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arXiv2026

Geometric Optics Approximation Sampling: A Reflector-Induced Transport Map Framework

In this paper, we propose Geometric Optics Approximation Sampling (GOAS), a reflector-induced transport-map framework for sampling from target measures. Once a reflecting surface is constructed, the associated transport map is explicitly determined by the physical law of reflection. As a concrete realization, we develop a supporting-hyperellipsoid construction that requires only a discrete approximation of the target measure and does not require gradient information of the target density. The formulation accommodates both density-based and sample-based target representations. A softmin smoothing technique is introduced to obtain a smooth approximate transport map from this piecewise hyperellipsoidal construction. We establish well-posedness and stability of the reflector-induced push-forward measure and derive quantitative error estimates in the maximum mean discrepancy metric, and convergence of continuous statistical observables, including fixed-order moments. Numerical experiments on an analytically tractable example, strongly non-Gaussian targets, sample-based target approximations, and Bayesian inverse problems demonstrate the accuracy and flexibility of GOAS.

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arXiv2026

Geometric Optics Approximation Sampling: A Far-Field Reflector-Induced Transport Framework

We develop a far-field geometric optics approximation sampling (GOAS) framework for constructing direct samplers from target measures. The method exploits the connection between the far-field reflector problem and optimal transport with logarithmic cost, leading to a natural primal--dual transport structure. The associated dual reflector provides a reciprocal backward transport and, in the invertible smooth setting, the inverse of the forward reflector map. For numerical realization, we adopt a supporting hyperparaboloid construction based on a discrete approximation of the target measure. This construction is gradient-free with respect to the target density and naturally accommodates both density-based and sample-based target representations. The resulting piecewise reflector admits two sampling realizations: a primal--dual consistency resampling strategy that operates directly on the nonsmooth reflector, and a softmin-regularized realization yielding an explicit smooth transport map through the physical law of reflection. We establish the well-posedness and stability of the reflector-induced sampling measure and derive Wasserstein error estimates. Numerical experiments on non-Gaussian targets and Bayesian inverse problems demonstrate the accuracy, stability, Wasserstein error behavior, and applicability of the proposed framework, and compare it with MCMC and polynomial transport-map methods.

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