Cayley-transform analysis and numerical validation of the convergent Born series for the Helmholtz equation
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866914498319745024 |
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| author | Jakobsen, Morten |
| author_facet | Jakobsen, Morten |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_20282 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Cayley-transform analysis and numerical validation of the convergent Born series for the Helmholtz equation Jakobsen, Morten Numerical Analysis 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. |
| title | Cayley-transform analysis and numerical validation of the convergent Born series for the Helmholtz equation |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/2604.20282 |