Low-Rank Eigenvector Compression of Posterior Covariance Matrices for Linear Gaussian Inverse Problems

We consider the problem of efficient computations of the covariance matrix of the posterior probability density for linear Gaussian Bayesian inverse problems. When the probability density of the noise and the prior are Gaussian, the solution of such a statistical inverse problem is also Gaussian. Therefore, the underlying solution is characterized by the mean and covariance matrix of the posterior probability density. However, the covariance matrix of the posterior probability density is dense and large. Hence, the computation of such a matrix is impossible for large dimensional parameter spaces as is the case for discretized PDEs. Low-rank approximations to the posterior covariance matrix were recently introduced as promising tools. Nevertheless, for transient problems the resulting approximation suffers from an increased dimensionality. We here exploit the structure of the discretized equations in such a way that spatial and temporal components can be separated and the growing complexity is dramatically reduced. In particular, the storage for an eigenvector low-rank approximation up to now was dominated by the computation and storage complexity of O(n(x)n(t)), where n(x) is the dimension of the spatial domain and n t is the dimension of the time domain. We develop a new approach that utilizes a low-rank in time algorithm together with the low-rank Hessian method. We reduce both the computational complexity and storage requirement from O(n(x)n(t)) to O(n(x) + n(t)). We use numerical experiments to illustrate the advantages of our approach.