4,301 research outputs found
A Multilevel Approach for Stochastic Nonlinear Optimal Control
We consider a class of finite time horizon nonlinear stochastic optimal
control problem, where the control acts additively on the dynamics and the
control cost is quadratic. This framework is flexible and has found
applications in many domains. Although the optimal control admits a path
integral representation for this class of control problems, efficient
computation of the associated path integrals remains a challenging Monte Carlo
task. The focus of this article is to propose a new Monte Carlo approach that
significantly improves upon existing methodology. Our proposed methodology
first tackles the issue of exponential growth in variance with the time horizon
by casting optimal control estimation as a smoothing problem for a state space
model associated with the control problem, and applying smoothing algorithms
based on particle Markov chain Monte Carlo. To further reduce computational
cost, we then develop a multilevel Monte Carlo method which allows us to obtain
an estimator of the optimal control with mean squared
error with a computational cost of
. In contrast, a computational cost
of is required for existing methodology to achieve
the same mean squared error. Our approach is illustrated on two numerical
examples, which validate our theory
Multilevel Sequential Monte Carlo with Dimension-Independent Likelihood-Informed Proposals
In this article we develop a new sequential Monte Carlo (SMC) method for
multilevel (ML) Monte Carlo estimation. In particular, the method can be used
to estimate expectations with respect to a target probability distribution over
an infinite-dimensional and non-compact space as given, for example, by a
Bayesian inverse problem with Gaussian random field prior. Under suitable
assumptions the MLSMC method has the optimal bound on the
cost to obtain a mean-square error of . The algorithm is
accelerated by dimension-independent likelihood-informed (DILI) proposals
designed for Gaussian priors, leveraging a novel variation which uses empirical
sample covariance information in lieu of Hessian information, hence eliminating
the requirement for gradient evaluations. The efficiency of the algorithm is
illustrated on two examples: inversion of noisy pressure measurements in a PDE
model of Darcy flow to recover the posterior distribution of the permeability
field, and inversion of noisy measurements of the solution of an SDE to recover
the posterior path measure
Hot new directions for quasi-Monte Carlo research in step with applications
This article provides an overview of some interfaces between the theory of
quasi-Monte Carlo (QMC) methods and applications. We summarize three QMC
theoretical settings: first order QMC methods in the unit cube and in
, and higher order QMC methods in the unit cube. One important
feature is that their error bounds can be independent of the dimension
under appropriate conditions on the function spaces. Another important feature
is that good parameters for these QMC methods can be obtained by fast efficient
algorithms even when is large. We outline three different applications and
explain how they can tap into the different QMC theory. We also discuss three
cost saving strategies that can be combined with QMC in these applications.
Many of these recent QMC theory and methods are developed not in isolation, but
in close connection with applications
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