4,840 research outputs found
On the Use of Non-Stationary Policies for Stationary Infinite-Horizon Markov Decision Processes
We consider infinite-horizon stationary -discounted Markov Decision
Processes, for which it is known that there exists a stationary optimal policy.
Using Value and Policy Iteration with some error at each iteration,
it is well-known that one can compute stationary policies that are
-optimal. After arguing that this
guarantee is tight, we develop variations of Value and Policy Iteration for
computing non-stationary policies that can be up to
-optimal, which constitutes a significant
improvement in the usual situation when is close to 1. Surprisingly,
this shows that the problem of "computing near-optimal non-stationary policies"
is much simpler than that of "computing near-optimal stationary policies"
Adaptive Finite Element Methods for Elliptic Problems with Discontinuous Coefficients
Elliptic partial differential equations (PDEs) with discontinuous diffusion
coefficients occur in application domains such as diffusions through porous
media, electro-magnetic field propagation on heterogeneous media, and diffusion
processes on rough surfaces. The standard approach to numerically treating such
problems using finite element methods is to assume that the discontinuities lie
on the boundaries of the cells in the initial triangulation. However, this does
not match applications where discontinuities occur on curves, surfaces, or
manifolds, and could even be unknown beforehand. One of the obstacles to
treating such discontinuity problems is that the usual perturbation theory for
elliptic PDEs assumes bounds for the distortion of the coefficients in the
norm and this in turn requires that the discontinuities are matched
exactly when the coefficients are approximated. We present a new approach based
on distortion of the coefficients in an norm with which
therefore does not require the exact matching of the discontinuities. We then
use this new distortion theory to formulate new adaptive finite element methods
(AFEMs) for such discontinuity problems. We show that such AFEMs are optimal in
the sense of distortion versus number of computations, and report insightful
numerical results supporting our analysis.Comment: 24 page
From Infinite to Finite Programs: Explicit Error Bounds with Applications to Approximate Dynamic Programming
We consider linear programming (LP) problems in infinite dimensional spaces
that are in general computationally intractable. Under suitable assumptions, we
develop an approximation bridge from the infinite-dimensional LP to tractable
finite convex programs in which the performance of the approximation is
quantified explicitly. To this end, we adopt the recent developments in two
areas of randomized optimization and first order methods, leading to a priori
as well as a posterior performance guarantees. We illustrate the generality and
implications of our theoretical results in the special case of the long-run
average cost and discounted cost optimal control problems for Markov decision
processes on Borel spaces. The applicability of the theoretical results is
demonstrated through a constrained linear quadratic optimal control problem and
a fisheries management problem.Comment: 30 pages, 5 figure
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