An efficient method to solve large linearizable inverse problems under Gaussian and separability assumptions

Abstract

Inverse problems where relationships are linear arise in many fields of science and engineering and, consequently, algorithms for solving them are widespread. However, when the size of the problem increases, the computational challenge becomes huge, hence, unless some simplifying assumptions are yielded, it becomes impossible to solve the problem. Our study addresses these issues for large linear(izable) inverse problems when the uncertainties can be modeled as Gaussian and the forward relationship and covariance matrices can be expressed in terms of Kronecker products. Under these conditions, we illustrate an algorithm capable of addressing very large problems with very limited storage requirements and much faster than the traditional approach. The result is a complete characterization of the posterior distribution in a probabilistic sense in terms of mean and covariance. We extend this method also to nonlinear problems where a Gauss-Newton algorithm is employed. Applications to reflection seismology, magnetic anomaly inversion and image restoration are presented.