Optimal Transportation for the Far-field Reflector Problem
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866913761134116864 |
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| author | Bao, Gang Zhang, Yixuan |
| author_facet | Bao, Gang Zhang, Yixuan |
| contents | The inverse reflector problem aims to design a freeform reflecting surface that can direct the light from a specified source to produce the desired illumination in the target area, which is significant in the field of geometrical non-imaging optics. Mathematically, it can be formulated as an optimization problem, which is exactly the optimal transportation problem (OT) when the target is in the far field. The gradient of OT is governed by the generalized Monge-Amp`ere equation that models the far-field reflector system. Based on the gradient, this work presents a Sobolev gradient descent method implemented within a finite element framework to solve the corresponding OT. Convergence of the method is established and numerical examples are provided to demonstrate the effectiveness of the method. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_21182 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Optimal Transportation for the Far-field Reflector Problem Bao, Gang Zhang, Yixuan Numerical Analysis The inverse reflector problem aims to design a freeform reflecting surface that can direct the light from a specified source to produce the desired illumination in the target area, which is significant in the field of geometrical non-imaging optics. Mathematically, it can be formulated as an optimization problem, which is exactly the optimal transportation problem (OT) when the target is in the far field. The gradient of OT is governed by the generalized Monge-Amp`ere equation that models the far-field reflector system. Based on the gradient, this work presents a Sobolev gradient descent method implemented within a finite element framework to solve the corresponding OT. Convergence of the method is established and numerical examples are provided to demonstrate the effectiveness of the method. |
| title | Optimal Transportation for the Far-field Reflector Problem |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/2503.21182 |