Abstract

The increasing availability of data presents an opportunity to calibrate unknown parameters which appear in complex models of phenomena in the biomedical, physical, and social sciences. However, model complexity often leads to parameter-to-data maps which are expensive to evaluate and are only available through noisy approximations. This paper is concerned with the use of interacting particle systems for the solution of the resulting inverse problems for parameters. Of particular interest is the case where the available forward model evaluations are subject to rapid fluctuations, in parameter space, superimposed on the smoothly varying large-scale parametric structure of interest. A motivating example from climate science is presented, and ensemble Kalman methods (which do not use the derivative of the parameter-to-data map) are shown, empirically, to perform well. Multiscale analysis is then used to analyze the behavior of interacting particle system algorithms when rapid fluctuations, which we refer to as noise, pollute the large-scale parametric dependence of the parameter-to-data map. Ensemble Kalman methods and Langevin-based methods (the latter use the derivative of the parameter-to-data map) are compared in this light. The ensemble Kalman methods are shown to behave favorably in the presence of noise in the parameter-to-data map, whereas Langevin methods are adversely affected. On the other hand, Langevin methods have the correct equilibrium distribution in the setting of noise-free forward models, while ensemble Kalman methods only provide an uncontrolled approximation, except in the linear case. Therefore a new class of algorithms, ensemble Gaussian process samplers, which combine the benefits of both ensemble Kalman and Langevin methods, are introduced and shown to perform favorably.

Keywords

  1. ensemble methods
  2. ensemble Kalman sampler
  3. Langevin sampling
  4. Gaussian process regression
  5. multiscale analysis

MSC codes

  1. 60H30
  2. 35B27
  3. 60G15
  4. 82C80
  5. 65C35
  6. 62F15

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Information & Authors

Information

Published In

cover image SIAM Journal on Applied Dynamical Systems
SIAM Journal on Applied Dynamical Systems
Pages: 1539 - 1572
ISSN (online): 1536-0040

History

Submitted: 8 April 2021
Accepted: 8 March 2022
Published online: 21 June 2022

Keywords

  1. ensemble methods
  2. ensemble Kalman sampler
  3. Langevin sampling
  4. Gaussian process regression
  5. multiscale analysis

MSC codes

  1. 60H30
  2. 35B27
  3. 60G15
  4. 82C80
  5. 65C35
  6. 62F15

Authors

Affiliations

Funding Information

Austrian Academy of Sciences : NST-0001
Alan Turing Institute https://doi.org/10.13039/100012338
Office of Naval Research https://doi.org/10.13039/100000006 : N00014-17-1-2079
National Science Foundation https://doi.org/10.13039/100000001 : DMS-1818977, AGS-1835860
Royal Society https://doi.org/10.13039/501100000288
Engineering and Physical Sciences Research Council https://doi.org/10.13039/501100000266 : EP/T001569/1

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