Abstract

We introduce a fast iterative and a fast direct algorithm for rapid solution of frequency-domain integral-equation formulations pertinent to 3-D electromagnetic modeling and inversion. The iterative algorithm is based on the adaptive integral/pre-corrected FFT method: It accelerates the matrix-vector products at each iteration by enclosing the arbitrary primary mesh of the anomalous domain of interest with a single auxiliary regular grid, approximating interactions among basis and testing functions via grid-based anterpolation, propagation, and interpolation, using 3-D FFTs to multiply grid-to-grid propagation matrices with vectors, and pre-correcting inaccurate approximations for nearby interactions. The direct algorithm is based on a H-matrix framework: it accelerates the matrix factorization by constructing a hierarchy of matrix blocks that represent interactions between pairs of subdomains, compressing admissible interactions between well-separated subdomains with localized auxiliary regular grids, computing low-rank approximations of grid-to-grid propagation matrices, and using H-matrix arithmetic. The performance of the two algorithms is compared for computing the electromagnetic response of propped hydraulic fractures. Distributed-memory parallelization of the iterative algorithm, which has recently been scaled up to solve large-scale problems with >109 degrees of freedom using >104 processes, enables effective calculation of the response of very complex and large-scale fractures. The direct algorithm remains accurate, however, even as (a) the fracture conductivity increases relative to the background, (b) the conditioning of the system of equations and the convergence of its iterative solution deteriorates, and (c) the fast iterative algorithm fails to yield accurate results.