## PDE & Applied Math Seminar

### Wednesdays 1:10 - 2:00 p.m. in Surge 268

### May 17^{th}, 2017: Kurt E. Anderson of UC Riverside

#### Ecological dynamics on river networks

I will review current and recent work in my lab exploring dynamics of natural populations and communities on river networks. Using advances in graph theory, we will explore how the structure of river networks influences important ecological outcomes such as population persistence, ecological stability, and community asynchrony.

### May 10^{th}, 2017: Chen-Yun Lin of University of Toronto

#### The De-shape algorithm: “context-free” fundamental frequency estimator

Fundamental frequency (f0) estimation (a.k.a. pitch detection) has been an important topic in many fields since the age of computers. While many context-specific attempts have been made, it has been difficult to develop a “context-free” f0 estimator. Motivated by cesptrum analysis, Lin-Su-Wu proposes an f0 estimator, named the de-shape algorithm, which works well in a variety of contexts.

**In this talk, I introduce a mathematical model that quantifies multiple component oscillatory signals with time-varying frequency and amplitude and with time-varying non-sinusoidal oscillatory pattern. I illustrate the proposed algorithm through several numerical results on simulated, medical, musical, and biological signals as well as explain how cesptrum analysis is used to eliminate the wave-shape function influence and to extract the f0 information.**

### May 3^{rd}, 2017: Lizheng Tao of UC Riverside

#### Linear stability for the 2D Boussinesq equations in R*T

In this talk, we will cover the recent development in the linear stability problem of the Boussinesq equations around the Couette flow. The goal is to show that, with the shear flow, the perturbation is decaying exponentially, regardless of the troublesome term \partial_x \theta.

### April 26^{th}, 2017: Christina Knox of UC Riverside

#### Recovery of both sound speed and source in photo-acoustic tomography

Photo-acoustic tomography is an imaging method that attempts to combine the high resolution of ultrasound and the high contrast capabilities of electromagnetic waves. In this talk we will first introduce the mathematical problems photo-acoustic tomography presents. Uniqueness results will briefly be discussed for the situation when sound speed is known and the source term is to be recovered. Then the case when both sound speed and source term are unknown will be considered. Partial uniqueness results in this case proved by Liu and Uhlmann will be presented along with an outline of the proof which relies on the temporal Fourier transform.

### November 16^{th}, 2016: Ari Stern of Washington University in St. Louis

#### Multisymplectic hybridizable discontinuous Galerkin methods

For Hamiltonian ODEs, symplectic numerical integrators exhibit superior numerical performance in a global sense. For Hamiltonian PDEs, a suitable numerical method should be "multisymplectic" -- but what does this mean? We answer this question using the "unified framework" of Cockburn et al. for hybridizable discontinuous Galerkin (HDG) methods, which turns out to be particularly well-suited to this problem. Specifically, we give necessary and sufficient conditions for an HDG method to be multisymplectic, and we examine these criteria for several popular methods. (Joint work with Robert McLachlan, Massey University, New Zealand.)

### November 8^{th}, 2016: Ching-Shan Chou of Ohio State

#### Computer Simulations of Yeast Mating Reveal Robustness Strategies for Cell-Cell Interactions

Cell-to-cell communication is fundamental to biological processes which require cells to coordinate their functions. In this talk, we will present the first computer simulations of the yeast mating process, which is a model system for investigating proper cell-to-cell communication. Computer simulations revealed important robustness strategies for mating in the presence of noise. These strategies included the polarized secretion of pheromone, the presence of the alpha-factor protease Bar1, and the regulation of sensing sensitivity.

### November 2^{nd}, 2016: Jim Kelliher of UC Riverside

#### The Performance of Computer Caches

I will discuss the mathematical model and resulting algorithm that was behind a software product I developed in the early 1990s to model the behavior of disk caches. The model was created using "operational analysis," a somewhat unfortunate name for a beautiful idea of Denning and Buzen's appearing in a paper they published in 1978. I will digress a little to discuss operational analysis, which is a way of developing and analyzing the efficacy of mathematical algorithms used to model non-stochastically, systems traditionally modeled stochastically.

### October 26^{th}, 2016: Mihaela Ifrim of UC Berkeley

#### Two dimensional water waves in holomorphic coordinates

We consider this problem expressed in position-velocity potential holomorphic coordinates. We will explain the set up of the problem(s) and try to present the advantages of choosing such a framework. Viewing this problem(s) as a quasilinear dispersive equation, we develop new methods which will be used to prove enhanced lifespan of solutions and also global solutions for small and localized data. The talk will try to be self contained.

### October 12^{th}, 2016: Supei Su of Chang'an University

#### Entropy-conservative and Entropy-stable Schemes for the Conservation Laws

Additional criteria are required to single out a unique physically relevant solution of the conservative equations'. Entropy condition plays a decisive role in the theory and numerics to identify the unique solution. In the numerics, the algorithm, satisfying the corresponding discrete entropy inequality, is called entropy-conservative and entropy-stable schemes. We will talk about the research background of entropy-conservative and entropy-stable finite difference/finite volume schemes for the conservation laws. Finally, our research results based on comparison principle, which is called CWENO-type entropy-conservative and entropy-stable schemes, are shown for presenting high resolution and high order.

### October 5^{th}, 2016: Kevin Costello of UC Riverside

#### Gathering Information in Networks

Imagine a large number of nodes, each holding some piece of information and all wanting to transmit to the same receiver. If they all broadcast at the same time, the messages may get so garbled as to be intelligible. So somehow we want to guarantee that each message gets through -- that at least once it's transmitted at a different time than all the other messages. Easy enough if the nodes are all talking to each other and can agree to take turns.

But what if the nodes aren't speaking to each other? Can we design a protocol that will make sure all the messages still get through? And if so, how long will it take? I will discuss several variants of this problem.

Joint work with Marek Chrobak (UCR) and some with Leszek Gasieniec and Dariusz Kowalski (Liverpool).

### September 28^{th}, 2016: Amir Moradifam of UC Riverside

#### Existence and Structure of Minimizers of Least Gradient Problems

Please click here for the abstract.

### June 1^{st}, 2016: Jianxian Qiu of Xiamen University

#### Weighted Essentially Non-Oscillatory limiters for Runge-Kutta Discontinuous Galerkin Methods

In the presentation we will describe our recent work on a class of new limiters, called WENO (weighted essentially non-oscillatory) type limiters, for Runge-Kutta discontinuous Galerkin (RKDG) methods. The goal of designing such limiters is to obtain a robust and high order limiting procedure to simultaneously obtain uniform high order accuracy and sharp, non-oscillatory shock transition for the RKDG method. We adopt the following framework: first we identify the "troubled cells", namely those cells which might need the limiting procedure; then we replace the solution polynomials in those troubled cells by reconstructed polynomials using WENO methodology which maintain the original cell averages (conservation), have the same orders of accuracy as before, but are less oscillatory. These methods work quite well in our numerical tests for both one and two dimensional cases, which will be shown in the presentation.

### Special Day/Time: May 27^{th}, 2016 4:10 - 5:00 p.m. :

Yang Yang of Purdue University

#### Thermoacoustic tomography in bounded domains and with planar detectors

Thermoacoustic tomography is a hybrid medical imaging modality where optical waves and ultrasound waves are coupled. The propagation of ultrasound waves is typically modeled as an inverse source problem for the acoustic wave equation from pointwise boundary measurement. In this talk we discuss two variations of the traditional model: one is a model in a bounded domain while the other a model with planar boundary measurement. Results on uniqueness, stability and reconstruction will be given in each of the models. This is based on joint work with Plamen Stefanov.

### May 25^{th}, 2016: Damir Kinzebulatov of Indiana University

Special Time/Place: 4:10 - 5:00 p.m. in Surge 284

#### A new approach to the L^p-theory of -\Delta+b\grad, and its applications to Feller processes with general drifts

In this talk we will give a brief overview of stochastic gradient pursuit and the closely related Kaczmarz method for solving linear systems, or more generally convex optimization problems. We will present some new results which tie these methods together and prove the best known convergence rates for these methods under mild Lipschitz conditions. The methods empirically and theoretically rely on probability distributions to dictate the order of sampling in the algorithms. It turns out that the choice of distribution may drastically change the performance of the algorithm, and the theory has only begun to explain this phenomenon.

### May 25^{th}, 2016: Chiu-Yen Kao of Claremont McKenna College

#### Computational Methods for Extremal Steklov Problems

We develop a computational method for extremal Steklov eigenvalue problems and apply it to study the problem of maximizing the p-th Steklov eigenvalue as a function of the domain with a volume constraint. In contrast to the optimal domains for several other extremal Dirichlet- and Neumann-Laplacian eigenvalue problems, computational results suggest that the optimal domains for this problem are very structured. We reach the conjecture that the domain maximizing the p-th Steklov eigenvalue is unique (up to dilations and rigid transformations), has p-fold symmetry, and an axis of symmetry. The p-th Steklov eigenvalue has multiplicity 2 if p is even and multiplicity 3 if p>=3 is odd.

### May 18^{th}, 2016: Deanna Needell of Claremont McKenna College

#### SGD and Randomized Projections methods for linear systems

In this talk we will give a brief overview of stochastic gradient pursuit and the closely related Kaczmarz method for solving linear systems, or more generally convex optimization problems. We will present some new results which tie these methods together and prove the best known convergence rates for these methods under mild Lipschitz conditions. The methods empirically and theoretically rely on probability distributions to dictate the order of sampling in the algorithms. It turns out that the choice of distribution may drastically change the performance of the algorithm, and the theory has only begun to explain this phenomenon.

### May 11^{th}, 2016: Jing Li of UC Riverside

#### Random Walks and the Fractional Laplacian

### April 27^{th}, 2016: Gisele Goldstein of University of Memphis

#### The ubiquitous presence of dynamic boundary conditions in science

### April 20^{th}, 2016: Yulong Xing of UC Riverside

#### L2 stable discontinuous Galerkin methods for one-dimensional two-way wave equations

**I will present a recent work confirming the best constant of a Moser-Trudinger type inequality conjectured by A. Chang and P. Yang in 1987. The proof is based on a new and powerful lower bound of total mass for mean field equations. Other applications of the lower bound include the classification of certain Onsager vortices on the sphere, radial symmetry of solutions to Gaussian curvature equation on the plane, classification of solutions for mean field equations on tori and the sphere, etc. The resolution of several open problems in these areas will be presented. The work is jointly done with Changfeng Gui from University of Connecticut and University of Texas at San Antonio. **

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### April 13^{th}, 2016: Amir Moradifam of UC Riverside

#### Moser-Trudinger type inequalities, and symmetry in mean field equations and Onsager vortices

**I will present a recent work confirming the best constant of a Moser-Trudinger type inequality conjectured by A. Chang and P. Yang in 1987. The proof is based on a new and powerful lower bound of total mass for mean field equations. Other applications of the lower bound include the classification of certain Onsager vortices on the sphere, radial symmetry of solutions to Gaussian curvature equation on the plane, classification of solutions for mean field equations on tori and the sphere, etc. The resolution of several open problems in these areas will be presented. The work is jointly done with Changfeng Gui from University of Connecticut and University of Texas at San Antonio. **

### March 28^{th}, 2016: Juan Cheng from Institute of Applied Physics and Computational Mathematics, Beijing

#### High order positivity-preserving numerical methods for compressible multi-material fluid flow and radiative transfer problem

**Numerical simulation of compressible multi-material flows and radiative transfer problems arises in many applications, including astrophysics, inertial confine fusion, underwater bubble dynamics, shock wave interactions with material interface and combustion, optical molecular imaging, shielding, and so on. The positivity-preserving property is an important and challenging issue for the numerical solution of these problems. In this talk, we will introduce our recent work on high order positivity-preserving essentially non-oscillatory (ENO) Lagrangian schemes solving compressible Euler equations and high order positivity-preserving discontinuous Galerkin (DG) schemes solving radiative transfer equations. The properties of positivity-preserving and high order accuracy are proven rigorously. One- and two-dimensional numerical results are provided to verify the designed characteristics of the positivity-preserving schemes. **

**This is a joint work with Chi-Wang Shu and Daming Yuan. .**

### *2:10 - 3:00 p.m. Thursday, February 25^{th}, 2016: Yifeng Yu from UC Irvine

#### INVERSE PROBLEMS, NON-ROUNDNESS AND FLAT PIECES OF THE EFFECTIVE BURNING VELOCITY FROM AN INVISCID QUADRATIC HAMILTON-JACOBI MODEL

**I will talk about some finer properties of the effective burning velocity from a combustion model introduced by Majda and Souganidis in 90's. We proved that when the dimension is two and the flow of the ambient fluid is either weak or very strong, the level set of the effective burning velocity has flat pieces. Implications on the effective flame front and other related inverse type problems will also be discussed. This is a joint work with Wenjia Jing and Hung Tran." .**

### February 17^{th}, 2016: Abbas Momeni from Carleton University

#### A characterization for solutions of the Monge-Kantorovich mass transport problem

**The Monge Kantorovich mass-transportation problem has been shown to be fundamental for various basic problems in analysis and geometry. In this talk I describe some of the historical developments of this problem and some new results regarding the existence and uniqueness. I will present a measure theoretical approach to study the solutions of the Monge-Kantorovich optimal mass transport problem. I also talk about a criterion for the uniqueness.**

### February 3^{rd}, 2016: Long Chen of UC Irvine

#### An interface-fitted mesh generator and virtual element methods for elliptic interface problem

**In this work, we develop a simple interface-fitted mesh algorithm which can produce an interface-fitted mesh in two and three dimension quickly. Elements in such interface-fitted mesh are not restricted to simplex but can be polygon or polyhedron. We thus apply virtual element methods to solve the elliptic interface problem in two and three dimensions. We present some numerical results to illustrate the effectiveness of our method. This is a joint work with Huayi Wei and Min Wen. **

### January 27^{th}, 2016: Changyou Wang of Purdue University

#### On nematic liquid crystal flows in dimensions two and three

**In this talk, I will discuss a simplified Ericksen-Leslie system modeling the hydrodynamic flow of nematic liquid crystals, which is coupling between Navier-Stokes equations and transported heat flow of harmonic maps. I will describe some existence results of global solutions in dimension two and three. **

### January 20^{th}, 2016: Thomas Schellhous of UC Riverside

#### Vortex Patches

**Classical vortex patches are solutions to the 2D Euler Equations for fluid motion whose vorticities are constant inside of a bounded region and zero everywhere else. The manner in which the region's boundary can deform over time has been studied since the late 1970s, with current research ongoing in various generalized forms. This talk will discuss the interesting history of the vortex patch problem along with some of the insights that have driven past progress. It will conclude with a description of current work being done in this area.**

### January 13^{th}, 2016: Amir Moradifam of UC Riverside

#### Mathematics of Photo-Acoustic Tomography

**Photo-acoustic tomography (PAT) aims to leverage the photo-acoustic coupling between optical absorption of light sources and ultrasound (US) emission to obtain high contrast reconstructions of optical parameters with the high resolution of sonic waves. Quantitative PAT often involves a two-step procedure: first the map of sonic emission is reconstructed from ultrasound boundary measurements; and second optical properties of biological tissues are evaluated. I will review existing results and then describe a practical measurement setting in which such a separation does not apply. We shall assume that the optical source and an array of ultrasonic transducers are mounted on a rotating frame (in two or three dimensions) so that the light source rotates at the same time as the ultrasound measurements are acquired. This is a joint work with G. Bal (Columbia University). **

### November 25^{th}, 2015: Jim Kelliher of UC Riverside

#### Some comments on the aggregation equation with Newtonian potential

**I will describe some work in progress with Elaine Cozzi and Gung-Min Gie on a generalization of the aggregation equation with Newtonian potential, which is equivalent to a commonly studied limiting base of the Keller-Segel system modeling chemotaxis. In particular, I will discuss the inviscid limit.**

### November 18^{th}, 2015: Yongki Lee of UC Riverside

#### Thresholds for shock formation in the hyperbolic Keller-Segel model and traffic flow models

**We investigate a class of non-local conservation laws with the nonlinear advection coupling both local and non-local mechanisms, which arise in several applications such as tra?ffic flows and the collective motion of cells. We identify sub-thresholds for finite time shock formation in traffi?c flow models and the hyperbolic Keller-Segel model.**

### November 4^{th}, 2015: Alejandro Vèlez-Santiago of UC Riverside

#### Global regularity for Robin problems in a class of "bad" domains

**In this talk, I will discuss some recent results concerning the global regularity for solutions of elliptic Robin boundary value problems in a class of non-smooth domains, which include domains with fractal boundary, and also some rough domains.**

### October 28^{th}, 2015: Zuoqiang Shi of Tsinghua University

#### A numerical method for Poisson equation on point cloud Presenter

**Simulating wave propagation is one of the fundamental problems in scientific computing. In this talk, we consider one-dimensional two-way wave equations, and investigate a family of L2 stable high order discontinuous Galerkin methods, which is defined through a general form of numerical fluxes. For these L2 stable methods, we systematically establish stability, energy conservation, error estimates, superconvergence and dispersion analysis. Numerical examples are presented to illustrate the accuracy and the long-term behavior of the methods under consideration.**

### October 21^{st}, 2015: Lizheng Tao of UC Riverside

#### Recent results on wave breaking solutions

**In this talk, we will present two recent results on breaking solutions to system related to water wave problem. The first system has a dispersive operator combining Hilbert transform and fractal Laplacian. The second system has a dispersion close to the Whitham's projected operator in 1974.**

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### October 14^{th}, 2015: Fernando Lopez-Garcia of UC Riverside

#### A decomposition of functions and weighted Korn inequality on John domains

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**PDE & AM seminar lead this quarter by:**