A Hierarchical direct solver for Lippmann-Schwinger equations based on boundary traces
Abstract
A high-order hierarchical direct solver for variable-media scattering in two dimensions is presented. Standard discretizations for these problems produce dense linear systems that are commonly solved iteratively, requiring a new global solve for each incident field. The method proposed here instead constructs a reusable scattering operator for the inhomogeneous medium, based on a hierarchical subdivision of the computational domain. First, high-accuracy leaf scattering operators are built from discretized local integral equations. Up the hierarchy, neighboring boxes are merged to build the scattering operators for successively larger boxes. A key feature of the method is that each scattering operator is represented as a map of incoming traces to outgoing traces. Consequently, volume unknowns appear only in the leaf computations; all higher levels of the hierarchy operate entirely on boundary traces. The hierarchical solver is presented first at the continuum level and discretized only afterward using high-order local volume quadrature, polynomial trace representations, and boundary integral techniques. For a discretization involving N degrees of freedom, the method requires O(N3/2) work and O(Nlog N) storage to build all scattering operators. Once these are available, any incoming field can be processed at cost O(N) if only the outgoing field is required, and cost O(Nlog N) if the full field is required. Numerical experiments demonstrate high-order accuracy on a range of challenging scattering problems including cavities, back-scattering, limited regularity, and high wavenumber, that pose significant challenges to iterative methods.
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BibTeX
@misc{cai2026hierarchical,
title = {{A Hierarchical direct solver for Lippmann-Schwinger equations based on boundary traces}},
author = {Yunhui Cai and Simon Dirckx and Joar Bagge and Per-Gunnar Martinsson},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.00494},
eprint = {2611.00494},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.00494},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A Hierarchical direct solver for Lippmann-Schwinger equations based on boundary traces
AU - Yunhui Cai
AU - Simon Dirckx
AU - Joar Bagge
AU - Per-Gunnar Martinsson
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2611.00494
UR - https://arxiv.org/abs/2611.00494
AB - A high-order hierarchical direct solver for variable-media scattering in two dimensions is presented. Standard discretizations for these problems produce dense linear systems that are commonly solved iteratively, requiring a new global solve for each incident field. The method proposed here instead constructs a reusable scattering operator for the inhomogeneous medium, based on a hierarchical subdivision of the computational domain. First, high-accuracy leaf scattering operators are built from discretized local integral equations. Up the hierarchy, neighboring boxes are merged to build the scattering operators for successively larger boxes. A key feature of the method is that each scattering operator is represented as a map of incoming traces to outgoing traces. Consequently, volume unknowns appear only in the leaf computations; all higher levels of the hierarchy operate entirely on boundary traces. The hierarchical solver is presented first at the continuum level and discretized only afterward using high-order local volume quadrature, polynomial trace representations, and boundary integral techniques. For a discretization involving N degrees of freedom, the method requires O(N^3/2) work and O(Nlog N) storage to build all scattering operators. Once these are available, any incoming field can be processed at cost O(N) if only the outgoing field is required, and cost O(Nlog N) if the full field is required. Numerical experiments demonstrate high-order accuracy on a range of challenging scattering problems including cavities, back-scattering, limited regularity, and high wavenumber, that pose significant challenges to iterative methods.
ER -