Deterministic and Heuristic Methods for Shape Reconstruction Problems Under Robin Boundary Conditions

Shape optimization problems with Robin boundary conditions have attracted considerable interest and remain a challenging task from a numerical point of view. In this paper, we focus on establishing a mathematical analysis of this class of inverse problems and on constructing both deterministic and heuristic methods for their numerical approximation. As a first step, we derive a constrained optimization problem from the considered ill-posed problem, then we show the optimal solution existence and uniqueness. Using the Lagrangian functional, we compute the first-order optimality conditions as well as the shape gradient and Hessian. For all our proposed numerical methods, we employ the finite element method to discretize the equations. The deterministic minimization algorithms considered are the conjugate gradient and the quasi-Newton techniques, for which we establish the convergence analysis. As heuristic methods, we adopt the differential evolution and genetic algorithms. In addition, we develop a hybrid minimization approach, a quasi-Newton method guided by differential evolution. At the end, we conduct several numerical tests to demonstrate the validity of our proposed methods and provide a comparative assessment of their performance.