积分算子特征值问题多尺度快速算法的若干研究

Many practical problems in science and engineering are formulated as eigen-problems of integral operators. For many years, numerical solutions of the eigen-problem have attracted much attention. The main bottleneck problem for the integral operators is that the matrix resulting from a discretization of an integral operator is a full matrix. Solving the eigen-problem of a full matrix requires significant amount of computational effort. Hence, fast algorithms for solving such a problem are highly desirable. In this proposal, taking advantage of the idea of fast methods for integral equations, which is using a sparse matrix to approximate the dense matrix, and the idea of two-grid for eigenvalue problems, which is reducing the eigenvalue problem to solve a linear algebraic system, we develop multiscale fast methods for eigen-problems for integral operators. We firstly develop a fast method with almost optimal convergence order by constructing a truncation strategy, which leads to the optimal complexity of algorithm. This exhibits the fast of the method. Then we use the idea of two-grid discretization scheme to develop fast multilevel augmentation methods for solving eigen-problems, based on multilevel decompositions of the approximate subspace aiming at efficiently solving linear systems of large scale obtained from discretization of integral operators.

积分算子特征值问题多尺度快速算法是计算数学领域的一个研究重点和热点，在数学、物理和工程上有很强的应用背景。目前这类问题的研究成果相当的少，其瓶颈问题是积分算子离散后的系数矩阵通常是满矩阵，导致计算量过大而使数值计算无法进行下去。构造计算量少效率高的积分算子特征值问题的快速算法对科学与工程计算具有重要的理论价值和应用价值。充分吸收多尺度快速算法的核心思想和两网格方法的核心思想的精华，将两个精华结合起来，构建一套积分算子特征值问题多尺度快速算法是本项目的研究重点。首先，根据积分算子的性质，构造截断策略，在保证计算收敛性前提下，对稠密矩阵进行压缩，使得计算复杂度达到最优，体现出算法的快速性。其次，对压缩后的非典型大规模矩阵特征值求解上，充分吸收两网格和多层扩充算法的精华，将特征值求解问题转化为方程求解问题，利用多尺度基底具有高低频层次性，构造大规模稀疏矩阵的特征值问题的多层扩充算法。

本项目旨在构造积分方程问题和积分算子特征值问题的多尺度快速算法。利用试探函数空间基底的多尺度性、紧支集性和消失矩性质，构造特征值问题的多尺度投影逼近框架。同时，针对积分算子的性质，以及系数矩阵的形状，分别构造截断策略，在保证计算收敛性前提下，对稠密矩阵进行压缩，使得计算复杂度达到最优，体现出算法的快速性。对压缩后的非典型大规模矩阵特征值求解上，充分吸收两网格和多层扩充算法的精华，将特征值求解问题转化为方程求解问题，构造高效率的积分算子特征值问题多尺度快速算法。研究思想和内容在积分方程、特征值问题、奇异摄动问题上都得到一系列的研究成果。