Schillings, Claudia and Sprungk, Björn and Wacker, Philipp
(2020)
*On the convergence of the Laplace approximation and
noise-level-robustness of Laplace-based Monte Carlo
methods for Bayesian inverse problems.*
Numerische Mathematik, 145
.
pp. 915-971.

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Official URL: https://doi.org/10.1007/s00211-020-01131-1

## Abstract

The Bayesian approach to inverse problems provides a rigorous framework for the incorporation and quantification of uncertainties in measurements, parameters and models. We are interested in designing numerical methods which are robust w.r.t. the size of the observational noise, i.e., methods which behave well in case of concentrated posterior measures. The concentration of the posterior is a highly desirable situation in practice, since it relates to informative or large data. However, it can pose a computational challenge for numerical methods based on the prior measure. We propose to employ the Laplace approximation of the posterior as the base measure for numerical integration in this context. The Laplace approximation is a Gaussian measure centered at the maximum a-posteriori estimate and with covariance matrix depending on the logposterior density. We discuss convergence results of the Laplace approximation in terms of the Hellinger distance and analyze the efficiency of Monte Carlo methods based on it. In particular, we show that Laplace-based importance sampling and Laplace-based quasi-Monte-Carlo methods are robust w.r.t. the concentration of the posterior for large classes of posterior distributions and integrands whereas prior-based importance sampling and plain quasi-Monte Carlo are not. Numerical experiments are presented to illustrate the theoretical findings.

Item Type: | Article |
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Subjects: | Mathematical and Computer Sciences > Mathematics > Applied Mathematics |

Divisions: | Department of Mathematics and Computer Science > Institute of Mathematics > Deterministic and Stochastic PDEs Group |

ID Code: | 2692 |

Deposited By: | Monika Drueck |

Deposited On: | 25 Jan 2022 15:54 |

Last Modified: | 12 Apr 2022 11:13 |

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