Cayley-transform analysis and numerical validation of the convergent Born series for the Helmholtz equation

TL;DR

This paper introduces a new operator-theoretic framework for the Convergent Born Series method applied to high-frequency Helmholtz problems, providing convergence guarantees and broadening its applicability.

Contribution

It develops a Cayley-transform based analysis that extends CBS applicability to complex wave equations without explicit Green's functions and includes numerical validation.

Findings

CBS solutions are accurate and stable across various contrasts and frequencies.

The Cayley-transform framework provides basis-independent convergence bounds.

Incorporating absorbing layers improves iteration contractivity without altering the differential operator.

Abstract

We develop an operator-theoretic framework for the Convergent Born Series (CBS) method applied to the Lippmann--Schwinger equation for high-frequency Helmholtz problems. In contrast to the Fourier-based analysis of Osnabrugge et al., our approach expresses the preconditioned Lippmann--Schwinger iteration entirely in terms of the resolvent of a self-adjoint background operator. This leads to a unitary Cayley-transform representation of the CBS iteration operator, from which we derive basis-independent bounds on its numerical range and a general convergence criterion valid on arbitrary bounded domains and for complex-valued wave numbers. Because the analysis does not rely on an explicit Green's function in the Fourier domain, the Cayley-transform framework extends naturally to a broader class of frequency-domain wave and diffusion equations whose fundamental solutions are not available in closed form. We further incorporate smoothly tapered complex-wavenumber absorbing layers that preserve the self-adjoint structure of the reference operator and enhance the contractivity of the iteration without modifying the differential operator. In addition to this theoretical generalization, we present a detailed numerical validation in which CBS solutions are benchmarked against PML-based finite-difference wavefield simulations. These experiments demonstrate that the operator-theoretic CBS formulation delivers accurate and stable results across a broad range of contrasts and frequencies, thereby significantly extending the applicability and theoretical foundation of the CBS method beyond previously analyzed settings.