**ISU PADS Seminar**

**Probability, Analysis, and Data Science Seminar**

Department of Mathematics at Iowa State University

Fall 2021

Wednesdays at 4:10pm-5:10pm (including 10 minutes Q&A) in person (Carver 401) or virtually (via Zoom at https://iastate.zoom.us/j/9974481906), depending on the speaker.

Organizers: Pablo Raúl Stinga and Ruoyu Wu

**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.

**Sep ****22**

**Sep 29**

**Oct 6**

**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 G_{N} between the servers and the task types. When G_{N} 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)**

**Oct 27**

**Nov 3 (4:10pm) ****Evan Camrud**** (ISU)**

**Nov 10**

**Nov 17**

**Dec 1**

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

**Past Seminars: Spring 2021**

**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.