两类随机哈密尔顿偏微分方程的保结构算法研究

In this project, we focus on the investigation of the structure-preserving numerical methods of two typical stochastic Hamiltonian partial differential equations---stochastic Schrödinger equation, stochastic Maxwell equations and their coupled system, which are widely used in condensed matter physics, quantum physics, molecular dynamics, electromagnetics, nano materials and many other fields of science and technology, in order to provide theoretical basis for large-scale computation in practical application. The main contents include: for the stochastic Schrödinger equation, stochastic Maxwell equation and their coupled system, we study the techniques to improve computational efficiency and to reduce computational complexity, and construct structure-preserving numerical methods, which inherit the geometric structure and physical characteristics of the underlying system; we investigate the inherent superiority mechanism of structure-preserving methods; we study the (strong、weak、in probability) convergence analysis, stability and ergodic theory of stochastic multi-symplectic numerical methods for stochastic Hamiltonian partial differential equations. The aim is to provide a systematic construction methodology and the related numerical analysis theory for the structure-preserving methods of these two kinds of stochastic Hamiltonian partial differential equations. The research results of this project will enrich the theoretical analysis and numerical simulation of the structure-preserving methods for stochastic Hamiltonian partial differential equations, and it has important theoretical significance and research value.

本项目将聚焦于在凝聚态物理、量子物理、分子动力学、电磁学和纳米材料等众多科学技术领域中具有广泛应用的随机薛定谔方程和随机麦克斯韦方程这两类典型随机哈密尔顿偏微分方程以及它们的耦合系统，开展保结构算法研究，为算法在实际应用中的大规模计算提供理论依据。研究内容主要包括：针对随机非线性薛定谔方程、随机麦克斯韦方程以及它们的耦合系统，研究提高计算效率、降低计算复杂度的方法，构造保持原系统几何结构、物理特征的保结构算法；研究保结构算法的内在优越性机理；研究随机哈密尔顿偏微分方程随机多辛算法的（强、弱、依概率）收敛性和收敛阶、稳定性及遍历性理论。本项目旨在为这两类随机哈密尔顿偏微分方程提供保结构算法的系统构造方法和相关的数值分析理论。本项目的研究成果将丰富随机哈密尔顿偏微分方程保结构算法的理论分析和数值模拟，具有重要的理论意义和研究价值。

在本项目研究中，我们发展了随机麦克斯韦方程和随机薛定谔方程等重要随机哈密尔顿偏微分方程高效保结构算法的构造方法、数值遍历性分析理论、收敛性与优越性机理分析理论、以及高效实现技术等；对于随机麦克斯韦方程，提出了该方程无穷维随机哈密尔顿系统的典则形式，证明了其相流保持无穷维随机辛几何结构，提出了一系列保持数学结构、遍历性的随机算法，给出了不变测度之间的误差估计，解决了算法收敛性方面的一个公开问题，实现了对三维随机麦克斯韦方程的高效、长时间数值模拟；对于随机薛定谔方程，提出了若干保持随机多辛几何结构的随机算法，以及能提高计算效率的自适应时间步全离散算法，基于大偏差原理，建立了数值解的渐近行为分析，得到了数值解和精确解重要物理量之间误差的概率估计。主要成果发表在《SIAM 系列刊物》(8篇)、《J. Differential Equations》(1篇)、《IMA J. Numer. Anal.》(2 篇)、《J. Comput. Phys.》(4篇)、《BIT》(2篇)、《Potential Anal.》(1篇)、《Inverse Problems》(1篇)等期刊上。