Quasi-Monte Carlo methods for uncertainty quantification of wave propagation and scattering problems modelled by the Helmholtz equation

TL;DR

This work develops and analyzes a forward uncertainty quantification framework for wave propagation governed by the Helmholtz equation with random coefficients, formulated on the exterior domain and approximated via a PML-FEM pipeline. Randomness is modeled through affine expansions of the coefficients with infinitely many independent parameters, which are truncated to a finite dimension $s$ for computation. The forward quantity of interest is the expectation of linear functionals of the solution, including the far-field pattern; the methodology couples dimension truncation, quasi-Monte Carlo quadrature (randomly shifted lattice and interlaced polynomial lattice rules), and PML-based PDE solves with $h$-FEM, yielding explicit error bounds that depend on $s$, $N$, $h$, and the wavenumber $k$, along with an exponential PML accuracy in the truncation boundary. Theoretical results cover parametric regularity, dimension truncation, QMC error, and FEM-PML discretization, with numerical experiments demonstrating near $\mathcal{O}(N^{-1})$ QMC convergence for the far-field in a 2D scattering problem featuring a randomly heterogeneous medium and a sound-soft obstacle. The framework provides a scalable, $k$-explicit approach for high-frequency UQ in wave scattering, with robust error control and applicability to practical computations of quantities like the far-field pattern.

Abstract

We analyse and implement a quasi-Monte Carlo (QMC) finite element method (FEM) for the forward problem of uncertainty quantification (UQ) for the Helmholtz equation with random coefficients, both in the second-order and zero-order terms of the equation, thus modelling wave scattering in random media. The problem is formulated on the infinite propagation domain, after scattering by the heterogeneity, and also (possibly) a bounded impenetrable scatterer. The spatial discretization scheme includes truncation to a bounded domain via a perfectly matched layer (PML) technique and then FEM approximation. A special case is the problem of an incident plane wave being scattered by a bounded sound-soft impenetrable obstacle surrounded by a random heterogeneous medium, or more simply, just scattering by the random medium. The random coefficients are assumed to be affine separable expansions with infinitely many independent uniformly distributed and bounded random parameters. As quantities of interest for the UQ, we consider the expectation of general linear functionals of the solution, with a special case being the far-field pattern of the scattered field. The numerical method consists of (a) dimension truncation in parameter space, (b) application of an adapted QMC method to compute expected values, and (c) computation of samples of the PDE solution via PML truncation and FEM approximation. Our error estimates are explicit in $s$ (the dimension truncation parameter), $N$ (the number of QMC points), $h$ (the FEM grid size) and (most importantly), $k$ (the Helmholtz wavenumber). The method is also exponentially accurate with respect to the PML truncation radius. Illustrative numerical experiments are given.

Figures (7)

• Figure 1: The domain $D_{R_0}$ is the (open) shaded region inside the ball $B_{R_0}$, excluding the closure of the obstacle $D$. The supports of $A-I$ and $n-1$ are in $D_{R_0}$. The support of the source term $f$ is in $\overline{D_{R_0}}$. The quantity of interest $G$ will be a linear functional on $D_R$ for $R \geq R_0$.
• Figure 2: The PML truncation problem is posed on the domain $D_{R_2}$ which is the region inside the ball $B_{R_2}$, excluding $\overline{D}$. The red gradient shading illustrates the radial cutoff function $\varphi_{\rm PML}\in C^3({\mathbb{R}}_+)$ satisfying \ref{['eq:sigma_prop']}; it is increasing in the annulus between radii $R_1$ and $R_2$.
• Figure 3: When formulating the plane-wave sound-soft scattering problem as an EDP, the blue gradient shading illustrates the radial cutoff function $\varphi_{\rm alt}\in C^2({\mathbb{R}}^d)$ satisfying \ref{['eq:psi_pw1']}. The support of the data $f^{\rm alt}$ as defined in \ref{['eq:pw_f']} is in the annulus between radii $R_0-\eta$ (solid red circle) and $R_0-\eta/2$ (dashed red circle).
• Figure 4: Computed total field $|u^T|$ and scattered field $|u^S|$ within radius $R_0=4.5$ for $k=48$. Top row: homogeneous field $n({\boldsymbol{x}},{\boldsymbol{0}}) = 1$. Bottom row: heterogeneous field $n({\boldsymbol{x}},{\boldsymbol{y}})$ with ${\boldsymbol{y}}=(\frac{1}{2},\frac{1}{2},\frac{1}{2},{\boldsymbol{0}})$.
• Figure 5: Far-field pattern for $k=12$ computed with the homogeneous field $n({\boldsymbol{x}},{\boldsymbol{0}})=1$ (top left) and with realizations of the heterogeneous field $n({\boldsymbol{x}},{\boldsymbol{\tfrac{1}{2}}})$ truncated to $s=1,2,3,4,5$ terms.

Theorems & Definitions (29)

• Definition 2.4: The exterior Dirichlet Problem (EDP)
• Definition 2.5: Exterior Dirichlet-to-Neumann map ${\rm DtN}_k$ on $\partial{B}_R$
• Definition 2.6: A particular class of nontrapping $D$, $A$, and $n$
• Lemma 2.7: Bounds on the solution of the EDP
• Definition 2.8: Plane-wave sound-soft scattering problem
• Lemma 2.10
• Remark 2.11
• Corollary 2.13
• Theorem 3.1
• Corollary 3.2