MiniSymposium-9: Recent Development on Numerical PDEs and Applications

Inverse problems seek to infer causal factors from observations. They arise naturally in a wide range of scientific fields. Computational methods for inverse problems are crucial and of unique flavor due to their ill-posedness. Recently, the enormous increase in computing power and the development of cutting-edge numerical methods have made it possible to simulate forward problems of growing complexity, which in turn leads to demand for novel computational methods to solve the resulting inverse problems. This mini-symposium will focus on new challenges that have appeared recently in inverse problems, including new applications in imaging sciences, life sciences, physical sciences and industry. It will bring together applied mathematicians and engineers to discuss the recent progress of computational methods in inverse problems. The mini-symposium is expected to promote the development of novel ideas and new research collaborations through knowledge dissemination and interdisciplinary discussion.

Organizers:

Hailiang Liu, Iowa State University

Songting Luo, Iowa State University

Saturday, October 2, 2021 at 2:40 – 4:00 pm (CST) (Session 1)

Weizhang Huang

University of Kansas

2:40 pm

A Well-balanced Positivity-preserving Quasi-Lagrange Moving Mesh DG Method for the Shallow Water Equations



In this talk we will present a high-order, well-balanced, positivity-preserving quasi-Lagrange moving mesh DG method for the numerical solution of the shallow water equations with non-flat bottom topography. The well-balance property is crucial to the ability of a scheme to simulate perturbation waves over the lake-at-rest steady state such as waves on a lake or tsunami waves in the deep ocean. The method combines a quasi-Lagrange moving mesh DG method, a hydrostatic reconstruction technique, and a change of unknown variables. We will discuss the strategies to use slope limiting, positivity-preservation limiting, and change of variables to ensure the well-balance and positivity-preserving properties. Compared to rezoning-type methods, the current method treats mesh movement continuously in time and has the advantages that it does not need to interpolate flow variables from the old mesh to the new one and places no constraint for the choice of an update scheme for the bottom topography on the new mesh. A selection of one- and two-dimensional examples are presented to demonstrate the well-balance property, positivity preservation, and high-order accuracy of the method and its ability to adapt the mesh according to features in the flow and bottom topography.

Yanzhi Zhang

Missouri University of Science and Technology

3:00 pm

Numerical Methods for the Tempered Fractional Laplacian and its Applications



The tempered fractional Laplacian has been widely used to study the coexistence and transition of anomalous to normal diffusion in many fields. However, the current mathematical and numerical studies of the tempered models still remain limited. In this talk, I will present an new finite difference method to discretize the tempered integral fractional Laplacian and apply it to study the tempered effects on the solution of problems arising in various applications. Compared to other existing methods, our method has higher accuracy and simpler implementation. Error analysis will be discussed along with numerical experiments. Since our method yields a (multilevel) Toeplitz stiffness matrix, one can design fast algorithms via the fast Fourier transform for efficient simulations. Finally, we apply it together with fast algorithms to study the tempered effects on the solutions of various tempered fractional PDEs, including the Allen–Cahn equation and Gray–Scott equations. 

Yi Jiang

Southern Illinois University-Edwardsville

3:20 pm

Numerical Study of Non-uniqueness for 2D Compressible Isentropic Euler Equations



In this talk, I will present our recent numerical study on a class of solutions with spiraling singularities in vorticity for two-dimensional, inviscid, compressible Euler systems, where the designed initial data has an algebraic singularity in vorticity at the origin. These are different from the multi-dimensional Riemann problems widely studied in the literature. Our computations provide numerical evidence of the existence of initial value problems with multiple solutions, thus revealing a fundamental obstruction toward the well-posedness of the governing equations. The compressible Euler equations are solved using the positivity-preserving discontinuous Galerkin method.

Manas Bhatnagar

Iowa State University

3:40 pm

Critical Thresholds in 1D Pressureless Euler Poisson Alignment Systems with Varying Background



The Euler Poisson equations describe important physical phenomena in many applications such as semiconductor modeling and plasma physics. In this talk, we will first give introduction to critical threshold phenomena: what it is and from where it was motivated. We will advance our understanding of critical threshold phenomena in Euler-Poisson systems in the presence of different forces. We will identify critical thresholds in damped Euler Poisson systems, with spatially varying background state and attractive forcing. We will give respective bounds for subcritical and supercritical regions in the space of initial configuration, thereby proving the existence of a critical threshold for each scenario. We will then also look into the more complicated and interesting repulsive forcing case, where usual comparison techniques do not work. Key tools include comparison with auxiliary systems, phase space analysis of the transformed system.

Sunday, October 3, 2021 at 11:00 am – 12:20 pm (CST) (Session 2)

Wei Zhu

University of Alabama

11:00 am

A Novel Second-order Image Restoration Model



In this talk, I will discuss a novel variational image restoration model whose regularizer is piecewisely defined according to the magnitude of image gradient. Augmented Lagrangian method is utilized to minimize the proposed functional and convergence analysis is also provided. Numerical experiments are presented to demonstrate the features of the new model.

Weihua Geng

Southern Methodist University

11:20 am

A Regularized Matched Interface and Boundary (rMIB) Method and its Biological Application in Computing Protein pKa Values



The pKa values are important quantities characterizing the ability of protein active sites to give up protons. In addition to NMR measurement, pKa can be calculated numerically by electrostatic free energy changes subject to the protonation and deprotonation of titration sites. To this end, the Poisson-Boltzmann (PB) model is an effective approach for the electrostatics. However, numerically solving PB equation is challenging due to the jump conditions across the dielectric interfaces, irregular geometries of the molecular surface, and charge singularities. Our recently developed matched interface and boundary (MIB) method treats these challenges rigorously, resulting in a solid second order MIBPB solver. Since the MIBPB solver uses Green's function based regularization of charge singularities by decomposing the solution into a singular component and a regularized component, it is particularly efficient in treating the accuracy-sensitive, numerous, and complicated charges distribution from the pKa calculation. Our numerical results demonstrate that accurate electrostatics potentials, forces, energies, and pKa values are achieved at coarse grid rapidly. In addition, the resulting software, which pipelines the entire pKa calculation procedure, is available to all potential users from the greater bioscience community.

Yuan Liu

Wichita State University

11:40 am

Sparse Grid Discontinuous Galerkin Methods for Time Dependent PDEs



In this talk, we will introduce a class of adaptive multiresolution discontinuous Galerkin methods for several time dependent PDEs including reaction-diffusion equations, wave equations and Schrodinger equations. The main ingredients of the sparse grid discontinuous Galerkin methods include L2 orthonormal Alpert’s multiwavelets and the interpolatory multiwavelets. By exploring the inherent mesh hierarchy and the nested polynomial approximation spaces, multiresolution analysis is able to accelerate the computation, and adjust the computational grid adaptively. Sparse tensor product is further introduced to decrease the computational cost in multi-dimensional space.

Xuping Tian

Iowa State University

12:00 pm

AEGD: Adaptive Gradient Descent with Energy



We propose AEGD, a new algorithm for first-order gradient-based optimization of non-convex objective functions, based on a dynamically updated ‘energy’ variable. The method is shown to be unconditionally energy stable, irrespective of the step size. We prove energy-dependent convergence rates of AEGD for both non-convex and convex objectives, which for a suitably small step size recovers desired convergence rates for the batch gradient descent. We also provide an energy-dependent bound on the stationary convergence of AEGD in the stochastic non-convex setting. The method is straightforward to implement and requires little tuning of hyper-parameters. Experimental results demonstrate that AEGD works well for a large variety of optimization problems: it is robust with respect to initial data, capable of making rapid initial progress. The stochastic AEGD shows comparable and often better generalization performance than SGD with momentum for deep neural networks.