随机Helmholtz型问题的数值方法

项目名称： 随机Helmholtz型问题的数值方法

项目编号： No.11471141

项目类型： 面上项目

立项/批准年度： 2015

项目学科： 数理科学和化学

项目作者： 张凯

作者单位： 吉林大学

项目金额： 62万元

中文摘要： 确定性Helmholtz问题的理论与计算方法已广泛应用于地质勘探,军事科学等诸多科学技术领域，但对于更具有实际意义的随机Helmholtz型问题还有待进一步研究。我们拟基于多级蒙特卡洛和稀疏网格等方法,研究几种不同随机Helmholtz型问题的快速数值算法。主要包括：（1）对于带随机系数（即随机波数）的Helmholtz内问题，采用多级蒙特卡洛方法，使得蒙特卡洛方法的分级样本数与最细空间剖分相匹配，从而减少计算所需样本量。（2）对于交界面带随机小扰动的光栅问题，采用形状泰勒展开和摄动分析的原理，研究随机问题解与数值解之间期望，方差等统计量相应的误差（依赖于振幅）。（3）对于边界带随机大扰动的Helmholtz外问题，对物理空间采用完全匹配层方法截断,对随机边界采用坐标变换和稀疏网格方法，减少自由度，从而加速求解过程。这些研究将加深人们对相关现象的认识，有着重要的理论和实用价值。

中文关键词： 随机系数；随机交界面或边界；随机Helmholtz型问题；多级蒙特卡洛方法；稀疏网格方法

英文摘要： The theoretical and computational results of deterministic Helmholtz problems have important applications in many fields of science and technology, such as geophysical exploration, non-destructive testing, radar and sonar and other defense technology, etc. In this project, we shall be concerned with several stochastic Helmholtz-type problems based on multilevel Mento-carlo and sparse grid techniques, which are of more practical significance and have received wide attentions recently in literature. Specifically, the following results will be derived: (1) For the interior Helmholtz problem with stochastic wavenumber, we shall adopt multilevel Mento-carlo method, which allows us to have the same overall convergence as the Mento-carlo method on the finest grid, but the computational costs are only a fraction of the latter. (2) For the grating problems with stochastic interface of small pertubation, based on the shape-Taylor expansion and pertubation theory, we shall quantify the mean field and the variance of the stochastic solution in terms of certain orders of the pertubation amplitude. (3) For the exterior Helmholtz problem with stochastic boundary of large pertubation, we shall adopt perfectly matched layer for the spatial domain, and derive the coordinate transformation with respect to the stochastic boudary and adopt sparse grid method in stochastic domain, which shrinks the degree of freedom and accelerates the computation. The theoretical and computational results achieved in this project will deepen our knowledge about corresponding phenomena, and also have significant practical impacts.

英文关键词： stochastic coefficients;stochastic interface or stochastic boundary;stochastic Helmholtz-type problem;multilevel Mento-carlo method;sparse grid method