Computational and Applied Mathematics Seminar
Spring 2022
Mondays at 4:10-5:00p.m, or 1:10-2:00pm (ZOOM or in-person (Room to be announced) talks)
The CAM Seminar is organized in the ISU Mathematics Department. It brings speakers from inside and outside of ISU, raising issues and exchanging ideas on topics of current interest in the are of computational and applied mathematics.
- April 25 (1:10-2:00pm, Carver 401, in-person)
Title: Fractional operators from the holomorphic functional calculus
Evan Camrud, Iowa State University
Abstract: We will introduce the holomorphic functional calculus and use it to construct fractional powers of (closed linear) operators.
ZOOM Link:
- January 31, Room 401
Title: Anderson accelerated A-H (Arrow-Hurwicz) Method for the Steady Incompressible Navier-Stokes Equations
Ahmed Zytoon, Iowa State University
Abstract:
The Arrow-Hurwicz algorithm is an iterative method for approximating the solution of saddle-point problems, which arise in many applications like CFD. Viewing the Arrow-Hurwicz algorithm as a fixed-point iteration allows the use of Anderson acceleration to improve and accelerate the convergence. We provide the analysis needed to show that the Arrow-Hurwicz algorithm satisfies the hypothesis of the Anderson acceleration. Also, We provide some numerical examples to compare the performance of the Arrow-Hurwicz algorithm with and without Anderson acceleration on steady state Navier-Stokes equations. The results indicate that the Anderson accelerated Arrow-Hurwicz algorithm converges significantly faster than the algorithm without acceleration with the right choice of parameters. We show the effect of adding a grad-div term to the Navier-Stokes equations on the convergence rates as well.
ZOOM Link:
- February 07, (In-Person) Carver 401, 1:10-2pm
Title: Discussion of the Scope of CAM Seminar
ZOOM Link:
- February 14
Title:
University
Abstract:
ZOOM Link:
- February 21, 4:10-5pm
Title: Parametric Fokker-Planck equation and its Wasserstein error estimates
Haomin Zhou, Georgia Institute of Technology
Abstract: In this presentation, I will introduce a system of ODEs that are constructed by using generative neural networks to spatially approximate the Fokker-Planck equation. We call this system Parametric Fokker-Planck equation. We design a semi-implicit time discretized scheme to compute its solution by only using random samples in space. The resulting algorithm allows us to approximate the solution of Fokker-Planck equation in high dimensions. Furthermore, we provide error bounds, in terms of Wasserstein metric, for the semi-discrete and fully discrete approximations. This presentation is based on a recent joint work with Wuchen Li (South Carolina), Shu Liu (Math GT) and Hongyuan Zha (CUHK-Shenzhen).
ZOOM Link:
- February 28, 1:10-2:00pm (Cancelled, to be rescheduled)
Title: Efficient natural gradient method for large-scale optimization problems
Levon Nurbekyan, UCLA
Abstract: We propose an efficient numerical method for computing natural gradient descent directions with respect to a generic metric in the state space. Our technique relies on representing the natural gradient direction as a solution to a standard least-squares problem. Hence, instead of calculating, storing, or inverting the information matrix directly, we apply efficient methods from numerical linear algebra to solve this least-squares problem. We treat both scenarios where the derivative of the state variable with respect to the parameter is either explicitly known or implicitly given through constraints. We apply the QR decomposition to solve the least-squares problem in the former case and utilize the adjoint-state method to compute the natural gradient descent direction in the latter case.
ZOOM Link:
- March 07
Title: Prediction of Molecular Binding/Unbidning with Impilcit Solvent: Geometrical Flow, Transition Paths, and Brownian Dynamics
Bo Li, University of California, San Diego
Abstract: Ligand-receptor binding and unbinding are fundamental molecular processes, and are particularly essential to drug efficacy, whereas water fluctuations impact the corresponding thermodynamics and kinetics. We develop a variational implicit-solvent model (VISM), a geometrical flow model, to calculate the potential of mean force (PMF) as well as the solute-solvent interfacial structures of dry and wet states for a model ligand-pocket system. We also combine our VISM with the string method for transition paths to obtain the dry-wet transition rates, and conduct two-state Brownian dynamics simulations of the ligand stochastic motion, providing the mean first-passage times for the ligand-pocket binding and unbinding. We find that the dewetting transition around the pocket is slowed down as the ligand approaches the pocket but is peaked suddenly once the ligand enters the pocket. In contrast to binding, the ligand unbinding involves a much larger timescale due to a high energy barrier at the pocket entrance. The dry-wet fluctuation slows down the binding but accelerates the unbinding process. Without any explicit description of individual water molecules, our predictions are in a very good, qualitative and semi-quantitative, agreement with existing explicit-water molecular dynamics simulations, providing a promising step in further efficient studies of the ligand-receptor binding/unbinding kinetics. This is joint work with Shenggao Zhou, R. Gregor Weiss, Li-Tien Cheng, Joachim Dzubiella, and J. Andrew McCammon.
ZOOM Link:
- March 21
Title: DeepParticle: deep-learning invariant measure by minimizing Wasserstein distance on data generated from an interacting particle method
Jack Xin, University of California, Irvine
Abstract: High dimensional partial differential equations (PDE) are challenging to compute by traditional mesh based methods especially when their solutions have large gradients or concentrations at unknown locations. Mesh free methods are more appealing, however they remain slow and expensive when a long time and resolved computation is necessary. We present DeepParticle, an integrated deep learning (DL), optimal transport (OT), and interacting particle (IP) approach through a case study of Fisher-Kolmogorov-Petrovsky-Piskunov front speeds in incompressible flows. PDE analysis reduces the problem to a computation of principal eigenvalue of an advection-diffusion operator. Stochastic representation via Feynman-Kac formula makes possible a genetic interacting particle algorithm that evolves particle distribution to a large time invariant measure from which the front speed is extracted. The invariant measure is parameterized by a physical parameter (the Peclet number). We learn this family of invariant measures by training a physically parameterized deep neural network on affordable data from IP computation at moderate Peclet numbers, then predict at a larger Peclet number when IP computation is expensive. The network is trained by minimizing a discrete Wasserstein distance from OT theory. The DL prediction serves as a warm start to accelerate IP computation especially for a 3-dimensional time dependent Kolmogorov flow with chaotic streamlines. Our methodology extends to a more general context of deep-learning stochastic particle dynamics. This is joint work with Zhongjian Wang (University of Chicago) and Zhiwen Zhang (University of Hong Kong).
ZOOM Link:
- March 28
Title:
University
Abstract:
ZOOM Link:
- April 04 (401 Carver, 1:10-2:00pm)
Title: Introduction to Singular Integral Operators
Paul Sacks, Iowa State University
Abstract: This talk will review the basic notions and tools used in the study of integral operators and integral equations, with special emphasis on singular integral operators of the type arising in connection with fractional differential equations
ZOOM Link:
- April 11 (1:10-2:00pm, Carver 401, in-person)
Title: Introduction to the fractional Laplacian
Pablo Raul Stinga, Iowa State University
Abstract: We introduce the fractional Laplacian and review some of its properties, including the nonlocal Dirichlet and Neumann problems, and some basic regularity estimates.
ZOOM Link:
- April 18
Title:
University
Abstract:
ZOOM Link: