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.

翻译：本文研究参数化线性方程组（可能具有非线性参数依赖性）的高效计算问题。当矩阵对参数高度敏感时，基于均值的预条件处理方法可能不再适用。针对这种情况，我们探索一种在预计算阶段于参数空间中布置多个预条件子的方法。为确定有限数量预条件子的最优布局，我们预先估计给定预条件子对应的期望迭代次数，并采用选址分配策略优化预条件子的空间分布。我们以高频外部狄利克雷散射的亥姆霍兹问题为例详细阐述该方法，通过灰箱高斯过程回归方法估计GMRES迭代的期望次数。我们在两个实际应用中验证该方法：具有参数化折射率域内的散射问题，以及参数化形状散射体的散射问题。数值算例表明，该方法可实现约一个数量级的计算时间节省。此外，我们研究了参数维度的影响以及维度各向异性对方法效能的重要性。