Diffusion Estimation from Multiscale Data by Operator Eigenpairs
Daan T. Crommelin, Eric Vanden‐Eijnden · Multiscale Modeling and Simulation · 2011
In this paper we present a new procedure for the estimation of diffusion processes from discretely sampled data. It is based on the close relation between eigenpairs of the diffusion operator [Formula: see text] and those of the conditional expectation operator [Formula: see text], a relation stemming from the semigroup structure [Formula: see text] for [Formula: see text]. It allows for estimation without making time discretization errors, an aspect that is particularly advantageous in the case of data with low sampling frequency. After estimating eigenpairs of [Formula: see text] via eigenpairs of [Formula: see text], we infer the drift and diffusion functions that determine [Formula: see text] by fitting [Formula: see text] to the estimated eigenpairs using a convex optimization procedure. We present numerical examples in which we apply the procedure to one- and two-dimensional diffusions, reversible as well as nonreversible. In the second part of the paper, we consider estimation of coarse-grained (homogenized) diffusion processes from multiscale data. We show that eigenpairs of the homogenized diffusion operator are asymptotically close to eigenpairs of the underlying multiscale diffusion operator. This implies that we can infer the correct homogenized process from data of the multiscale process, using the estimation procedure discussed in the first part of the paper. This is illustrated with numerical examples.