Summary
Proceedings of the 2013 International Symposium on Nonlinear Theory and its Applications
2013
Session Number:C1L-A
Session:
Number:345
Inference in nonlinear dynamical systems using transport maps
Youssef Marzouk, Tarek Moselhy,
pp.345-345
Publication Date:
Online ISSN:2188-5079
Summary:
We present a new approach to Bayesian inference that entirely avoids Markov chain simulation or sequential importance resampling, by constructing a map that pushes forward the prior measure to the posterior measure [1]. Existence and uniqueness of a suitable measure-preserving map is established by formulating the problem in the context of optimal transportation [2]. The map is computed efficiently through the solution of a stochastic optimization problem, using a sample-average approximation approach. Advantages of a map-based representation include analytical expressions for posterior moments and the ability to generate arbitrary numbers of independent and uniformly-weighted posterior samples without additional evaluations of the dynamical model.We then focus on various means of explicitly parameterizing the map in order to take advantage of low-dimensional structure present in many problems of parameter inference and state estimation. Numerical demonstrations include largescale inverse problems involving partial differential equations and sequential data assimilation (filtering) in nonlinear and chaotic dynamical systems of increasing complexity.
References:
[1] T. Moselhy, Y. Marzouk, Bayesian inference with optimal maps, Journal of Computational Physics 231 (2012), 7815-7850.
[2] C. Villani, Optimal Transport: Old and New, Springer (2009).