Frugal FWI

Seismic waveform inversion aims at obtaining detailed estimates of subsurface medium parameters, such as soundspeed, from seismic data. A formulation in the frequency-domain leads to an optimization problem constrained by a Helmholtz equation with many right-hand- sides. Application of this technique in 3D precludes the use of factorization techniques to solve the Helmholtz equation due the large number of gridpoints and the bandwidth of the matrix. While many sophisticated pre-conditioned iterative techniques have been developed for the Helmholtz equation, they often include model-specific tuning parameters and are thus not very attractive for inversion since the medium parameters change from one iteration to the next. In this paper, we propose a method for 3D seismic waveform inversion that addresses both the need to efficiently solve the Helmholtz equation as well as the computational cost induced by the many right-hand-sides. To solve the Helmholtz equation, we consider a simple generic preconditioned iterative method (CARP-CG) that is well-suited for inversion because of its robustness. We extend this method to a block-iterative method that can efficiently handle multiple right-hand sides. To reduce the computational cost of of the overall optimization procedure, we use recently proposed techniques from stochastic optimization that allow us to work with approximate gradient information. These approximations are obtained by evaluating only a small portion of the right-hand sides and/or by solving the PDE approximately. We propose heuristics to adaptively determine the required accuracy of the PDE solves and the sample-size and illustrate the algorithms on synthetic benchmark models. This is joint work with Tristan van Leeuwen.