Georgia Institute of Technology College of Sciences

Uncertainty quantification for linear inverse problems remains a challenging task, especially for problems with a very large number of unknown parameters (e.g., dynamic inverse problems), for problems where computation of the square root and inverse of the prior covariance matrix are not feasible, and for hierarchical problems where the mean is not known a priori. This work exploits Krylov subspace methods to develop and analyze new techniques for large-scale uncertainty quantification in inverse problems. We assume that generalized Golub-Kahan based methods have been used to compute an estimate of the solution, and we describe efficient methods to explore the posterior distribution. We present two methods that use the preconditioned Lanczos algorithm to efficiently generate samples from the posterior distribution. Numerical examples from dynamic photoacoustic tomography and atmospheric inverse modeling, including a case study from NASA's Orbiting Carbon Observatory 2 (OCO-2) satellite, demonstrate the effectiveness of the described approaches.