DIFFUSION ESTIMATION FROM MULTISCALE DATA BY OPERATOR EIGENPAIRS
DIFFUSION ESTIMATION FROM MULTISCALE DATA BY OPERATOR EIGENPAIRS
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DOI：
10.1137/100795917
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发表时间：
2011-01-01
影响因子：
1.6
通讯作者：
Vanden-Eijnden, Eric
中科院分区：
文献类型：
--
作者：
Crommelin, Daan;Vanden-Eijnden, Eric
本文提出了一种新的离散采样数据扩散过程的估计方法。它是基于扩散算子L的本征对与条件期望算子P-t的本征对之间的密切关系，这种关系源于半群结构P-t = exp（tL），其中t >= 0。它允许在不产生时间离散化误差的情况下进行估计，这是在具有低采样频率的数据的情况下特别有利的方面。在通过P-t的本征对估计L的本征对之后，我们通过使用凸优化过程将L拟合到估计的本征对来推断确定L的漂移和扩散函数。我们提出的数值例子中，我们适用于一维和二维扩散，可逆以及不可逆的程序。在本文的第二部分，我们考虑估计粗粒度（均匀化）的扩散过程从多尺度数据。我们证明了均匀化扩散算子的本征对渐进地接近于底层多尺度扩散算子的本征对。这意味着，我们可以推断出正确的均匀化过程的多尺度过程的数据，使用本文第一部分中讨论的估计过程。这是用数值例子说明。
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 L and those of the conditional expectation operator P-t, a relation stemming from the semigroup structure P-t = exp(tL) for t >= 0. 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 L via eigenpairs of P-t, we infer the drift and diffusion functions that determine L by fitting L 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.