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.

中文翻译：

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研究热点

《SIAM 评论》，第 65 卷，第 4 期，第 1029-1029 页，2023 年 11 月。
本期的两个研究热点重点介绍了针对具有挑战性的偏微分方程 (PDE) 系统获得更加现实的解决方案的技术。尽管来自应用数学的不同领域，这两篇论文都旨在利用先验信息来提高偏微分方程的保真度和实用解决方案。如果模型违反了已知的基础物理现象所满足的约束或数值稳定性要求所施加的约束，那么该模型的预测能力如何？从根本上讲，只要先验已知这些约束，人们就希望避免非线性不稳定性、非物理解和数值方法发散，但这种追求往往说起来容易做起来难。在本期的第一个研究热点“保界方案分析和设计的几何拟线性化框架”中，作者 Kailiang Wu 和 Chi-Wang Shu 扩展了可以对解施加边界约束的偏微分方程组的范围。例如，特殊相对论磁流体动力学方程的解具有以光速为上限的流体速度。这些约束方程以及作者所阐述的许多其他方程都是非线性的，因此执行起来具有挑战性。作者将此类问题提升到更高维度的空间，其优点是基于底层凸区域的几何特性，用更高维度的线性约束来表示原始非线性约束。作者阐明了这种提升何时会产生等效表示——几何拟线性化 (GQL)——并推导了在实践中构建基于 GQL 的边界保持方法的三种技术。所得框架的适用性基于非线性约束的形式，在本例中基于凸可行区域，但提供了满足更一般约束的潜在路径。第二个研究焦点涉及从数据估计未知的、空间变化的偏微分方程系统参数。作者 David Aristoff 和 Wolfgang Bangerth 特别感兴趣的是此类逆问题的贝叶斯公式，因为这些公式产生了未知参数的预测分布。获得这样的分布对于不确定性量化和其他下游用途非常有益，但随着未知参数维度的增长，贝叶斯反演很快在计算上变得不切实际。更困难的是验证所获得的分布。在“偏微分方程系数贝叶斯反演的基准”中，作者试图通过 64 维基准问题的全面规范来推进该领域和对现有技术的理解。作者提供了对潜在物理问题、数据生成过程、可能性和先验的完整描述，以及开源、用于定义问题的简单代码的多语言版本。作者还提供了对该问题进行全面数值检查的结果，包括来自后验分布的 30 个 CPU 年的样本，以及问题的低维和高维扩展。该基准对于想要测试新算法和采样方法有效性的研究人员应该有所帮助。