Topological sensitivity analysis for the narrow escape problem

Abstract

The topological sensitivity analysis method has been recognized as a promising, fast, and accurate approach for solving topology optimization and inverse problems. It is based on developing an asymptotic expansion of a design functional with respect to the creation of a small hole inside the computational domain. In this work, we extend this method to the narrow escape problem. The biological process is governed by a parabolic diffusion equation. We derive a sensitivity analysis for the parabolic problem solution with respect to the creation of a small absorbing boundary subset. We develop a rigorous mathematical framework that is valid in two- and three-dimensional space. It provides an asymptotic formula that describes the behavior of the perturbed solution with respect to the location and size of an arbitrary perturbed boundary subset. The Sobolev capacity notion has been employed to measure the smallness of the boundary subset and to describe the asymptotic behavior with respect to the perturbation size. The performed mathematical analysis is general and can be adapted for a large class of partial differential equations. The obtained asymptotic formula can serve as a useful tool to perform numerical algorithms for solving optimization and control problems.