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Links : Interdisciplinary Center for Quantitative Modeling in Biology (ICQMB) seminar

UCR Math Department seminar

Winter 2022 Schedule

Jan 5 10:00 (Wed) Organizational meeting

Jan 12 10:00 (Wed) Konstantinos Mamis, North Carolina State University

Jan 19 10:00 (Wed) Sam Punshon-Smith, Institute for Advanced Study

Jan 26 10:00 (Wed) Kyongmin Yeo, IBM T.J. Watson Research Center

Feb 2 10:00 (Wed) Yoonsang Lee, Dartmouth College

Feb 9 10:00 (Wed) He Yang, Augusta University

Feb 16 10:00 (Wed) Ruchi Guo, UC Irvine

Feb 23 10:00 (Wed) Zheng Sun, University of Alabama

Mar 2 10:00 (Wed) Hyung Ju Hwang, Pohang University of Science and Technology

Mar 9 10:00 (Wed) Changho Kim, UC Merced

Talk Titles & Abstracts:

Jan 12, 2022, 10:00 - 10:50 AM PT

Dr. Konstantinos Mamis, North Carolina State University

Title In search of a Fokker-Planck description for dynamical systems driven by correlated noises

Abstract The inclusion of stochastic processes (commonly called noises in engineering) as excitations in dynamical systems, is an essential part for the realistic mathematical modeling required in physics and engineering, as well as in demanding biological applications like the accurate prediction of pandemics. My talk will focus on formulating evolution equations for the probability density functions (PDFs) of the response to nonlinear stochastic dynamical systems. In the well-studied case where stochastic excitation is delta-correlated, Gaussian white noise, the response PDF evolution equation is the exact, classical Fokker-Planck equation. However, most sources of randomness in the environment are smoothly-correlated, and thus cannot be realistically modeled as white noise, see e.g. excitations from sea waves, wind loads or earthquakes. In these cases, the convenient Fokker-Planck description no longer holds. The aim of our work is to remedy this, by deriving approximate, Fokker-Planck-like equations, for the case of correlated excitations. The new PDF evolution equations that we derive fall into the category of nonlinear Fokker-Planck equations, and their solutions are in very good agreement with results from direct simulations of the corresponding stochastic differential equations, even for random excitations with high noise intensities and large correlation times. This constitutes our main contribution to the field, since the PDF evolution equations in the existing literature are valid only near the white noise limit.

Jan 19, 2022, 10:00 - 10:50 AM PT

Dr. Sam Punshon-Smith, Institute for Advanced Study

Title: Positive Lyapunov exponent for the Galerkin-Navier-Stokes equations with stochastic forcing

Abstract: In this talk I will discuss a recently introduced method (joint with Jacob Bedrossian and Alex Blumenthal) for obtaining strictly positive lower bounds on the top Lyapunov exponent of high-dimensional, stochastic differential equations such as the weakly-damped Lorenz-96 (L96) model or Galerkin truncations of the 2d Navier-Stokes equations (GNSE). This hallmark of chaos has long been observed in these models, however, no mathematical proof had previously been made for any type of deterministic or stochastic forcing. The method we proposed combines (A) a new identity connecting the Lyapunov exponents to a Fisher information of the stationary measure of the Markov process tracking tangent directions (the so-called "projective process"); and (B) an L1-based hypoelliptic regularity estimate to show that this (degenerate) Fisher information is an upper bound on some fractional regularity. For L96 and GNSE, we then further reduce the lower bound of the top Lyapunov exponent to proving that the projective process satisfies Hörmander's condition. I will also discuss the recent work with Jacob Bedrossian on verifying this condition for the 2d Galerkin-Navier-Stokes equations in a rectangular, periodic box of any aspect ratio using some special structure of matrix Lie algebras associated to the Euler equations and ideas from computational algebraic geometry. The methods here are broadly applicable to other high dimensional truncations of chaotic stochastic partial differential equations.

Feb, 02, 2022, 10:00 - 10:50 AM PT

Dr. Yoonsang Lee, Dartmouth College

Title: Hierarchical Learning to Solve Partial Differential Equations Using Physics-Informed Neural Networks

Abstract: The Neural network-based approach to solving partial differential equations has attracted considerable attention due to its simplicity and flexibility to represent the solution of the partial differential equation. In training a neural network, the network tends to learn global features corresponding to low-frequency components while high-frequency components are approximated at a much slower rate (F-principle). For a class of equations in which the solution contains a wide range of scales, the network training process can suffer from slow convergence and low accuracy due to its inability to capture the high-frequency components. In this work, we propose a hierarchical approach to improve the convergence rate and accuracy of the neural network solution to partial differential equations. The proposed method comprises multi-training levels in which a newly introduced neural network is guided to learn the residual of the previous level approximation. By the nature of neural networks' training process, the high-level correction is inclined to capture the high-frequency components. We validate the efficiency and robustness of the proposed hierarchical approach through a suite of linear and nonlinear partial differential equations.

Feb 9, 2022, 10:00 - 10:50 AM PT

Dr. He Yang, Augusta University

Title: Discontinuous Galerkin Methods for Relativistic Vlasov-Maxwell System and Khokhlov-Zabolotskaya-Kuznetzov Equation

Abstract: In this talk, I will introduce our proposed discontinuous Galerkin (DG) methods to solve Relativistic Vlasov-Maxwell (RVM) system and Khokhlov-Zabolotskaya-Kuznetzov (KZK) equation. The RVM system is a kinetic model that describes the dynamics of plasma when the charged particles move in the relativistic regime and their collisions are not important. In the first part of my talk, I will introduce our proposed methods which preserve the structural properties of the RVM system, i.e., positivity of the particle number density function, mass and energy conservation. I will introduce some theoretical results and implementation issues of our methods. Numerical experiments, including streaming Weibel instability and wakefield acceleration, will also be presented to demonstrate the performance of our methods.

KZK equation is a model that describes the propagation of the ultrasound beams in the thermoviscous fluid. Accurate numerical methods to simulate the KZK equation are important to its broad applications in medical ultrasound simulations. In the second part of my talk, I will discuss our proposed local discontinuous Galerkin method to solve the KZK equation. I will show numerical stability and a series of numerical experiments including the focused circular short tone burst excitation and the propagation of unfocused sound beams, which show that our scheme leads to accurate solutions and outperforms the benchmark solutions in the literature.

Feb 16, 2022, 10:00 - 10:50 AM PT

Dr. Ruchi Guo, UC Irvine

Title: A Deep Direct Sampling Method for Electrical Impedance and Diffuse Optical Tomography

Abstract: Electrical impedance tomography (EIT) and Diffuse Optical Tomography (DOT) are promising techniques for non-invasive and radiation-free type of medical imaging. They all can be considered as inverse boundary value problems to identify PDE coefficients. But a high-quality reconstruction is always challenging due to its severe ill-posedness. Based on the idea of direct sampling methods (DSMs), we present a framework to construct deep neural networks for solving these two problems. It is able to capture the underlying mathematical structure from background projection of boundary measurement to coefficient distribution. The resulting Deep DSM (DDSM) is easy for implementation and its offline-online decomposition inherits efficiency from the original DSM that does not need any optimization process in reconstruction. Additionally, it is capable of systematically incorporating multiple Cauchy data pairs to achieve high-quality reconstruction and is also highly robust to large noise.

Feb 23, 2022, 10:00 - 10:50 AM PT

Dr. Zheng Sun, University of Alabama

Title Discontinuous Galerkin methods for Wasserstein type gradient flows

Abstract The gradient flow structure, describing the evolution of a system driven by a decaying energy functional, commonly appears in models for interacting gasses, chemotactic migrations, and population dynamics. The unknowns for these problems usually represent densities or concentrations and are expected to be positive (nonnegative). For numerical simulation of Wasserstein type gradient flows, failing to preserve positivity may lead to instability associated with spurious energy growth or undefined quantities in the equation, which can cause the blow up of the algorithm. In this talk, we present high-order discontinuous Galerkin methods for scalar Wasserstein gradient flows and cross-diffusion gradient flow systems. The proposed semidiscrete schemes admit discrete entropy inequalities that mimic the behavior of continuum equations. Positivity can be preserved at the fully discrete level by using scaling limiters. Numerical tests are performed to confirm the high-order accuracy for smooth problems and to demonstrate the effectiveness for preserving long time asymptotics.

Mar 2, 2022, 10:00 - 10:50 AM PT

Dr. Hyung Ju Hwang, Pohang University of Science and Technology

Title: Neural network approach in mathematical biology

Abstract: We discuss a neural network approach to two problems which may arise in mathematical biology. First, we look into the chemotaxis model and the neural net can be used to find approximate solutions of the PDE. Also, the Neural Net approximation can be easily applied to the inverse problem. It was confirmed that even when the coefficients of the PDE equation were unknown, a prediction with high accuracy was achieved. Next, we focus on how to approximate traveling wave solutions for various kinds of partial differential equations via neural networks. We propose a novel method to approximate both the traveling wave solution and the unknown wave speed via a neural network and an additional free parameter. Lastly, we talk on real-world implications of a rapidly-responsive COVID-19 spread model via deep learning. The methodology could also be employed for a short-term prediction of COVID-19, which could help the government prepare for a new outbreak.