EFFICIENT NUMERICAL SOLUTION OF GEOMETRIC INVERSE PROBLEMS INVOLVING MAXWELL'S EQUATIONS USING SHAPE DERIVATIVES AND AUTOMATIC CODE GENERATION

We propose a novel approach using shape derivatives to solve sharp interface geometric inverse optimization problems governed by Maxwell's equations. A tracking-type target functional determines the distance between the solution of a 3D time-dependent Maxwell problem and given measured data in an L-2-norm. Minimization is conducted using H-1-gradient information based on shape derivatives, which is related to the shape Hessian of the problem regularization. We describe the underlying formulas and the derivation of appropriate upwind fluxes and arrive at shape gradients for general tracking-type objectives and conservation laws. Subsequently, an explicit boundary gradient formulation based on variational forms is given for the problem at hand. Using such variational forms as domain specific programming languages, the FEniCS environment can then automatically generate the solvers, leading to structure exploiting data efficient transient adjoints. Checkpointing strategies are not necessary. Numerical results of up to 1.2 . 10(9) state unknowns demonstrate the practicability of the proposed approach.