Efficient management of parallelism in object-oriented numerical software libraries
Modern software tools for scientific computing
Estimation Techniques for Distributed Parameter Systems
Estimation Techniques for Distributed Parameter Systems
Computational Methods for Inverse Problems
Computational Methods for Inverse Problems
A variational finite element method for source inversion for convective-diffusive transport
Finite Elements in Analysis and Design - Special issue: 14th Robert J. Melosh competition
A Computational Strategy for the Solution of Large Linear Inverse Problems in Geophysics
IPDPS '03 Proceedings of the 17th International Symposium on Parallel and Distributed Processing
Inverse Problem Theory and Methods for Model Parameter Estimation
Inverse Problem Theory and Methods for Model Parameter Estimation
SLEPc: A scalable and flexible toolkit for the solution of eigenvalue problems
ACM Transactions on Mathematical Software (TOMS) - Special issue on the Advanced CompuTational Software (ACTS) Collection
SC '05 Proceedings of the 2005 ACM/IEEE conference on Supercomputing
Lagrange Multiplier Approach to Variational Problems and Applications
Lagrange Multiplier Approach to Variational Problems and Applications
Extreme-scale UQ for Bayesian inverse problems governed by PDEs
SC '12 Proceedings of the International Conference on High Performance Computing, Networking, Storage and Analysis
Journal of Computational Physics
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We consider the problem of estimating the uncertainty in large-scale linear statistical inverse problems with high-dimensional parameter spaces within the framework of Bayesian inference. When the noise and prior probability densities are Gaussian, the solution to the inverse problem is also Gaussian and is thus characterized by the mean and covariance matrix of the posterior probability density. Unfortunately, explicitly computing the posterior covariance matrix requires as many forward solutions as there are parameters and is thus prohibitive when the forward problem is expensive and the parameter dimension is large. However, for many ill-posed inverse problems, the Hessian matrix of the data misfit term has a spectrum that collapses rapidly to zero. We present a fast method for computation of an approximation to the posterior covariance that exploits the low-rank structure of the preconditioned (by the prior covariance) Hessian of the data misfit. Analysis of an infinite-dimensional model convection-diffusion problem, and numerical experiments on large-scale three-dimensional convection-diffusion inverse problems with up to 1.5 million parameters, demonstrate that the number of forward PDE solves required for an accurate low-rank approximation is independent of the problem dimension. This permits scalable estimation of the uncertainty in large-scale ill-posed linear inverse problems at a small multiple (independent of the problem dimension) of the cost of solving the forward problem.