Geometric optics approximation sampling

Abstract

In this article, we propose a new dimensionality-independent and gradient-free sampler, called Geometric Optics Approximation Sampling, which is based on the reflector antenna system. The core idea is to construct a reflecting surface that redirects rays from a source with a predetermined simpler measure towards a output domain while achieving a desired distribution defined by the projection of a complex target measure of interest. Given a such reflecting surface, one can generate arbitrarily many independent and uncorrelated samples from the target measure simply by dual re-simulating or rays tracing the reflector antenna system and then projecting the traced rays onto target domain. In order to obtain a desired reflecting surface, we use the method of supporting paraboloid to solve the reflector antenna problem that does not require a gradient information regarding the density of the target measure. Furthermore, within the supporting paraboloid method, we utilize a low-discrepancy sequence or a random sequence to discretize the target measure, which in turn yields a dimensionality-independent approach for constructing the reflecting surface. Meanwhile, we present a dual re-simulation or ray tracing method based on its dual reflecting surface, which enables drawing samples from the target measure using the reflector antenna system obtained through the dimensionality-independent method. Several examples and numerical experiments comparing with measure transport samplers as well as traditional Markov chain Monte Carlo simulations are provided in this paper to demonstrate the efficiency and applicability of our geometric optics approximation sampling, especially in the context of Bayesian inverse problems. Additionally, these numerical results confirm the theoretical findings.

Figures (10)

• Figure 1.1: Draw samples $Q(y)$ from the non-Gaussian target distribution with Funnel geometry $\mu_t$ using geometric optics approximation sampling method in $\mathbb{R}^3$
• Figure 3.1: Uniform sequence (a), Hammersley sequence (b) and random sequence (c) for discretizing target distribution.
• Figure 3.2: The dual reflector antenna system
• Figure 5.1: Comparison of GOAS and MCMCs for non-Gaussian distribution sampling. True densities and the kernel density estimations obtained from the GOAS and MCMCs simulation (the first five columns), as well as the computational time in seconds and number of model evaluations provided by different methods regarding the ESS (the sixth column) for sampling the MoG distribution.
• Figure 5.2: Kernel density estimations using $5\times 10^3$ samples obtained by GOAS with the increasing number of parabolas $K$ and TM with Hermite polynomials degree $P=1,\; P=2,\; P=3$ for Biochemical Oxygen Demand model (a)-(c) and Euler Bernoulli beam problem (d)-(f). The estimates of the MCMC simulation, i.e., the red line, serves as the "true" density.

Theorems & Definitions (39)

• Definition 2.1: supporting paraboloid
• Definition 2.2: convex reflecting surface
• Definition 2.3: subdifferential
• Remark 2.1
• Lemma 2.1
• proof
• Definition 2.4: weak solution
• Theorem 2.2
• Theorem 2.3: existence, uniqueness and regularity
• Theorem 2.4: stability regarding target domain