this work concerns the zeroth-order global minimization of continuous nonconvex functions with a unique global minimizer and possibly multiple local minimizers. we formulate a theoretical framework for inexact proximal point (ipp) methods for global optimization, establishing convergence guarantees under mild assumptions when either deterministic or stochastic estimates of proximal operators are used. the quadratic regularization in the proximal operator and the scaling effect of a positive parameter create a concentrated landscape of an associated gibbs measure that is practically effective for sampling. the convergence of the expectation under the gibbs measure is established, providing a theoretical foundation for evaluating proximal operators inexactly using sampling-based methods such as monte carlo (mc) integration. in addition, we propose a new approach based on tensor train (tt) approximation. this approach employs a randomized tt cross algorithm to efficiently construct a low-rank tt approximation of a discretized function using a small number of function evaluations, and we provide an error analysis for the tt-based estimation. we then propose two practical ipp algorithms, tt-ipp and mc-ipp. the tt-ipp algorithm leverages tt estimates of the proximal operators, while the mc-ipp algorithm employs mc integration to estimate the proximal operators. both algorithms are designed to adaptively balance efficiency and accuracy in inexact evaluations of proximal operators. the effectiveness of the two algorithms is demonstrated through experiments on diverse benchmark functions and various applications.

i will review a general framework for developing numerical methods working with non-parametrically defined surfaces for various problems involving. the main idea is to formulate appropriate extensions of a given problem defined on a surface to ones in the narrow band of the surface in the embedding space. the extensions are arranged so that the solutions to the extended problems are equivalent, in a strong sense, to the surface problems that we set out to solve. such extension approaches allow us to analyze the well-posedness of the resulting system, develop, systematically and in a unified fashion, numerical schemes for treating a wide range of problems involving differential and integral operators, and deal with similar problems in which only point clouds sampling the surfaces are given.