**
Department of Mathematicsand Statistics
**

Mathematics and Science Center, Room 368

146 Library Drive

Rochester,
MI
48309-4479

**(location map)**

phone: (248) 370-3430

fax: (248) 370-4184

**Hours:**

Monday–Friday: 8:00–11:59 a.m. and 1:00–5:00 p.m.

# Department Colloquium

## 2022-2023 Colloquium

Unless indicated otherwise (*), the talks will be held 12-12:50 p.m. Tuesday in 372 MSC, with refreshment and conversation from 11:30 a.m. - noon in 368 MSC.

**April 04, Gary McDonald, Disparity testing when there is uncertainty in data sources.**

This colloquia talk centers on a new class of statistical inference problems arising from disparity assessments in the financial, insurance, and related industries where individual information is incomplete with respect to race/ethnicity. The evolution of such problems will be discussed along with the formulation of the statistical model and strategies for relevant parameter estimation. Both frequentist and Bayesian approaches will be formulated and computational methods given to permit implementation with real-world problems involving both small sample and large sample cases. This work has taken place over the past five years with the contributions of three Dept. of Math & Stat students. The talk should be of interest to a "general" audience as well as to "technical specialists".

**March 21, Ebrahim Sarabi, Miami University, Twice Epi-Differentiability: Past, Present, and Future**

In this talk, we discuss various aspects of twice epi-differentiablity of extended-real-valued functions and its remarkable applications in parametric optimization, second-order variational analysis, and local convergence analysis of the Newton method. We begin with presenting the history of this concept and then proceed with its evolution in the last three decades. In particular, we demonstrate that this property often holds for various classes of functions, important for applications to optimization problems. Finally, we discuss how a generalization of this concept leads us to achieve a characterization of continuous differentiability of the projection mapping for a large class of sets.

**March 14, Loic Cappanera (University of Houston), Robust and efficient numerical methods for incompressible flows with variable density.**

The modeling and approximation of incompressible flows with variable density are important for a large range of applications in biology, engineering and geophysics. Our main goal here is to develop and analyze numerical methods that use time-independent stiffness matrices and that can be used with high order finite element and spectral methods. First, we introduce a semi-implicit scheme based on projection methods and the use of the momentum, equal to the density times the velocity, as primary unknown. We analyze the stability and convergence properties of the method and establish a priori error estimates. A fully explicit version of the scheme is then proposed. Its robustness and convergence properties are studied with a pseudo spectral code over various setups involving large ratio of density, gravity and surface tension effects, or manufactured solutions. Applications to magnetohydrodynamics instabilities in industrial setups such as liquid metal batteries will be presented shortly. Eventually, a novel method based on artificial compressibility techniques is introduced and its performances are compared to our projection-based method.

**Feb 14, Santosh Kottalgi (Ansys), Understanding the increasing role of numerical simulation to address product design challenges using Ansys software**

Product design process has changed significantly in the last 20 years because of increasing use of simulation technologies, improved IT hardware offering and pressure to reduce time and cost. A simulation that was used as a failure analysis tool during the service life stage is now used almost in all stages of the product life cycle, from design to manufacturing to servicing to recycling. The strength of simulation is improved by added focus from researchers on accurate and comprehensive numerical methods to represent real-world physics as closely as possible. It is no wonder that simulation models are used as Digital twins of physical models in many cases. Ansys as a market leader in simulation solution providers is working on addressing these complex challenges and the talk will discuss some of the market trends and technologies. The author will also provide guidance to students to find their career paths as simulation analysts, developers, researchers, and educators.

The talk is from noon - 12:50 p.m. in MSC 372.

Santosh Kottalgi's talk will be also live on Zoom.

**Nov 8 Matthew Toeniskoetter (Oakland University) Overrings of a 2-Dimensional RLR**

Given a two-dimensional regular local ring D, there is a rich classical theory of the regular local rings birationally dominating it (between it and its field of fractions). These rings are in one-to- one correspondence with the divisorial valuation rings birationally dominating D, and they form a tree structure called the quadratic tree through the process of blowing-up. Much less is known about the non-Noetherian rings between D and its field of fractions. In this talk, we work towards a classification of the integrally closed rings between D and its field of fractions. We consider subspaces of the Zariski-Riemann space of valuation rings dominating D, and we relate the topological properties of the subspace with the ring-theoretic properties of the ring it produces. We give a new construction of a type of ring first proved by Nagata: a 1-dimensional integrally closed local ring birationally dominating D that's not integrally closed. We also describe new examples of one- and two-dimensional vacant domains (domains with a unique Kronecker function ring) that are not Prüfer. This is joint work with B. Heinzer, A. Loper, and

B. Olberding.

(*) The talk is from 11:30 a.m. - 12:20 p.m. in MSC 372.

Matthew Toeniskoetter's talk will be also live on Zoom.

**Oct 25 Jun Hu (Oakland University) A general sequential learning procedure with illustrations**

Sequential learning builds sampling schemes in which the required sample size is not fixed in advance and instead, observations are collected successively according to some predefined stopping rule. In this talk, we propose a broad and general sequential learning procedure, which incorporates four different types of sampling schemes: (i) the classic Anscombe-Chow-Robbins purely sequential sampling scheme; (ii) the ordinary accelerated sequential sampling scheme; (iii) the relatively new k-at-a-time sequential sampling scheme; and (iv) the new k-at-a-time improved accelerated sequential sampling scheme. The second-order efficiency of this general sequential learning procedure is fully investigated.

We will implement this sequential learning procedure to handle three fundamental statistical inference problems as possible illustrations, namely, (i) minimum risk point estimation, (ii) bounded variance point estimation, and (iii) point estimation in linear regression. An extensive set of simulations are presented to validate our theoretical findings. And real data analyses are included to highlight its practical applicability.

Jun Hu's talk will be also live on Zoom.

**Oct 18 Yongjin Lu (Oakland University) Large time behavior of nonlinear partial differential equations subject to external force**

In this presentation, we address the problem of long-time behavior and the associated stabilization of solutions to nonlinear partial differential equations (PDE) when they are subject to external force. The equations under study include a system of nonlinear PDEs that couples Navier-Stokes equation with wave equation to describe the interaction between a solid submerged in surrounding fluid and its constitutive equation: the Navier-Stokes equation. We study the technically interesting and practically realistic problem of stabilizing the coupled dynamics of FSI to a non-trivial equilibrium driven by a time-independent external force. To achieve this goal, feedback control mechanisms that depend on the equilibrium and applied to the fluid and solid domains are proposed. A natural problem to study next is the large-time behavior of the solution when the system is subject to a time dependent external force. In this direction, we established the existence of pullback attractor for the constitutive equation, the Navier-Stokes equation, of FSI, when it is subject to a time dependent external force with relaxed compactness assumption. We also showed that the pullback attractor has a finite fractal dimension using the trace formula.

Yongjin Lu's talk will be also live via Zoom.

**Oct 11 Hon Yiu So (Oakland University) Semiparametric inference in one-shot device with competing risks**

One-shot devices mean one-time products. Typical one-shot devices include airbags, fire-extinguishers, missiles, etc. Those devices' observations are either successes or failures at the time of test/use. So, there is usually a considerable loss of information, as we cannot observe the exact failure time. In addition, those one-shot devices contain multiple components. For example, airbags contain crash sensors and air inflation chemicals, and missiles have accelerators and explosives. Malfunctioning in any element will result in device failures. Then, engineers will inspect the failed devices to identify the specific cause of failure. With such complexity, estimating those life characteristics becomes a complex problem.

This talk will focus on the estimation problem of One-shot devices under constant stress accelerated life-test. To avoid model misspecification, we proposed a semiparametric method. It can analyze the relationship between the lifetime of the parts and the stress level without any assumption about the component's lifetime. A link function relating to stress levels and lifetime is then applied to extrapolate the lifetimes of units from accelerated conditions to normal operating conditions.

Hon Yiu So's talk will be also live via Zoom.

**Sep 27 Gary McDonald (Oakland University) Approaches to the problem of ranking populations (or choosing the "best")**

The subject area of statistical ranking and selection procedures will be introduced. The so-called “indifference-zone” procedures and “subset selection” procedures will be described and their properties presented. These methodologies are applicable in comparing two or more populations with the goal of selecting (or isolating) the “best” population with a user specified level of confidence. In this context “best” is based on the ordering of a parameter characterizing each of the populations. For example, the “best” population could be defined as that one possessing the largest mean. Parametric, nonparametric (distribution-free), and Bayesian methods will be included. An analysis of motor vehicle traffic fatality rates will be given illustrating the use of a distribution-free subset selection procedure in a two-way block design context. The analysis of the traffic fatality rates will also be addressed from a Bayesian perspective.

Time permitting, some discussion will be given to research issues still remaining with one or more of these methodologies.

Gary McDonald's talk will be also live via Zoom.

Room: 910 1377 8430

Passcode: 894238

## 2021-2022 Colloquium

Unless indicated otherwise (*), the talks will be held 12-12:50 p.m. Tuesday in 372 MSC, with refreshment and conversation from 11:30 a.m. - noon in 368 MSC.

**Apr 19 Zhimin Zhang (Wayne State University) Some Recent Development in Superconvergence: LDG, DDG, IFEM, and IFVM (*)**

Superconvergence phenomenon is well understood for the h-version finite element method and researchers in this old field have accumulated a vast literature during the past half century. However, the relevant systematic study for discontinuous Galerkin, finite volume, and spectral methods is lacking. We believe that the scientific community would also benefit from the study of superconvergence phenomenon of those methods. Recently, some efforts have been made to expand the territory of the superconvergence. In this talk, I will summarize some recent development on superconvergence study for these methods. At the same time, some current issues and un-solved problems will also be addressed.

(*) The talk is from 10:30 - 11:30 a.m., in MSC 372.

**Apr 12 Liang (Jason) Hong (The University of Texas at Dallas) Instantaneous and limiting behavior of an n-node blockchain under cyber-attacks from a single hacker**

We investigate the instantaneous and limiting behavior of an n-node blockchain which is under continuous monitoring of the IT department of a company, but faces non-stop cyber attacks from a hacker. The blockchain is functional as far as no data stored on it has been changed, deleted, or locked. Once the IT department detects the attack from the hacker, it will immediately re-set the blockchain, rendering all previous efforts of the hacker in vain. The hacker will not stop until the blockchain is dysfunctional. For arbitrary distributions of the hacking times and detecting times, we derive the limiting functional probability, instantaneous functional probability, and mean functional time of the blockchain. We also show that all these quantities are increasing functions of the number of the nodes, substantiating the intuition that more nodes a blockchain has, the harder it is for a hacker to succeed in a cyber attack.

**Apr 5 Alex Yong (University of Illinois at Urbana-Champaign) Newell-Littlewood numbers**

The Newell-Littlewood numbers are defined in terms of the Littlewood-Richardson coefficients from algebraic combinatorics. Both appear in representation theory as tensor product multiplicities for a classical Lie group. This talk concerns the question:

Which multiplicities are nonzero?

In 1998, Klyachko established common linear inequalities defining both the eigencone for sums of Hermitian matrices and the saturated Littlewood-Richardson cone. We prove some analogues of Klyachko's nonvanishing results for the Newell-Littlewood numbers.

This is joint work with Shiliang Gao (UIUC), Gidon Orelowitz (UIUC), and Nicolas Ressayre (Universite Claude Bernard Lyon I). The presentation is based on arXiv:2005.09012, arXiv:2009.09904, and arXiv:2107.03152.

**March 29 Sarah Beetham (Oakland University) The peculiar nature of particle-laden turbulence**

Turbulent, disperse two-phase flows are pervasive in nature and industry. In many systems, the disperse phase (e.g., solid particles, liquid droplets, gas bubbles) modifies the turbulence in the carrier phase, giving rise to complicated flow features such as dense clusters (or bubble clouds) and regions nearly void of particles. This heterogeneity predicates a wide range of length- and time-scales, making fully-resolved computations at scales of interest intractable, even on modern super computers. Thus, the Reynolds Averaged Navier--Stokes (RANS) equations, which depend heavily upon modeling, continue to be the primary tool for large-scale computations of both single and multiphase turbulence. Despite their prevalence, developing accurate models, especially for the multiphase RANS equations, has remained a challenge. This is primarily due to the large parameter space characterizing such flows, making brute-force modeling approaches unfeasible and extensions from single-phase turbulence inadequate. In this talk, a few interesting examples of multiphase flows will be highlighted, followed by the introduction of a data-driven methodology based on sparse regression to enable modeling of these peculiar flows.

**Nov 23 Fernando Charro (Wayne State University) The Monge-Ampère equation: Classical local applications and recent nonlocal developments**

This talk will present the classical, local Monge-Ampère equation and its applications to optimal transport and differential geometry. We will discuss the degeneracy of the equation and the challenges it poses for the regularity of solutions. Finally, we will consider a nonlocal analog of the Monge-Ampère operator, introduced in collaboration with Luis Caffarelli.

**Nov 16 Yunier Bello-Cruz (Northern Illinois University) Infeasibility and error bound imply finite convergence of alternating projections**

In this talk, we combine two ingredients in order to get a rather surprising result on one of the most studied, elegant and powerful tools for solving convex feasibility problems, the method of alternating projections (MAP). Going back to names such as Kaczmarz and von Neumann, MAP has the ability to track a pair of points realizing minimum distance between two given closed convex sets. Unfortunately, MAP may suffer from arbitrarily slow convergence, and sublinear rates are essentially only surpassed in the presence of some Lipschitzian error bound, which is our first ingredient. The second one is a seemingly unfavorable and unexpected condition, namely, infeasibility. For two non-intersecting closed convex sets satisfying an error bound, we establish finite convergence of MAP. In particular, MAP converges in finitely many steps when applied to a polyhedron and a hyperplane in the case in which they have empty intersection. Moreover, the farther the target sets lie from each other, the fewer are the iterations needed by MAP for finding a best approximation pair. Insightful examples and further theoretical and algorithmic discussions accompany our results, including the investigation of finite termination of other projection methods.

**Nov 9 Tamas Horvath (Oakland University) Space-Time (Embedded-)Hybridized Discontinuous Galerkin Method for incompressible flow problems**

The Space-time (Embedded-)Hybridized Discontinuous Galerkin methods allow for an arbitrarily high order approximation in space and time, even on time-varying domains. Moreover, they are known to be pressure-robust, meaning that the approximation error in the velocity is independent of the pressure. Two essential ingredients are required for pressure-robustness: exact enforcement of the incompressibility constraint and H(div)-conformity of the finite element solution.

In this talk, we present analytical results, and we apply the method for fluid-rigid body interactions. We introduce a sliding grid technique for the rotational movement that can handle arbitrary rotation. The numerical examples will include galloping and fluttering motion.

**Sept 7 Li Li (Oakland University) Support of elements in cluster algebras**

The theory of cluster algebra is a branch in mathematics emerged in the year 2000, which grows rapidly and has far-reaching implications in many fields including representation theory, geometry, combinatorics, mirror symmetry of string theory, statistical physics, etc. Lots of research of cluster algebras focuses on construction of their natural bases. In this talk, we will study the properties of Newton polytopes and some possibly non-convex regions that contain the support of those basis elements, and illustrate several applications of these properties.