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PreprintCC BY 4.0arXiv

A Mixed Elliptic--Hyperbolic Monge--Ampère Problem: Modeling and a Least-Squares Solver

  • G. V. Carvalho
  • M. J. H. Anthonissen
  • J. H. M. ten Thije Boonkkamp
  • L. Kusch
  • K. Mitra
  • W. L. IJzerman
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Abstract

We consider a transport problem motivated by freeform optical design of a parallel-to-far-field reflector. Classical formulations rely on either elliptic or hyperbolic Monge--Ampère equations, leading to globally convex/concave or saddle-shape reflector geometries, respectively. However, the design of smooth reflector arrays may require optical surfaces combining both types of curvature. To address this, we introduce a mixed elliptic--hyperbolic Monge--Ampère system obtained by partitioning the source domain into one elliptic and one hyperbolic subregion, and we develop a least-squares method for its numerical approximation. Numerical examples demonstrate the ability of the proposed framework to compute both isolated and periodic mixed-type transport solutions, yielding smooth reflector configurations.

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Cite

BibTeXRIS
BibTeX
@misc{carvalho2026mixed,
  title = {{A Mixed Elliptic--Hyperbolic Monge--Ampère Problem: Modeling and a Least-Squares Solver}},
  author = {G. V. Carvalho and M. J. H. Anthonissen and J. H. M. ten Thije Boonkkamp and L. Kusch and K. Mitra and W. L. IJzerman},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.23544},
  eprint = {2610.23544},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.23544},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - A Mixed Elliptic--Hyperbolic Monge--Ampère Problem: Modeling and a Least-Squares Solver
AU  - G. V. Carvalho
AU  - M. J. H. Anthonissen
AU  - J. H. M. ten Thije Boonkkamp
AU  - L. Kusch
AU  - K. Mitra
AU  - W. L. IJzerman
PY  - 2026
DA  - 2026/10/19
PB  - arXiv
DO  - 10.48550/arxiv.2610.23544
UR  - https://arxiv.org/abs/2610.23544
AB  - We consider a transport problem motivated by freeform optical design of a parallel-to-far-field reflector. Classical formulations rely on either elliptic or hyperbolic Monge--Ampère equations, leading to globally convex/concave or saddle-shape reflector geometries, respectively. However, the design of smooth reflector arrays may require optical surfaces combining both types of curvature. To address this, we introduce a mixed elliptic--hyperbolic Monge--Ampère system obtained by partitioning the source domain into one elliptic and one hyperbolic subregion, and we develop a least-squares method for its numerical approximation. Numerical examples demonstrate the ability of the proposed framework to compute both isolated and periodic mixed-type transport solutions, yielding smooth reflector configurations.
ER  -