1 research outputs found
Uncertainty quantification in large Bayesian linear inverse problems using Krylov subspace methods
For linear inverse problems with a large number of unknown parameters,
uncertainty quantification remains a challenging task. In this work, we use
Krylov subspace methods to approximate the posterior covariance matrix and
describe efficient methods for exploring the posterior distribution. Assuming
that Krylov methods (e.g., based on the generalized Golub-Kahan
bidiagonalization) have been used to compute an estimate of the solution, we
get an approximation of the posterior covariance matrix for `free.' We provide
theoretical results that quantify the accuracy of the approximation and of the
resulting posterior distribution. Then, we describe efficient methods that use
the approximation to compute measures of uncertainty, including the
Kullback-Liebler divergence. We present two methods that use preconditioned
Lanczos methods to efficiently generate samples from the posterior
distribution. Numerical examples from tomography demonstrate the effectiveness
of the described approaches.Comment: 26 pages, 4 figures, 2 tables. Under revie