Evaluation of an iterative framework for geophysical electromagnetic forward and inverse modelling problems

Summary Modelling and inversion of controlled-source electromagnetic data requires elaborate numerical tools. The major challenge is the high computational cost of computing solutions to numerous forward problems (for the forward responses as well as the sensitivity matrix). Forward modelling is accomplished using either a direct or an iterative solver. Current modelling suites predominantly employ direct solution methods in the forward operator since multiple solutions are easily accessible using inexpensive and quick forward-backward substitution after an initial resource-demanding matrix factorisation step. Iterative techniques, on the other hand, require little resources for single forward solutions, and are yet very time consuming if solutions for many right-hand sides are to be computed. Evaluations of different solution techniques for modelling and inverse problems are only sparsely investigated. In light of this, we integrated an iterative solver as alternative in the forward and and inversion operators of the open-source software custEM and pyGIMLi. In particular, we implemented a two-level iterative scheme where the outer solver employs a generalised conjugate residual algorithm preconditioned with a highly efficient block-based preconditioner for square blocks. The inner-level solver is either of the same type as the outer solver, but preconditioned with the auxiliary-space Maxwell preconditioner, or may alternatively be a direct solver. In this paper, we evaluate the described iterative forward operator for forward modelling tasks for the Marlim R3D model for a single as well as numerous right-hand side vectors and compare the performance to the direct solver MUMPS. We further investigate the solver’s applicability on small and medium-sized computing platforms. We then examine the iterative solver for inversions of synthetic land-based and semi-airborne data in terms of computational requirements. Our results demonstrate that forward modelling tasks are best performed using an iterative approach for single source problems. Moreover, simulations of large and complex problems are accessible on even on small computing platforms such as laptops in very reasonable time. For inversions, the iterative forward operator, in particular the mixed iterative-direct-based one, performs equally well in terms of time as the direct one while reducing the memory demands for the computations of the forward responses and the data sensitivities.