58 research outputs found
Implicit sampling for path integral control, Monte Carlo localization, and SLAM
The applicability and usefulness of implicit sampling in stochastic optimal
control, stochastic localization, and simultaneous localization and mapping
(SLAM), is explored; implicit sampling is a recently-developed
variationally-enhanced sampling method. The theory is illustrated with
examples, and it is found that implicit sampling is significantly more
efficient than current Monte Carlo methods in test problems for all three
applications
Small-noise analysis and symmetrization of implicit Monte Carlo samplers
Implicit samplers are algorithms for producing independent, weighted samples
from multi-variate probability distributions. These are often applied in
Bayesian data assimilation algorithms. We use Laplace asymptotic expansions to
analyze two implicit samplers in the small noise regime. Our analysis suggests
a symmetrization of the algo- rithms that leads to improved (implicit) sampling
schemes at a rel- atively small additional cost. Computational experiments
confirm the theory and show that symmetrization is effective for small noise
sampling problems
Symmetrized importance samplers for stochastic differential equations
We study a class of importance sampling methods for stochastic differential
equations (SDEs). A small-noise analysis is performed, and the results suggest
that a simple symmetrization procedure can significantly improve the
performance of our importance sampling schemes when the noise is not too large.
We demonstrate that this is indeed the case for a number of linear and
nonlinear examples. Potential applications, e.g., data assimilation, are
discussed.Comment: Added brief discussion of Hamilton-Jacobi equation. Also made various
minor corrections. To appear in Communciations in Applied Mathematics and
Computational Scienc
Gaussian approximations in filters and smoothers for data assimilation
We present mathematical arguments and experimental evidence that suggest that Gaussian approximations of posterior distributions are appropriate even if the physical system under consideration is nonlinear. The reason for this is a regularizing effect of the observations that can turn multi-modal prior distributions into nearly Gaussian posterior distributions. This has important ramifications on data assimilation (DA) algorithms in numerical weather prediction because the various algorithms (ensemble Kalman filters/smoothers, variational methods, particle filters (PF)/smoothers (PS)) apply Gaussian approximations to different distributions, which leads to different approximate posterior distributions, and, subsequently, different degrees of error in their representation of the true posterior distribution. In particular, we explain that, in problems with medium' nonlinearity, (i) smoothers and variational methods tend to outperform ensemble Kalman filters; (ii) smoothers can be as accurate as PF, but may require fewer ensemble members; (iii) localization of PFs can introduce errors that are more severe than errors due to Gaussian approximations. In problems with strong' nonlinearity, posterior distributions are not amenable to Gaussian approximation. This happens, e.g. when posterior distributions are multi-modal. PFs can be used on these problems, but the required ensemble size is expected to be large (hundreds to thousands), even if the PFs are localized. Moreover, the usual indicators of performance (small root mean square error and comparable spread) may not be useful in strongly nonlinear problems. We arrive at these conclusions using a combination of theoretical considerations and a suite of numerical DA experiments with low- and high-dimensional nonlinear models in which we can control the nonlinearity.Office of Naval Research [N00173-17-2-C003, PE-0601153N]; Alfred P. Sloan Research Fellowship; National Science Foundation [DMS-1619630]Open access journalThis item from the UA Faculty Publications collection is made available by the University of Arizona with support from the University of Arizona Libraries. If you have questions, please contact us at [email protected]
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Data assimilation: mathematics for merging models and data
When you describe a physical process, for example, the weather on Earth, or an engineered system, such as a self-driving car, you typically have two sources of information. The first is a mathematical model, and the second is information obtained by collecting data. To make the best predictions for the weather, or most effectively operate the self-driving car, you want to use both sources of information. Data assimilation describes the mathematical, numerical and computational framework for doing just that
Localization for MCMC: sampling high-dimensional posterior distributions with local structure
We investigate how ideas from covariance localization in numerical weather
prediction can be used in Markov chain Monte Carlo (MCMC) sampling of
high-dimensional posterior distributions arising in Bayesian inverse problems.
To localize an inverse problem is to enforce an anticipated "local" structure
by (i) neglecting small off-diagonal elements of the prior precision and
covariance matrices; and (ii) restricting the influence of observations to
their neighborhood. For linear problems we can specify the conditions under
which posterior moments of the localized problem are close to those of the
original problem. We explain physical interpretations of our assumptions about
local structure and discuss the notion of high dimensionality in local
problems, which is different from the usual notion of high dimensionality in
function space MCMC. The Gibbs sampler is a natural choice of MCMC algorithm
for localized inverse problems and we demonstrate that its convergence rate is
independent of dimension for localized linear problems. Nonlinear problems can
also be tackled efficiently by localization and, as a simple illustration of
these ideas, we present a localized Metropolis-within-Gibbs sampler. Several
linear and nonlinear numerical examples illustrate localization in the context
of MCMC samplers for inverse problems.Comment: 33 pages, 5 figure
Parameter estimation by implicit sampling
Implicit sampling is a weighted sampling method that is used in data
assimilation, where one sequentially updates estimates of the state of a
stochastic model based on a stream of noisy or incomplete data. Here we
describe how to use implicit sampling in parameter estimation problems, where
the goal is to find parameters of a numerical model, e.g.~a partial
differential equation (PDE), such that the output of the numerical model is
compatible with (noisy) data. We use the Bayesian approach to parameter
estimation, in which a posterior probability density describes the probability
of the parameter conditioned on data and compute an empirical estimate of this
posterior with implicit sampling. Our approach generates independent samples,
so that some of the practical difficulties one encounters with Markov Chain
Monte Carlo methods, e.g.~burn-in time or correlations among dependent samples,
are avoided. We describe a new implementation of implicit sampling for
parameter estimation problems that makes use of multiple grids (coarse to fine)
and BFGS optimization coupled to adjoint equations for the required gradient
calculations. The implementation is "dimension independent", in the sense that
a well-defined finite dimensional subspace is sampled as the mesh used for
discretization of the PDE is refined. We illustrate the algorithm with an
example where we estimate a diffusion coefficient in an elliptic equation from
sparse and noisy pressure measurements. In the example, dimension\slash
mesh-independence is achieved via Karhunen-Lo\`{e}ve expansions
Limitations of polynomial chaos expansions in the Bayesian solution of inverse problems
Polynomial chaos expansions are used to reduce the computational cost in the
Bayesian solutions of inverse problems by creating a surrogate posterior that
can be evaluated inexpensively. We show, by analysis and example, that when the
data contain significant information beyond what is assumed in the prior, the
surrogate posterior can be very different from the posterior, and the resulting
estimates become inaccurate. One can improve the accuracy by adaptively
increasing the order of the polynomial chaos, but the cost may increase too
fast for this to be cost effective compared to Monte Carlo sampling without a
surrogate posterior
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