38 research outputs found
A random map implementation of implicit filters
Implicit particle filters for data assimilation generate high-probability
samples by representing each particle location as a separate function of a
common reference variable. This representation requires that a certain
underdetermined equation be solved for each particle and at each time an
observation becomes available. We present a new implementation of implicit
filters in which we find the solution of the equation via a random map. As
examples, we assimilate data for a stochastically driven Lorenz system with
sparse observations and for a stochastic Kuramoto-Sivashinski equation with
observations that are sparse in both space and time
Path integral formulation of stochastic optimal control with generalized costs
Path integral control solves a class of stochastic optimal control problems
with a Monte Carlo (MC) method for an associated Hamilton-Jacobi-Bellman (HJB)
equation. The MC approach avoids the need for a global grid of the domain of
the HJB equation and, therefore, path integral control is in principle
applicable to control problems of moderate to large dimension. The class of
problems path integral control can solve, however, is defined by requirements
on the cost function, the noise covariance matrix and the control input matrix.
We relax the requirements on the cost function by introducing a new state that
represents an augmented running cost. In our new formulation the cost function
can contain stochastic integral terms and linear control costs, which are
important in applications in engineering, economics and finance. We find an
efficient numerical implementation of our grid-free MC approach and demonstrate
its performance and usefulness in examples from hierarchical electric load
management. The dimension of one of our examples is large enough to make
classical grid-based HJB solvers impractical
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
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
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
Performance bounds for particle filters using the optimal proposal
Particle filters may suffer from degeneracy of the particle weights. For the simplest "bootstrap" filter, it is known that avoiding degeneracy in large systems requires that the ensemble size must increase exponentially with the variance of the observation log-likelihood. The present article shows first that a similar result applies to particle filters using sequential importance sampling and the optimal proposal distribution and, second, that the optimal proposal yields minimal degeneracy when compared to any other proposal distribution that depends only on the previous state and the most recent observations. Thus, the optimal proposal provides performance bounds for filters using sequential importance sampling and any such proposal. An example with independent and identically distributed degrees of freedom illustrates both the need for exponentially large ensemble size with the optimal proposal as the system dimension increases and the potentially dramatic advantages of the optimal proposal relative to simpler proposals. Those advantages depend crucially on the magnitude of the system noise
Small-Noise Analysis and Symmetrization of Implicit Monte Carlo Samplers
Implicit samplers are algorithms for producing independent, weighted samples from multivariate 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 algorithms that leads to improved implicit sampling schemes at a relatively small additional cost. Computational experiments confirm the theory and show that symmetrization is effective for small noise sampling problems.© 2016 Wiley Periodicals, Inc
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