A block rational Krylov method for 3-D time-domain marine controlled-source electromagnetic modelling

Abstract

We introduce a novel block rational Krylov method to accelerate 3-D time-domain marine controlled-source electromagnetic modelling with multiple sources. This method approximates the time-varying electric solutions explicitly in terms of matrix exponential functions. A main attraction is that no time stepping is required, while most of the computational costs are concentrated in constructing a rational Krylov basis. We optimize the shift parameters defining the rational Krylov space with a positive exponential weight function, thereby producing smaller approximation errors at later times and reducing iteration numbers. Furthermore, for multisource modelling problems, we adopt block Krylov techniques to incorporate all source vectors in a single approximation space. The method is tested on two examples: a layered seafloor model and a 3-D hydrocarbon reservoir model with seafloor bathymetry. The modelling results are found to agree very well with those from 1-D semi-analytic solutions and finite-element time-domain solutions using a backward Euler scheme, respectively. Numerical benchmarks show that the block method benefits from better memory and cache efficiency, resulting in about 1.26- to 1.48-fold speedup compared to non-block methods. Further efficiency gains are achieved through optimized rational Krylov techniques, allowing our approach to outperform classical time stepping schemes.