An adaptive Levin method for complicated domains

TL;DR

The paper tackles accurate numerical evaluation of multivariate oscillatory integrals $\int_{\Omega} f(\mathbf{x}) e^{i g(\mathbf{x})} \ d\Omega$ over general domains by introducing an adaptive multivariate Levin method applied on transfinite triangular meshes. It reformulates the integral via the Levin PDE $\mathcal{L}[\mathbf{p}]=\nabla\cdot\mathbf{p}+i\nabla g\cdot\mathbf{p}=f$ and uses the divergence theorem to reduce the problem to boundary integrals, which are computed with a univariate adaptive Levin method, even in the presence of resonance points and stationary points. The discretization uses monomial bases with collocation and truncated SVD, with higher-order collocation to improve accuracy, and phase-function techniques to extend applicability to a broad class of oscillatory integrals. Resonance points on the boundary are handled effectively, enabling near frequency-independent performance. Numerical experiments on Helmholtz-type integrals demonstrate accuracy and parallelizability on complex geometries, confirming the approach’s robustness and practical impact for high-frequency oscillatory problems on general domains.

Abstract

In this paper we describe an adaptive Levin method for numerically evaluating integrals of the form $\int_Ωf(\mathbf x) \exp(i g(\mathbf x)) \,dΩ$ over general domains that have been meshed by transfinite elements. On each element, we apply the multivariate Levin method over adaptively refined sub-elements, until the integral has been computed to the desired accuracy. Resonance points on the boundaries of the elements are handled by the application of the univariate adaptive Levin method. When the domain does not contain stationary points, the cost of the resulting method is essentially independent of the frequency, even in the presence of resonance points.

Figures (2)

• Figure 1: Real part of the integrand of (\ref{['eqn:example']}) with $\omega=100$.
• Figure 2: The results of the numerical experiments computing the integral (\ref{['eqn:example']}). The top plot gives the error in the value of the integral computed via the multivariate adaptive Levin method as a function of $\omega$. The bottom plot gives the running time in milliseconds as a function of $\omega$.