带高维随机参数的偏微分方程的新型模型化简算法

Numerical methods for partial differential equations (PDEs) with random parameters are crucial in the area of uncertainty quantification. Since conducting uncertainty quantification for practical problems can involve high-dimensional parameters, solving PDEs with high-dimensional parameters becomes an important topic. High-dimensional random parameters typically cause two main difficulties in this area: first, traditional stochastic spectral methods normally meet bottlenecks for solving parametrically high-dimensional problems; second, high-dimensional parameters typically also lead to large rank structures in spatial approximations, which cause difficulties in applying spatial model reduction techniques. To make progress in solving these challenging (parametrically) high-dimensional and (spatially) large rank problems, we in this project develop new efficient model reduction techniques, which are based on a combination of parameter space decomposition and spatial model reduction methods. First, while high-dimensional parameter spaces can be decomposed through ANOVA (Analysis of Variance) methods, we will investigate the corresponding decomposition in spatial approximations and develop novel efficient spatial model reduction methods. Moreover, since domain decomposition methods for PDEs can also decompose the large rank structures in spatial approximations, we will design new reduced order coupling models for the domain decomposition methods, to achieve overall efficient model reduction algorithms.

求解带随机参数的偏微分方程是不确定性量化领域中的核心问题之一。由于高维参数常出现在实际问题中，为带高维随机参数的偏微分方程设计高效的数值解法已成为当前的一个重要研究问题。高维随机参数在此研究方向上会带来两个方面的主要挑战：首先，传统的随机谱方法很难处理高维参数问题；其次，高维参数也会带来物理空间的大秩结构，从而导致物理空间模型化简的困难。本项目将结合参数空间拆分与物理空间模型化简方法，设计新的高效模型化简算法以实现在处理这种“高维大秩”问题上的进步。首先，高维参数空间可由方差分析方法进行拆分，我们将研究其相应的物理空间的大秩结构的拆分，并设计新的高效物理空间模型化简算法。此外，偏微分方程（物理）区域分解方法可拆分物理空间的大秩结构，我们也将为区域分解方法的耦合模型设计新的高效模型化简方法，以实现最终求解的高效算法。

本项目针对不确定性量化领域核心问题之一的带随机参数的偏微分方程，主要研究了其高效模型化简方法，取得良好的研究效果。对于带高维随机参数的偏微分方程，我们研究了参数空间拆分以及物理空间的简化基方法。针对贝叶斯推断中涉及的偏微分方程模型，我们设计了新型基于后验分布的方差分析简化基方法以推断高维参数。同时，由于高斯过程回归模型的计算效率通常会随着参数空间维度的增加而下降，我们开发了基于方差分析展开的高斯过程以构造带高维参数的偏微分方程的替代模型。由于复杂偏微分方程的离散解空间通常维度很高且秩很大，直接模型化简有时会失效。为此，我们研究了区域分解对解空间的影响,并设计了深度区域分解方法。