Fast alogorithms for Bayesian uncertainty quantification in large-scale linear inverse problems based on low-rank partial Hessian approximations
- ORNL
- University of Texas, Austin
- Sandia National Laboratories (SNL)
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 lowrank 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 3D 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.
- Research Organization:
- Oak Ridge National Laboratory (ORNL)
- Sponsoring Organization:
- ORNL LDRD Director's R&D
- DOE Contract Number:
- AC05-00OR22725
- OSTI ID:
- 1006461
- Journal Information:
- SIAM Journal on Scientific Computing, Journal Name: SIAM Journal on Scientific Computing Journal Issue: 1 Vol. 33; ISSN 1064-8275; ISSN SJOCE3
- Country of Publication:
- United States
- Language:
- English
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