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Path Integral Formulation of Stochastic Optimal Control with Generalized Costs?
"... Abstract: Path integral control solves a class of stochastic optimal control problems with a Monte Carlo (MC) method for an associated HamiltonJacobiBellman (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 ..."
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Abstract: Path integral control solves a class of stochastic optimal control problems with a Monte Carlo (MC) method for an associated HamiltonJacobiBellman (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 gridfree 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 gridbased HJB solvers impractical. 1.
Implicit sampling for path integral control, Monte Carlo localization and online SLAM. in review
, 2013
"... 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 recentlydeveloped variationallyenhanced sampling method. The theory is illustrated with examples, and ..."
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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 recentlydeveloped variationallyenhanced 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. 1
Smallnoise analysis and symmetrization of implicit Monte Carlo samplers
, 2014
"... 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 s ..."
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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. 1
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 t ..."
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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 constructing a surrogate posterior. 1