Matrix probing: a randomized preconditioner for the wave-equation Hessian

TL;DR

This paper introduces a randomized method to approximate the inverse wave-equation Hessian using curvelet-based trial functions, enabling efficient seismic imaging and inversion.

Contribution

It develops a novel expansion scheme for the inverse Hessian's symbol and demonstrates its effectiveness as a low-complexity preconditioner in seismic inversion.

Findings

The method accurately approximates the inverse Hessian with fewer applications of the normal operator.

Randomized trial functions in curvelet space improve parameter fitting.

Numerical experiments confirm the preconditioner's effectiveness in seismic imaging.

Abstract

This paper considers the problem of approximating the inverse of the wave-equation Hessian, also called normal operator, in seismology and other types of wave-based imaging. An expansion scheme for the pseudodifferential symbol of the inverse Hessian is set up. The coefficients in this expansion are found via least-squares fitting from a certain number of applications of the normal operator on adequate randomized trial functions built in curvelet space. It is found that the number of parameters that can be fitted increases with the amount of information present in the trial functions, with high probability. Once an approximate inverse Hessian is available, application to an image of the model can be done in very low complexity. Numerical experiments show that randomized operator fitting offers a compelling preconditioner for the linearized seismic inversion problem.