Fast calculations of the gradient and the Hessian in the time-domain global electromagnetic induction inverse problem

Summary The deterministic approach to the inverse problem of the time-domain electromagnetic induction in a spherical Earth requires the calculation of the first derivative of a misfit function in every step of the minimization process. In addition, an a-posteriori error analysis can benefit from the knowledge of the Hessian, the matrix of the second derivatives. We present the derivation of the formulas for the fast calculation of the misfit gradient, the Hessian, and the Hessian-vector product, based on the solution of an adjoint problem. We validate our results on a synthetic model against a slow finite-difference scheme.