Research Spotlights

Abstract

SIAM Review, Volume 65, Issue 4, Page 1029-1029, November 2023.
This issue's two Research Spotlights highlight techniques for obtaining ever more realistic solutions to challenging systems of partial differential equations (PDEs). Although borne from different fields of applied mathematics, both papers aim to leverage prior information to improve the fidelity and practical solution of PDEs. How predictive is a model if it violates constraints known to be satisfied by the underlying physical phenomena or otherwise imposed by numerical stability requirements? Fundamentally, one desires to avoid nonlinear instabilities, nonphysical solutions, and numerical method divergence whenever these constraints are known a priori, but this pursuit is often easier said than done. In this issue's first Research Spotlight, “Geometric Quasilinearization Framework for Analysis and Design of Bound-Preserving Schemes,” authors Kailiang Wu and Chi-Wang Shu extend the range of systems of PDEs for which bound constraints can be imposed on solutions. For example, solutions of the special relativistic magnetohydrodynamic equations have fluid velocities upper bounded by the speed of light. Such constraint equations, and many others illustrated by the authors, are nonlinear and hence challenging to enforce. The authors lift such problems into a higher-dimensional space with the benefit of representing the original nonlinear constraints with higher-dimensional linear constraints based on the geometric properties of the underlying convex regions. The authors illuminate when such lifting results in an equivalent representation---a geometric quasilinearization (GQL)---and derive three techniques for constructing GQL-based bound-preserving methods in practice. The applicability of the resulting framework is based on the form of the nonlinear constraint, in this case based on convex feasible regions, but provides a potential path forward for satisfying even more general constraints. The second Research Spotlight addresses the estimation of unknown, spatially varying PDE system parameters from data. Of particular interest to authors David Aristoff and Wolfgang Bangerth are Bayesian formulations for such inverse problems since these formulations yield predictive distributions on the unknown parameters. Obtaining such a distribution can be highly beneficial for uncertainty quantification and other downstream uses, but Bayesian inversion quickly becomes computationally impractical as the dimension of the unknown parameters grows. More difficult still is validating the obtained distributions. In “A Benchmark for the Bayesian Inversion of Coefficients in Partial Differential Equations” the authors seek to advance the field and understanding of the state of the art through a comprehensive specification of a 64-dimensional benchmark problem. The authors provide a complete description of the underlying physical problem, data-generating process, likelihood, and prior, as well as open-source, multilanguage versions of the simple code to define the problem. The authors also provide the results of a comprehensive numerical examination of the problem, including 30 CPU-years worth of samples from the posterior distribution, and lower- and higher-dimensional extensions of the problem. The benchmark should be helpful for researchers wanting to test the efficacy of new algorithms and sampling approaches.