10,364 research outputs found
Analytic Regularity and GPC Approximation for Control Problems Constrained by Linear Parametric Elliptic and Parabolic PDEs
This paper deals with linear-quadratic optimal control problems constrained by a parametric or stochastic elliptic or parabolic PDE. We address the (difficult) case that the state equation depends on a countable number of parameters i.e., on with , and that the PDE operator may depend non-affinely on the parameters. We consider tracking-type functionals and distributed as well as boundary controls. Building on recent results in [CDS1, CDS2], we show that the state and the control are analytic as functions depending on these parameters . We
establish sparsity of generalized polynomial chaos (gpc) expansions of both, state and control, in terms of the stochastic coordinate sequence of the random inputs, and prove convergence rates of best -term truncations of these expansions. Such truncations are the key for subsequent computations since they do {\em not} assume that the stochastic input data has a finite expansion. In the follow-up paper [KS2], we explain two methods how such best -term truncations can practically be computed, by greedy-type algorithms
as in [SG, Gi1], or by multilevel Monte-Carlo methods as in
[KSS]. The sparsity result allows in conjunction with adaptive wavelet Galerkin schemes for sparse, adaptive tensor discretizations of control problems constrained by linear elliptic and parabolic PDEs developed in [DK, GK, K], see [KS2]
Dynamically optimal treatment allocation using Reinforcement Learning
Devising guidance on how to assign individuals to treatment is an important
goal in empirical research. In practice, individuals often arrive sequentially,
and the planner faces various constraints such as limited budget/capacity, or
borrowing constraints, or the need to place people in a queue. For instance, a
governmental body may receive a budget outlay at the beginning of a year, and
it may need to decide how best to allocate resources within the year to
individuals who arrive sequentially. In this and other examples involving
inter-temporal trade-offs, previous work on devising optimal policy rules in a
static context is either not applicable, or sub-optimal. Here we show how one
can use offline observational data to estimate an optimal policy rule that
maximizes expected welfare in this dynamic context. We allow the class of
policy rules to be restricted for legal, ethical or incentive compatibility
reasons. The problem is equivalent to one of optimal control under a
constrained policy class, and we exploit recent developments in Reinforcement
Learning (RL) to propose an algorithm to solve this. The algorithm is easily
implementable with speedups achieved through multiple RL agents learning in
parallel processes. We also characterize the statistical regret from using our
estimated policy rule by casting the evolution of the value function under each
policy in a Partial Differential Equation (PDE) form and using the theory of
viscosity solutions to PDEs. We find that the policy regret decays at a
rate in most examples; this is the same rate as in the static case.Comment: 67 page
A game interpretation of the Neumann problem for fully nonlinear parabolic and elliptic equations
We provide a deterministic-control-based interpretation for a broad class of
fully nonlinear parabolic and elliptic PDEs with continuous Neumann boundary
conditions in a smooth domain. We construct families of two-person games
depending on a small parameter which extend those proposed by Kohn and Serfaty
(2010). These new games treat a Neumann boundary condition by introducing some
specific rules near the boundary. We show that the value function converges, in
the viscosity sense, to the solution of the PDE as the parameter tends to zero.
Moreover, our construction allows us to treat both the oblique and the mixed
type Dirichlet-Neumann boundary conditions.Comment: 58 pages, 2 figure
Quantitative Homogenization of Elliptic PDE with Random Oscillatory Boundary Data
We study the averaging behavior of nonlinear uniformly elliptic partial
differential equations with random Dirichlet or Neumann boundary data
oscillating on a small scale. Under conditions on the operator, the data and
the random media leading to concentration of measure, we prove an almost sure
and local uniform homogenization result with a rate of convergence in
probability
Continuous dependence results for Non-linear Neumann type boundary value problems
We obtain estimates on the continuous dependence on the coefficient for
second order non-linear degenerate Neumann type boundary value problems. Our
results extend previous work of Cockburn et.al., Jakobsen-Karlsen, and
Gripenberg to problems with more general boundary conditions and domains. A new
feature here is that we account for the dependence on the boundary conditions.
As one application of our continuous dependence results, we derive for the
first time the rate of convergence for the vanishing viscosity method for such
problems. We also derive new explicit continuous dependence on the coefficients
results for problems involving Bellman-Isaacs equations and certain quasilinear
equation
Mesoscopic higher regularity and subadditivity in elliptic homogenization
We introduce a new method for obtaining quantitative results in stochastic
homogenization for linear elliptic equations in divergence form. Unlike
previous works on the topic, our method does not use concentration inequalities
(such as Poincar\'e or logarithmic Sobolev inequalities in the probability
space) and relies instead on a higher (, ) regularity theory
for solutions of the heterogeneous equation, which is valid on length scales
larger than a certain specified mesoscopic scale. This regularity theory, which
is of independent interest, allows us to, in effect, localize the dependence of
the solutions on the coefficients and thereby accelerate the rate of
convergence of the expected energy of the cell problem by a bootstrap argument.
The fluctuations of the energy are then tightly controlled using subadditivity.
The convergence of the energy gives control of the scaling of the spatial
averages of gradients and fluxes (that is, it quantifies the weak convergence
of these quantities) which yields, by a new "multiscale" Poincar\'e inequality,
quantitative estimates on the sublinearity of the corrector.Comment: 44 pages, revised version, to appear in Comm. Math. Phy
A modified semi--implict Euler-Maruyama Scheme for finite element discretization of SPDEs with additive noise
We consider the numerical approximation of a general second order
semi--linear parabolic stochastic partial differential equation (SPDE) driven
by additive space-time noise. We introduce a new modified scheme using a linear
functional of the noise with a semi--implicit Euler--Maruyama method in time
and in space we analyse a finite element method (although extension to finite
differences or finite volumes would be possible). We prove convergence in the
root mean square norm for a diffusion reaction equation and diffusion
advection reaction equation. We present numerical results for a linear reaction
diffusion equation in two dimensions as well as a nonlinear example of
two-dimensional stochastic advection diffusion reaction equation. We see from
both the analysis and numerics that the proposed scheme has better convergence
properties than the standard semi--implicit Euler--Maruyama method
Uniform shift estimates for transmission problems and optimal rates of convergence for the parametric Finite Element Method
Let \Omega \subset \RR^d, , be a bounded domain with
piecewise smooth boundary and let be an open subset of a
Banach space . Motivated by questions in "Uncertainty Quantification," we
consider a parametric family of uniformly strongly
elliptic, second order partial differential operators on . We
allow jump discontinuities in the coefficients. We establish a regularity
result for the solution u: \Omega \times U \to \RR of the parametric,
elliptic boundary value/transmission problem , , with
mixed Dirichlet-Neumann boundary conditions in the case when the boundary and
the interface are smooth and in the general case for . Our regularity and
well-posedness results are formulated in a scale of broken weighted Sobolev
spaces \hat\maK^{m+1}_{a+1}(\Omega) of Babu\v{s}ka-Kondrat'ev type in
, possibly augmented by some locally constant functions. This implies
that the parametric, elliptic PDEs admit a shift theorem that
is uniform in the parameter . In turn, this then leads to
-quasi-optimal rates of convergence (i.e. algebraic orders of convergence)
for the Galerkin approximations of the solution , where the approximation
spaces are defined using the "polynomial chaos expansion" of with respect
to a suitable family of tensorized Lagrange polynomials, following the method
developed by Cohen, Devore, and Schwab (2010)
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