标题： 参数Helmholtz方程的分布式预条件处理 显示英文标题

标题： Distributed preconditioning for the parametric Helmholtz equation

摘要： 在这项工作中，我们研究了具有可能非线性参数依赖的参数化线性方程组的有效计算方法。当矩阵对参数高度敏感时，基于均值的预条件处理可能不够。 针对这种情况，我们探索了一种方法，在预计算步骤中在参数空间中放置多个预条件器。为了确定有限数量的预条件器的最佳放置位置，我们先验地估计相对于给定预条件器的预期迭代次数，并使用定位-分配策略优化预条件器的放置。我们详细阐述了高频率外部Dirichlet散射的Helmholtz问题的方法，并通过灰箱高斯过程回归方法估计GMRES迭代的预期次数。我们在两个实际应用中展示了我们的方法：具有参数折射率的域中的散射以及具有参数化形状的散射体的散射。利用这些数值例子，我们展示了我们的方法如何带来大约一个数量级的运行时间节省。此外，我们还研究了参数维度和维度各向异性对其有效性的影响。 显示英文摘要

摘要： In this work, we address the efficient computation of parameterized systems of linear equations, with possible nonlinear parameter dependence. When the matrix is highly sensitive to the parameters, mean-based preconditioning might not be enough. For this scenario, we explore an approach in which several preconditioners are placed in the parameter space during a precomputation step. To determine the optimal placement of a limited number of preconditioners, we estimate the expected number of iterations with respect to a given preconditioner a priori and use a location-allocation strategy to optimize the placement of the preconditioners. We elaborate on our methodology for the Helmholtz problem with exterior Dirichlet scattering at high frequencies, and we estimate the expected number of GMRES iterations via a gray-box Gaussian process regression approach. We illustrate our approach in two practical applications: scattering in a domain with a parametric refractive index and scattering from a scatterer with parameterized shape. Using these numerical examples, we show how our methods leads to runtime savings of about an order of magnitude. Moreover, we investigate the effect of the parameter dimension and the importance of dimension anisotropy on their efficacy.