Vol. 10, No. 2, 2015

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ISSN: 2157-5452 (e-only)
ISSN: 1559-3940 (print)
Parameter estimation by implicit sampling

Matthias Morzfeld, Xuemin Tu, Jon Wilkening and Alexandre J. Chorin

Vol. 10 (2015), No. 2, 205–225
Abstract

Implicit sampling is a weighted sampling method that is used in data assimilation to sequentially update state estimates of a stochastic model based on noisy and incomplete data. Here we apply implicit sampling to sample the posterior probability density of parameter estimation problems. The posterior probability combines prior information about the parameter with information from a numerical model, e.g., a partial differential equation (PDE), and noisy data. The result of our computations are parameters that lead to simulations that are compatible with the data. We demonstrate the usefulness of our implicit sampling algorithm with an example from subsurface flow. For an efficient implementation, we make use of multiple grids, BFGS optimization coupled to adjoint equations, and Karhunen–Loève expansions for dimensional reduction. Several difficulties of Markov chain Monte Carlo methods, e.g., estimation of burn-in times or correlations among the samples, are avoided because the implicit samples are independent.

Keywords
importance sampling, implicit sampling, Markov chain Monte Carlo
Mathematical Subject Classification 2010
Primary: 86-08, 65C05
Milestones
Received: 23 June 2015
Accepted: 25 June 2015
Published: 4 September 2015
Authors
Matthias Morzfeld
Department of Mathematics
University of California, Berkeley
Evans Hall
Berkeley, CA 94720
United States
Lawrence Berkeley National Laboratory
1 Cyclotron Road
Berkeley, CA 94720
United States
Xuemin Tu
Department of Mathematics
University of Kansas
1460 Jayhawk Boulevard
Lawrence, KS 66045
United States
Jon Wilkening
Department of Mathematics
University of California, Berkeley
Evans Hall
Berkeley, CA 94720
United States
Lawrence Berkeley National Laboratory
1 Cyclotron Road
Berkeley, CA 94720
United States
Alexandre J. Chorin
Department of Mathematics
University of California, Berkeley
Evans Hall
Berkeley, CA 94720
United States
Lawrence Berkeley National Laboratory
1 Cyclotron Road
Berkeley, CA 94720
United States