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

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. Acknowledgments. LD would like to thank Rami Nammour and William Symes for introducing him to their work. LD, PDL, and NB are supported by a grant from Total SA.