几类非光滑问题的基于区域分解技术的算法研究

In this project, we will consider numerical iterative algorithms based on domain decomposition techniques and nonsmooth reformulations for the solutions of several nonlinear and nonsmooth problems, arising from widely application areas, such as material science, continuum mechanics, electrodynamics, electromagnetics, hydrology, economy and finance, as well as image processing, optimum design, optimum control and many others. These problems are usually described as nonsmooth elliptic partial differential equations, linear and nonlinear complementarity problems, variational inequalities, HJB equations and constrained optimizations. The aim of this project is to further investigate domain decomposition and multigrid methods for solving the above mentioned nonsmooth problems. The algorithms will be constructed based on the equivalent nonsmooth equations of the problems. New progress will be achieved at some key points. For instance, to make the algorithms more scalable, some techniques, such as introducing preconditioner, absorbing boundary condition on the artificial boundary and finding a better solution guess from the coarser spaces, will be adopted. Numerical and theoretical analysis will be presented for the convergence of the proposed nonsmooth algorithms, including the convergence rate and the (quasi-)optimal computational complexity of the algorithms..The research will provide referential experiences and large-scale computing support in areas of the research we concerned and be a great benefit to efficient computations and simulations of the nonlinear phenomena in the relevant applications.

本项目研究几类非线性非光滑问题的数值迭代算法。这些问题在材料力学、连续介质力学、电动力学、电磁学、水文学、经济金融以及图像处理、最优设计和最优控制等领域有着广泛应用背景，通常以非光滑椭圆偏微分方程、线性与非线性互补问题、变分不等式、HJB方程和约束最优化等数学模型的形式出现。本项目针对这些非光滑问题等价的非光滑方程（组），探讨基于区域分解技术的可应用于大规模计算的区域分解和多重网格算法，在一些关键问题上取得突破。比如，通过在算法中引入预处理子、吸收边界传输条件或粗空间，使得算法具有较好的可扩展性，以适应于求解大规模问题。项目还将在理论上研究这类非光滑算法的收敛速度和（拟）最优计算复杂性。.此项目的研究将对所涉及的诸类非线性非光滑问题的大规模计算提供可借鉴的经验和支持，有助于对相关应用问题的非线性现象的有效计算和模拟。

本项目研究几类非线性非光滑问题的数值迭代算法。这些问题在材料力学、连续介质力学、电动力学、电磁学、水文学、经济金融以及图像处理、最优设计和最优控制等领域有着广泛应用背景，通常以非光滑椭圆偏微分方程、线性与非线性互补问题、变分不等式、HJB方程和约束最优化等数学模型的形式出现。本项目针对这些非光滑问题等价的非光滑方程（组）以及更具挑战性的全局最优化问题，探讨基于区域分解技术的可应用于大规模计算的区域分解和多重网格算法，使得算法具有较好的可扩展性，以适应于求解大规模问题。项目还将在理论上研究这类非光滑算法的收敛速度和（拟）最优计算复杂性。.此项目的研究将对所涉及的诸类非线性非光滑问题的大规模计算提供可借鉴的经验和支持，有助于对相关应用问题的非线性现象的有效计算和模拟。