ANALYSIS OF SPDES ARISING IN PATH SAMPLING PART II: THE NONLINEAR CASE

By Martin Hairer, Andrew M. Stuart, Jochen Voss. 2006

141 citations recorded by the source corpus.

In many applications, it is important to be able to sample paths of SDEs conditional on observations of various kinds. This paper studies SPDEs which solve such sampling problems. The SPDE may be viewed as an infinite-dimensional analogue of the Langevin equation used in finite-dimensional sampling. In this paper, conditioned nonlinear SDEs, leading to nonlinear SPDEs for the sampling, are studied. In addition, a class of preconditioned SPDEs is studied, found by applying a Green’s operator to the SPDE in such a way that the invariant measure remains unchanged; such infinite dimensional evolution equations are important for the development of practical algorithms for sampling infinite dimensional problems. The resulting SPDEs provide several significant challenges in the theory of SPDEs. The two primary ones are the presence of nonlinear boundary conditions, involving first order derivatives, and a loss of the smoothing property in the case of the pre-conditioned SPDEs. These challenges are overcome and a theory of existence, uniqueness and ergodicity is developed in sufficient generality to subsume the sampling problems of interest to us. The Gaussian theory developed in Part I of this paper considers Gaussian SDEs, leading to linear Gaussian SPDEs for sampling. This Gaussian theory is used as the basis for deriving nonlinear SPDEs which affect the desired sampling in the nonlinear case, via a change of measure.

Subfields: Environmental Engineering, Statistical and Nonlinear Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/ddd6368d2cfa0b50e28c54b0fb0acaa641faaac9

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Citations in the corpus

Besov priors for Bayesian inverse problems (2011, unknown): “ore π ≪ π0 and π has Radon-Nikodym derivative given by (12) as (noting that by (13) and sinceR µ0(X) = 1, we have X×Y exp −Φ(u;y) π0(du,dy) = 1) dπ dπ0 (u;y) = exp −Φ(u;y) . This then by Lemma 5.3 of [18], implies that µy(du) = π(du,dξ|ξ= y) is absolutely continuous with respect to µ0(y) = π0(du,dξ|”
Hybrid Monte Carlo on Hilbert spaces (2011, unknown): “This issue was addressed in continuous time for the preconditioned Langevin SPDE in the paper [12], by use of the concept of “asymptotic strong Feller”; it is likely that similar ideas could be used in the discrete time setting of MCMC.”
Inverse problems: A Bayesian perspective (2010, methodology): “The approach to Bayes’ rule in infinite dimensions that we adopt in Theorem 6.31 was used to study a specific problem arising in signal processing in Hairer et al. (2007).”
Posterior consistency via precision operators for Bayesian nonparametric drift estimation in SDEs (2012, unknown): “By Lemma 5.3 of [17], the posterior is well-defined if Z > 0 a.s.”
The Bayesian Approach to Inverse Problems (2013, background): “The particular SPDE which arises as a diffusion limit of our Markov chain is studied in [HSV07], and is called the preconditioned SPDE.”
Bayesian inverse problems for functions and applications to fluid mechanics (2009, methodology): “1 arising in the study of conditioned diffusion processes, using the Gaussian Kalman–Bucy smoother measure from [32] as the reference measure.”
Diffusion limits of the random walk metropolis algorithm in high dimensions (2010, background): “In this latter case we expect to get a different (nonpreconditioned) π -invariant SPDE as the limit when the dimension goes to infinity (see [15, 17] for analysis of these SPDEs) and a different (more severe) restriction on the scaling of the proposal variance with N ; however, we conjecture that th”
MAP estimators and their consistency in Bayesian nonparametric inverse problems (2013, methodology): “hit u+ at time 1 so that u(T) = u+, and denote this measure . Let 0 denote the Brownian bridge measure obtained in the case f0. By applying the approach to determining bridge diusion measures in [17] we obtain, from (6.2), the expression d d 0 (u) /exp Z T u(t) dt+ 1 ˙2 F u+ : (6.4) Since u+ is x”
MCMC Methods for Functions: ModifyingOld Algorithms to Make Them Faster (2012, background): “The SPDE has invariant measure μ0 for γ = 0 (when it is an infinite dimensional OU process) and μ for γ = 1 [25, 22, 16].”