## Abstract

In this paper we study algorithms to find a Gaussian approximation to a target measure defined on a Hilbert space of functions; the target measure itself is defined via its density with respect to a reference Gaussian measure. We employ the Kullback-Leibler divergence as a distance and find the best Gaussian approximation by minimizing this distance. It then follows that the approximate Gaussian must be equivalent to the Gaussian reference measure, defining a natural function space setting for the underlying calculus of variations problem. We introduce a computational algorithm which is well-adapted to the required minimization, seeking to find the mean as a function, and parameterizing the covariance in two different ways: through low rank perturbations of the reference covariance and through Schrödinger potential perturbations of the inverse reference covariance. Two applications are shown: to a nonlinear inverse problem in elliptic PDEs and to a conditioned diffusion process. These Gaussian approximations also serve to provide a preconditioned proposal distribution for improved preconditioned Crank-Nicolson Monte Carlo-Markov chain sampling of the target distribution. This approach is not only well-adapted to the high dimensional setting, but also behaves well with respect to small observational noise (resp., small temperatures) in the inverse problem (resp., conditioned diffusion).

Original language | English |
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Pages (from-to) | A2733-A2757 |

Journal | SIAM Journal on Scientific Computing |

Volume | 37 |

Issue number | 6 |

DOIs | |

Publication status | Published - 1 Jan 2015 |

## Keywords

- Gaussian distributions
- Inverse problems
- Kullback-Leibler divergence
- MCMC
- Relative entropy

## ASJC Scopus subject areas

- Computational Mathematics
- Applied Mathematics