Iowa State University

Mathematics Department Seminar



Computational and Applied Mathematics Seminar

Fall 2020

Mondays at 4:10 p.m. in 401 Carver

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.

Archive of previous CAM seminars



Spatial-temporal functional imputation for satellite data and application to National Resource
Inventory (NRI) survey

Zhengyuan Zhu
ISU, Department of Statistics

ABSTRACT:  Remote sensing data from satellite are used in a variety of disciplines. In many applications a seamless dataset is needed. However, most satellite data have large amount of missing data due to a number of factors such as cloud cover, other abnormal atmospheric conditions, and sensor specific problems. In this talk we introduce a general spatiotemporal satellite image imputation method based on sparse functional data analysis techniques. The latent spatiotemporal process is imputed from observations consisting of a few longitudinally repeated satellite images, which are themselves contaminated with noise and partially observed due to cloud coverage and other reasons. Under this new observation model we provide theoretical justifications for the proposed imputation approach. Practical analyses on Landsat data were conducted to illustrate and validate our algorithm. A comparison with existing gap-filling methods shows that our proposed algorithm significantly outperforms the other methods in terms of having smaller prediction errors. The proposed algorithm is used in an NRI application to map surface water using LandSat data. 

Past Talks Spring 2020

01/15 Carve 401

Energy Stable and Positivity Preserving Scheme for the Quantum Diffusion Equation

Xiaokai Huo

CEMSE Division
King Abdullah University of Science and Technology

ABSTRACT:  The quantum diffusion equation is a fourth order parabolic equation. The lack of maximum principle for this equation brings difficulties in solving it numerically while preserve the positivity of solutions. In this talk, we develop a new numerical scheme for the quantum diffusion equation in general dimensions and prove it to be energy stable and positivity-preserving. The difficulty in proving the positivity-preserving property is dealt by reformulating the scheme into an equivalent optimization problem and prove the solutions to the optimization problem cannot vanish, which is because the energy functional develops singularities at zero. We will also give some numerical examples in one and two dimensions to verify the energy stable and positivity-preserving properties.


Efficient, positive, and energy stable schemes for multi-dimensional Poisson-Nernst-Planck systems

Wumaier Maimaitiyiming
Iowa State, Math

ABSTRACT:  In this talk, we present positive and energy-dissipating schemes for solving the time-dependent multi-dimensional system of Poisson-Nernst-Planck (PNP) equations. Such equations arise in the modeling of biological membrane channels and semiconductor devices. The PNP system is a strongly coupled system of nonlinear equations, also, as a gradient flow can take long time evolution to reach steady states. Hence, designing efficient and stable methods with comprehensive numerical analysis for the PNP system is highly desirable. We first reformulate the system by using Slotboom variables, such reformulation converts the drift-diffusion operator into a self-adjoint elliptic operator. The new form can be more efficiently solved and suitable for keeping the solution positivity. Our numerical schemes are based on the new formulation. The semi-implicit time discretization results in a well-posed elliptic system, which is shown to be energy dissipating and preserves solution positivity for arbitrary time steps. Our first order (in time) fully-discrete scheme preserves solution positivity and mass conservation unconditionally, and energy dissipation with only a mild O(1) time step restriction. The scheme also preserves the steady-state. We further introduce a second-order (in both time and space) scheme, which has the same computational complexity as the first-order scheme. For such a second-order scheme, we use an accuracy preserving local scaling limiter to restore solution positivity when necessary. A sequence of three-dimensional numerical tests is carried out to verify our theoretical findings.



Variational Methods for Wasserstein Gradient Flows

Li Wang
University of Minnesota

ABSTRACT:  We develop variational methods for nonlinear equations with a gradient
flow structure. Such equations arise in applications of a wide range,
such as porous median flows, material science, animal swarms, and
chemotaxis. Our method builds on the JKO framework, which evolves the
equation as a gradient flow with respect to the Wasserstein metric. As
a result, our method has built-in positivity preserving, entropy
decreasing properties, and overcomes stability issue due to the strong
nonlinearity and degeneracy. We further modify the variational
formulation by adding a Fisher information regularization so that
second order information can be used to accelerate the convergence.


Communication-Efficient Network-Distributed Optimization with Differential-Coded Compressors

Jia (Kevin) Liu
Dept. of Computer Science
Iowa State University

ABSTRACT:  Network-distributed optimization has attracted significant attention in recent years due to its ever-increasing applications. However, the classic decentralized gradient descent (DGD) algorithm is communication-inefficient for large-scale and high-dimensional network-distributed optimization problems. To address this challenge, many compressed DGD-based algorithms have been proposed. However, most of the existing works have high complexity and assume compressors with bounded noise power. To overcome these limitations, in this paper, we propose a new differential-coded compressed DGD (DC-DGD) algorithm. The key features of DC-DGD include: i) DC-DGD works with general SNR-constrained compressors, relaxing the bounded noise power assumption; ii) The differential-coded design entails the same convergence rate as the original DGD algorithm; and iii) DC-DGD has the same low-complexity structure as the original DGD due to a self-noise-reduction effect. Moreover, the above features inspire us to develop a hybrid compression scheme that offers a systematic mechanism to minimize the communication cost. Finally, we conduct extensive experiments to verify the efficacy of the proposed DC-DGD and hybrid compressor.



SQG on Bounded Domains

Logan Stokols
UT Austin

ABSTRACT: The surface quasi-geostrophic (SQG) equation on R^2 was shown in the late ’00s to be well posed with smooth solutions. Recently, Constantin and Ignatova proposed a model for SQG on bounded open subsets of R^2, defined in terms of the Dirichlet Laplacian. This model is particularly complex because it involves a nonlocal operator on a bounded domain. We will discuss this model, including physical motivation, existence, and regularity.