Abstract
In this work, we present a new numerical framework for the efficient solution of the time-harmonic elastic wave equation at multiple frequencies. We show that multiple frequencies (and multiple right-hand sides) can be incorporated when the discretized problem is written as a matrix equation. This matrix equation can be solved efficiently using the preconditioned IDR(s) method. We present an efficient and robust way to apply a single preconditioner using MSSS matrix computations. For 3D problems, we present a memory-efficient implementation that exploits the solution of a sequence of 2D problems. Realistic examples in two and three spatial dimensions demonstrate the performance of the new algorithm.
Highlights
The understanding of the earth subsurface is a key task in geophysics, and Full-Waveform Inversion (FWI) is a computational approach that matches the intensity of reflected shock waves with simulation results in a least squares sense; cf. [44] and the references therein for an overview of state-of-the-art FWI algorithms
We present an efficient solver of the time-harmonic elastic wave equation that results from a finite element discretization, cf. [11, 15]
We note that Induced Dimension Reduction (IDR) outperforms BiCGStab in terms of number of iterations
Summary
The understanding of the earth subsurface is a key task in geophysics, and Full-Waveform Inversion (FWI) is a computational approach that matches the intensity of reflected shock waves (measurements) with simulation results in a least squares sense; cf. [44] and the references therein for an overview of state-of-the-art FWI algorithms. In order to design an efficient optimization algorithm, a fast numerical solution of the elastic wave equation is required at every iteration of the optimization loop This so-called forward problem is the focus of this work. The forward problem requires the fast numerical solution of the discretized time-harmonic elastic wave equation at multiple wave frequencies and for multiple source terms. In this context, many efficient numerical solution methods have been proposed mostly for the (acoustic) Helmholtz equation [23, 25, 26, 33].
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