivation of the mathematical model, or also in observational measurements. In this talk, we focus on the study of convergence/mixing rates for stochastic/random dynamical systems towards statistical equilibrium. Our approach uses the weak Harris theorem combined with a generalized coupling technique to obtain such rates for infinite-dimensional stochastic systems in a suitable Wasserstein distance. We emphasize the importance of obtaining these results via algorithms that are well-defined in infinite dimensions. This allows to obtain convergence rates that are robust with respect to finite-dimensional approximations, thus beating the curse of dimensionality. In particular, we show two scenarios where this approach is applied in the context of stochastic fluid flows. First, to show that Markov kernels constructed from a suitable numerical discretization of the 2D stochastic Navier-Stokes equations converge towards the invariant measure of the continuous system. Second, to approximate the posterior measure obtained via a Bayesian approach to inverse PDE problems, particularly when applied to advection-diffusion type PDEs. Here we present an alternative proof of mixing rates for the exact preconditioned Hamiltonian Monte Carlo algorithm in infinite dimensions, a result that was still an open problem until quite recently. This talk is based on joint works with Nathan Glatt-Holtz (Tulane U).

Apr 14 (4:10pm) - Chao Zhu (University of Wisconsin-Milwaukee)

Title: Regime-Switching Jump-Diffusion Processes with Countable Regimes

Abstract: This work focuses on a class of regime-switching jump diffusion processes, which is a two component Markov processes $(X(t),\Lambda(t))$, where the analog component $X(t) \in R^{d}$ models the state of interest while the switching component $\Lambda(t)\in \{1,2,\dots\}$ can be used to describe the structural changes of the state or random factors that are not represented by the usual jump diffusion formulation. Considering the corresponding stochastic differential equations, our focus is on treating those with non-Lipschitz coefficients. We first show that there exists a unique strong solution to the corresponding stochastic differential equation. Then Feller and strong Feller properties and exponential ergodicity are investigated.  This is a joint work with Khwanchai Kunwai, Fubao Xi and George Yin.

Apr 21 (4:10pm) - David Herzog (ISU)

Title: A functional law of the iterated logarithm for weakly hypoelliptic diffusions 

Abstract: This is part one of 2-3 talks which will be continued in the fall. In this part, we discuss the method of stochastic characteristics to help motivate laws of the iterated logarithm. This method is often used to solve second-order linear boundary-valued PDEs, such as the Dirichlet or Poisson Problem in a bounded domain in R^n. If the operator L defining the equation is uniformly elliptic in a neighborhood of the domain and the boundary is smooth, then we can always use this method with ease to find an expression for the unique classical solution, provided the data is smooth enough. However, if we drop the ellipticity assumption, say L is only hypoelliptic or "weakly" hypoelliptic (i.e. the randomness is degenerate), then the candidate expression typically makes sense, but it may no longer satisfy the equation. The problem moving from the elliptic to hypoelliptic settings is that the natural elliptic approximation of L may not approximate the defining stochastic characteristics near the boundary, even though the processes defined by these operators are close in a very strong sense. In order to determine when it does approximate L near the boundary, we establish laws of the iterated logarithm for the diffusion at time zero.

Apr 28 (4:10pm) - Sungchan Park (ISU)

Title: Adjusted Linear Preferential Attachment Model with Covariates and Existence of MLE

Abstract: Barabási–Albert (BA) model, or Linear Preferential Attachment Model is an appealing mechanism for modeling a growing network with the property that the more connected a node is, the more likely it is to receive new links. Despite its simplicity and useful results explaining the power-law degree distributions and the first mover advantage, BA model highly underestimates the possibility that late comers may beat the first movers. So, Bianconi-Barabási (BB) model introduced the concept of fitness, an intrinsic and quantitative measure of each node's ability to stay in front of the competition. On the other hand, however, BB model does not offer any idea where the intrinsic ability comes from. We introduced a new model with covariates and parameters that assesses how attributes of each node affect preferential attachment. In particular, we showed the unique MLE exists for the univariate case and found a sufficient condition that the unique MLE exists for multivariate cases. We also explored numerical examples of the model and its MLE.

May 5 (4:10pm) - Nathan Glatt-Holtz (Tulane University)

Title: A Bayesian Approach to Quantifying Uncertainty in Divergence Free Flows.

Abstract: We treat a statistical regularization of the ill-posed inverse problem of estimating a divergence free flow field $u$ from the partial and noisy observation of a passive scalar $\theta$ which is advected by $u$. Our solution is a Bayesian posterior distribution, that is a probability measure $\mu$ of the space of divergence free flow fields which precisely quantifies uncertainties in $u$ once one species models for measurement error and a prior knowledge for $u$.

In this talk we survey some of our recent work which analyzes $\mu$ both analytically and numerically. In particular we discuss a posterior contraction (consistency) result as well as some Markov Chain Monte Carlo (MCMC) algorithms which we have developed, refined and rigorously analyzed to effectively sample from $\mu$. This is joint work with Jeff Borggaard, Justin Krometis and Cecilia Mondaini.