580 research outputs found
On dimension reduction in Gaussian filters
A priori dimension reduction is a widely adopted technique for reducing the
computational complexity of stationary inverse problems. In this setting, the
solution of an inverse problem is parameterized by a low-dimensional basis that
is often obtained from the truncated Karhunen-Loeve expansion of the prior
distribution. For high-dimensional inverse problems equipped with smoothing
priors, this technique can lead to drastic reductions in parameter dimension
and significant computational savings.
In this paper, we extend the concept of a priori dimension reduction to
non-stationary inverse problems, in which the goal is to sequentially infer the
state of a dynamical system. Our approach proceeds in an offline-online
fashion. We first identify a low-dimensional subspace in the state space before
solving the inverse problem (the offline phase), using either the method of
"snapshots" or regularized covariance estimation. Then this subspace is used to
reduce the computational complexity of various filtering algorithms - including
the Kalman filter, extended Kalman filter, and ensemble Kalman filter - within
a novel subspace-constrained Bayesian prediction-and-update procedure (the
online phase). We demonstrate the performance of our new dimension reduction
approach on various numerical examples. In some test cases, our approach
reduces the dimensionality of the original problem by orders of magnitude and
yields up to two orders of magnitude in computational savings
Statistical Inference for Spatiotemporal Partially Observed Markov Processes via the R Package spatPomp
We consider inference for a class of nonlinear stochastic processes with
latent dynamic variables and spatial structure. The spatial structure takes the
form of a finite collection of spatial units that are dynamically coupled. We
assume that the latent processes have a Markovian structure and that
unit-specific noisy measurements are made. A model of this form is called a
spatiotemporal partially observed Markov process (SpatPOMP). The R package
spatPomp provides an environment for implementing SpatPOMP models, analyzing
data, and developing new inference approaches. We describe the spatPomp
implementations of some methods with scaling properties suited to SpatPOMP
models. We demonstrate the package on a simple Gaussian system and on a
nontrivial epidemiological model for measles transmission within and between
cities. We show how to construct user-specified SpatPOMP models within
spatPomp
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Chaotic dynamics and the role of covariance inflation for reduced rank Kalman filters with model error
The ensemble Kalman filter and its variants have shown to be robust for data assimilation in high dimensional geophysical models, with localization, using ensembles of extremely small size relative to the model dimension. However, a reduced rank representation of the estimated covariance leaves a large dimensional complementary subspace unfiltered. Utilizing the dynamical properties of the filtration for the backward Lyapunov vectors, this paper explores a previously unexplained mechanism, providing a novel theoretical interpretation for the role of covariance inflation in ensemble-based Kalman filters. Our derivation of the forecast error evolution describes the dynamic upwelling of the unfiltered error from outside of the span of the anomalies into the filtered subspace. Analytical results for linear systems explicitly describe the mechanism for the upwelling, and the associated recursive Riccati equation for the forecast error, while nonlinear approximations are explored numerically
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