Distributed preconditioning for the parametric Helmholtz equation

TL;DR

The paper tackles the costly repetition of solving parameterized Helmholtz systems by distributing multiple preconditioners across the parameter space and predicting solver effort with a gray-box Gaussian process surrogate. A location-allocation heuristic optimizes preconditioner placement using an a priori iteration bound and a dimension-aware anisotropy model, enabling continuous centers rather than only grid points. The approach achieves about an order-of-magnitude reduction in total runtime on high-frequency scattering problems, with particularly large gains at higher frequencies, and demonstrates the importance of exploiting anisotropy to avoid the curse of dimensionality. The framework integrates affine and parameterized-domain settings, showing practical impact for uncertainty quantification and PDE-constrained contexts where repeated solves arise.

Abstract

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.

Figures (5)

• Figure 1: Plot of $g(\alpha)$ up to a constant from 0 to $0.6$.
• Figure 2: Result of the location-allocation algorithm; Fig. \ref{['fig:1a']} shows the preconditioner attributions together with the underlying partition of $W$, and Fig. \ref{['fig:1b']} shows a contour plot of the underlying distance function $m(\cdot)$.
• Figure 3: Tabulated results in seconds for affine expansion with $\eta_i=\frac{1}{4}$
• Figure 3: Effect of different $N_{pc}$ values for the parameterized shape problem. The red curve shows the greedy initialization, the blue curve the location-allocation result from Algorithm \ref{['alg:location-allocation']}, and the black dashed curve the result of the location-allocation algorithm until convergence. $N_{pc}^*$ is the value where Algorithm \ref{['alg:location-allocation']} terminates.
• Figure 4: Comparison of the RMSE and disagree ratio for the stabilizing predictions (SP) stopping criterion. The parameter dimension is $N=15$ with $\vartheta=\frac{1}{2}\vartheta_{\max}$ and $\alpha=2$. The solid line represents the RMSE (left axis), the dashed line the disagree ratio, and the dotted line a trailing average of the disagree ratio (right axis). The solid gray line marks the $1\%$ threshold in the stopping criterion.

Theorems & Definitions (5)

• Remark 3.2
• Remark 3.3: Concentration of measure for isotropic parameter space
• Remark 4.1: Nonlinearity of $g(\alpha)$
• Remark 4.2: Different estimates
• Remark 4.3: Parallel training