Compressive Wave Computation

TL;DR

This paper introduces a novel method for large-scale wave simulation using random eigenfunction decomposition, demonstrating that sparse wavefields can be accurately computed with significantly fewer eigenfunctions, enabling efficient parallel computation.

Contribution

It proposes a compressive wave computation framework based on L1-Helmholtz recovery, with theoretical guarantees for sparse wavefields in 1D, and demonstrates practical efficiency through numerical experiments.

Findings

Accurate wavefield computation with as few as 10% eigenfunctions.

Theoretical bounds on the number of eigenfunctions needed, proportional to log(N) log(log(N)).

Potential for parallelization in large-scale wave simulations and seismic inversion.

Abstract

This paper considers large-scale simulations of wave propagation phenomena. We argue that it is possible to accurately compute a wavefield by decomposing it onto a largely incomplete set of eigenfunctions of the Helmholtz operator, chosen at random, and that this provides a natural way of parallelizing wave simulations for memory-intensive applications. This paper shows that L1-Helmholtz recovery makes sense for wave computation, and identifies a regime in which it is provably effective: the one-dimensional wave equation with coefficients of small bounded variation. Under suitable assumptions we show that the number of eigenfunctions needed to evolve a sparse wavefield defined on N points, accurately with very high probability, is bounded by C log(N) log(log(N)), where C is related to the desired accuracy and can be made to grow at a much slower rate than N when the solution is sparse. The PDE estimates that underlie this result are new to the authors' knowledge and may be of independent mathematical interest; they include an L1 estimate for the wave equation, an estimate of extension of eigenfunctions, and a bound for eigenvalue gaps in Sturm-Liouville problems. Numerical examples are presented in one spatial dimension and show that as few as 10 percents of all eigenfunctions can suffice for accurate results. Finally, we argue that the compressive viewpoint suggests a competitive parallel algorithm for an adjoint-state inversion method in reflection seismology.