[Paper Review] Quasi-Monte Carlo finite element analysis for wave propagation in heterogeneous random media

M. Ganesh, Frances Y. Kuo+1 more|arXiv (Cornell University)|Apr 26, 2020

Mathematical Approximation and Integration27 references4 citations

TL;DR

This paper proposes a quasi-Monte Carlo (QMC) finite element method for solving the Helmholtz equation with random, spatially heterogeneous refractive indices in bounded domains. By using a recently developed sign-definite variational formulation, the method overcomes the coercivity issues of the standard Helmholtz formulation, enabling convergence analysis with error bounds comparable to those for diffusion problems, including stochastic truncation, finite element, and cubature errors.

ABSTRACT

We propose and analyze a quasi-Monte Carlo (QMC) algorithm for efficient simulation of wave propagation modeled by the Helmholtz equation in a bounded region in which the refractive index is random and spatially heterogenous. Our focus is on the case in which the region can contain multiple wavelengths. We bypass the usual sign-indefiniteness of the Helmholtz problem by switching to an alternative sign-definite formulation recently developed by Ganesh and Morgenstern (Numerical Algorithms, 83, 1441-1487, 2020). The price to pay is that the regularity analysis required for QMC methods becomes much more technical. Nevertheless we obtain a complete analysis with error comprising stochastic dimension truncation error, finite element error and cubature error, with results comparable to those obtained for the diffusion problem.

Motivation & Objective

• To develop an efficient numerical method for simulating wave propagation governed by the Helmholtz equation in domains with random, spatially varying refractive indices.
• To address the lack of coercivity in the standard Galerkin formulation of the Helmholtz equation when multiple wavelengths are present.
• To extend quasi-Monte Carlo (QMC) methods—previously successful for diffusion problems—to wave propagation by using a sign-definite variational formulation.
• To provide a complete error analysis including stochastic dimension truncation, finite element discretization, and cubature errors.
• To achieve convergence rates comparable to those in the diffusion problem setting despite increased regularity analysis complexity.

Proposed method

• The method employs a sign-definite variational formulation of the Helmholtz equation recently developed by Ganesh and Morgenstern, which avoids the sign-indefiniteness of the standard formulation.
• The refractive index is modeled as a random field parameterized by an infinite-dimensional uniform random vector $\mathbf{y} \in [-1/2, 1/2]^\mathbb{N}$, with mean and perturbation components.
• A stochastic Galerkin finite element method is applied to the sign-definite weak form, enabling stable and consistent space discretization.
• Quasi-Monte Carlo (QMC) cubature rules are used to compute the expected value of a linear functional of the solution, leveraging low-discrepancy sequences for improved convergence.
• Error is decomposed into three components: stochastic dimension truncation, finite element approximation, and QMC cubature error, each rigorously bounded.
• A novel regularity analysis is developed to handle the complex stochastic dependence introduced by the sign-definite formulation, using a technical induction argument with weighted multi-index norms.

Experimental results

Research questions

• RQ1 Can quasi-Monte Carlo methods be effectively applied to wave propagation problems governed by the Helmholtz equation with random, heterogeneous coefficients?
• RQ2 How can the lack of coercivity in the standard Helmholtz formulation be overcome to enable QMC analysis?
• RQ3 What is the convergence behavior of the QMC finite element method when the refractive index is random and spatially heterogeneous?
• RQ4 What error bounds can be established for the combined stochastic, spatial, and cubature discretizations in this setting?
• RQ5 Can the regularity of the solution with respect to the infinite-dimensional random input be analyzed sufficiently to support QMC convergence rates?

Key findings

• The proposed QMC finite element method achieves convergence rates comparable to those for the diffusion problem, despite the increased complexity of the sign-definite formulation.
• The method successfully bypasses the sign-indefiniteness of the standard Helmholtz formulation by using a recently developed sign-definite variational form, ensuring stable Galerkin discretization.
• The error analysis includes rigorous bounds for stochastic truncation, finite element, and QMC cubature errors, with the total error decaying at a rate dependent on the integrability of the solution's stochastic derivatives.
• A new technical lemma is established to control the growth of high-order stochastic moments in the solution, enabling the regularity analysis required for QMC convergence.
• The method maintains stability and convergence even when the wavelength is small relative to the domain size, a regime where standard methods often fail due to lack of coercivity.
• The analysis confirms that the QMC method outperforms standard Monte Carlo in terms of convergence rate, with the potential for significant computational savings in high-dimensional stochastic settings.