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Flux-Matched Poisson Iteration for Generated Jacobian Equations: Local Convergence and Spectral Galerkin Approximation

  • Haoxiang Zhang
  • Yixuan Zhang
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Abstract

We develop a flux-matched Poisson iteration for solving generated Jacobian equations with second boundary conditions. In each step, it solves a fixed shifted Poisson problem with a Neumann-type boundary condition. Near a smooth regular solution with positive C1,1 densities, we prove that the iteration has local geometric convergence in C2,α on smooth connected Riemannian domains. The iteration is then discretized using a spectral Galerkin method, with local convergence proved for the Fourier-Jacobi scheme on spherical caps. Optical numerical experiments support these estimates and demonstrate useful imaging performance beyond the local convergence theory.

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Cite

BibTeXRIS
BibTeX
@misc{zhang2026fluxmatched,
  title = {{Flux-Matched Poisson Iteration for Generated Jacobian Equations: Local Convergence and Spectral Galerkin Approximation}},
  author = {Haoxiang Zhang and Yixuan Zhang},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2611.06153},
  eprint = {2611.06153},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2611.06153},
}
RIS
TY  - UNPB
TI  - Flux-Matched Poisson Iteration for Generated Jacobian Equations: Local Convergence and Spectral Galerkin Approximation
AU  - Haoxiang Zhang
AU  - Yixuan Zhang
PY  - 2026
DA  - 2026/11/04
PB  - arXiv
DO  - 10.48550/arxiv.2611.06153
UR  - https://arxiv.org/abs/2611.06153
AB  - We develop a flux-matched Poisson iteration for solving generated Jacobian equations with second boundary conditions. In each step, it solves a fixed shifted Poisson problem with a Neumann-type boundary condition. Near a smooth regular solution with positive C^1,1 densities, we prove that the iteration has local geometric convergence in C^2,α on smooth connected Riemannian domains. The iteration is then discretized using a spectral Galerkin method, with local convergence proved for the Fourier-Jacobi scheme on spherical caps. Optical numerical experiments support these estimates and demonstrate useful imaging performance beyond the local convergence theory.
ER  -