Geometric optics approximation sampling: near-field case
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866915556228071424 |
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| author | Sun, Zejun Zheng, Guang-Hui |
| author_facet | Sun, Zejun Zheng, Guang-Hui |
| contents | In this paper, we propose a novel gradient-free and dimensionality-independent sampler, the Geometric Optics Approximation Sampling (GOAS), based on a near-field reflector system. The key idea involves constructing a reflecting surface that redirects rays from a source with a prescribed simple distribution toward a target domain, achieving the desired target measure. Once this surface is constructed, an arbitrary number of independent, uncorrelated samples can be drawn by re-simulating (ray-tracing) the reflector system, i.e., push-forward samples from the source distribution under a reflecting map. To compute the reflecting surface, we employ an enhanced supporting ellipsoid method for the near-field reflector problem. This approach does not require gradient information of the target density and discretizes the target measure using either a low-discrepancy or random sequence, ensuring dimensionality independence. Since the resulting surface is non-smooth (being a union of ellipsoidal sheets) but continuous, we apply a softmin smoothing technique to enable sampling. Theoretically, we define the geometric optics approximation measure as the push-forward of the source measure through the reflecting map. We prove that this measure is well-defined and stable with respect to perturbations of the target domain, ensuring robustness in sampling. Additionally, we derive error bounds between the numerical geometric optics approximation measure and the target measure under the Hellinger metric. Our numerical experiments validate the theoretical claims of GOAS, demonstrate its superior performance compared to MCMC for complex distributions, and confirm its practical effectiveness and broad applicability in solving Bayesian inverse problems. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_01655 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Geometric optics approximation sampling: near-field case Sun, Zejun Zheng, Guang-Hui Numerical Analysis Statistics Theory In this paper, we propose a novel gradient-free and dimensionality-independent sampler, the Geometric Optics Approximation Sampling (GOAS), based on a near-field reflector system. The key idea involves constructing a reflecting surface that redirects rays from a source with a prescribed simple distribution toward a target domain, achieving the desired target measure. Once this surface is constructed, an arbitrary number of independent, uncorrelated samples can be drawn by re-simulating (ray-tracing) the reflector system, i.e., push-forward samples from the source distribution under a reflecting map. To compute the reflecting surface, we employ an enhanced supporting ellipsoid method for the near-field reflector problem. This approach does not require gradient information of the target density and discretizes the target measure using either a low-discrepancy or random sequence, ensuring dimensionality independence. Since the resulting surface is non-smooth (being a union of ellipsoidal sheets) but continuous, we apply a softmin smoothing technique to enable sampling. Theoretically, we define the geometric optics approximation measure as the push-forward of the source measure through the reflecting map. We prove that this measure is well-defined and stable with respect to perturbations of the target domain, ensuring robustness in sampling. Additionally, we derive error bounds between the numerical geometric optics approximation measure and the target measure under the Hellinger metric. Our numerical experiments validate the theoretical claims of GOAS, demonstrate its superior performance compared to MCMC for complex distributions, and confirm its practical effectiveness and broad applicability in solving Bayesian inverse problems. |
| title | Geometric optics approximation sampling: near-field case |
| topic | Numerical Analysis Statistics Theory |
| url | https://arxiv.org/abs/2403.01655 |