On Friday, October 3, a seminar will be held at 11:00 a.m. in Room 087 (Building 4), entitled “Solution of Inverse Problems of the Heat Equation using Shape Optimization and a Fast Parabolic Boundary Element Solver”.

The seminar will be given by Prof. Johannes Tausch, Southern Methodist University.

Abstract

A typical inverse problem is to determine the shape of an inclusion from measurements of temperatures and heat fluxes on the exterior boundary. This is known to be a severely ill posed problem. One way to improve the conditioning is to incorporate transient temperature information which leads to an inverse problem of the transient heat equation.

Our approach is to minimize the difference of the boundary data obtained with a given shape and the measured data. In addition, descent based optimization will be applied to find the optimal shape. To that end, the concept of the shape gradient is introduced, which describes the change of the solution of the heat equation with respect to changing the geometry of the domain. We will then show that the shape gradient can be computed by solving a state and an adjoint equation, which are forward problems of the heat equation.

The latter problems are be solved using parabolic boundary integral equations. Here a fast method is used to efficiently evaluate the time convolutions of parabolic layer potentials.

The talk will conclude with numerical experiments that illustrate the efficiency and reconstruction quality of the approach.