This paper develops and analyzes a general iterative framework for solving parameter-dependent and random diffusion problems. It is inspired by the multi-modes method of [7,8] and the ensemble method of [19] and extends those methods into a more general and unified framework. The main idea of the framework is to reformulate the underlying problem into another problem with a parameter-independent diffusion coefficient and a parameter-dependent (and solution-dependent) right-hand side, a fixed-point iteration is then employed to compute the solution of the reformulated problem. The main benefit of the proposed approach is that an efficient direct solver and a block Krylov subspace iterative solver can be used at each iteration, allowing to reuse the $LU$ matrix factorization or to do an efficient matrix-matrix multiplication for all parameters, which in turn results in significant computation saving. Convergence and rates of convergence are established for the iterative method both at the variational continuous level and at the finite element discrete level under some structure conditions. Several strategies for establishing reformulations of parameter-dependent and random diffusion problems are proposed and their computational complexity is analyzed. Several 1-D and 2-D numerical experiments are also provided to demonstrate the efficiency of the proposed iterative method and to validate the theoretical convergence results.
翻译:本文开发并分析了解决依赖参数和随机扩散问题的一般迭接框架,它受[7,8]的多模式方法[7,8]和[19]的混合方法的启发,并将这些方法推广到一个更加全面和统一的框架。框架的主要想法是用一个依赖参数的传播系数和一个依赖参数的(和依赖溶的)右侧,将根本问题改造成另一个问题,然后使用一个固定点的迭接法来计算重订问题的解决方案。拟议方法的主要好处是,在每次迭代时,都可以使用一个高效的直接求解器和一个块的Krylov 子空间迭接解器,允许重新使用$LU的矩阵因子化,或对所有参数进行高效的矩阵矩阵矩阵乘法的乘法,这反过来又可以节省大量计算费用。为迭接法的迭接法在变化连续水平和某些结构条件下的有限元素离异的计算方法确定了趋同率和趋同率。若干重新确定依赖参数和随机扩散问题的战略,并提出了其计算方法的趋同性,还分析了其计算方法的趋同性。