多尺度偏微分方程的数值均匀化及相关方法

Multiscale problems are ubiquitous in many research areas such as geophysics, materials science, etc. Multiscale analysis and simulation is one of the fundamental topics of scientific computation. Due to the complex nonlinear interplay of different spatial and temporal scales, the resolution of all characteristic scales is prohibitively expensive and the cost of direct simulation often far exceeds the capabilities of available computing resources. The goal of multiscale methods such as numerical homogenization is to achieve optimal balance between computational cost and accuracy with reduced degrees of freedom. Deep connections have been identified between numerical homogenization and direct methods such as multiresolution decomposition and multilevel methods. The objective of the proposed work is to study some fundamental mathematical questions and algorithmic challenges for multiscale numerical methods, thence to provide solid theoretical justification and advanced techniques for the efficient solution of some important multiscale application problems. We will establish and enrich the theoretical framework of multiscale numerical methods for: the robustness and convergence of those multiscale numerical methods with respect to coefficient oscillation and contrast ratio, the adaptive construction of multiscale numerical methods, and the numerical homogenization models and related multilevel Monte Carlo methods for multiscale stochastic PDEs. Based on those theoretical developments, we will apply those rigorously justified multiscale numerical methods to important application problems such as multiscale optimal control and multiscale wave propagation.

多尺度问题广泛存在于地球物理，材料科学等领域。多尺度分析和模拟是科学计算的基础课题之一。空间和时间尺度复杂的非线性耦合使得直接模拟的计算量往往会远超可用的计算资源。以数值均匀化为代表的多尺度方法的目标是在降低自由度的基础上取得计算量和精度的最佳平衡。数值均匀化与多分辨率分解／多水平方法之间存在着深刻的内在联系。本项目研究数值均匀化及相关方法的若干核心数学问题和算法挑战，从而为一些重要多尺度应用问题的高效求解提供坚实的理论支撑和技术指导。我们将深入研究数值均匀化和相关多分辨率分解/多水平方法对系数振荡和系数比例的健壮性和收敛性，进而发展自适应的多尺度数值方法，探索多尺度随机偏微分方程的数值均匀化模型和多水平蒙特卡洛方法，从而构建和丰富多尺度数值方法的理论体系。在此基础上，将具有理论支撑的多尺度数值方法等应用到多尺度最优控制和多尺度波动方程等重要的实际问题。

本项目着重研究数值均匀化方法及相关多分辨率分解/多水平方法等。以数值均匀化为代表的多尺度方法的目标是在降低自由度的基础上取得计算量和精度的最佳平衡。数值均匀化与多分辨率分解/多水平方法之间存在着深刻的内在联系。本项目研究数值均匀化及相关方法的若干核心数学问题和算法挑战，从而为一些重要多尺度应用问题的高效求解提供坚实的理论支撑和技术指导。在本项目的资助下，我们结合算子自适应小波和多水平子空间校正构造了多尺度特征值问题的快速算法，发展了非线性多尺度偏微分方程的迭代数值均匀化方法，研究了确定性和随机双曲型偏微分方程的多水平蒙特卡洛方法，设计了多尺度神经网络来高效求解多尺度偏微分方程，还应用于多尺度椭圆方程约束的最优控制问题、多尺度微磁模型等。本项目的研究将为设计和应用多尺度方法，以及深入研究多尺度问题，打下坚实的理论基础。