10,364 research outputs found

    Analytic Regularity and GPC Approximation for Control Problems Constrained by Linear Parametric Elliptic and Parabolic PDEs

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    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 σj\sigma_j with jNj\in\N, 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 σj\sigma_j. We establish sparsity of generalized polynomial chaos (gpc) expansions of both, state and control, in terms of the stochastic coordinate sequence σ=(σj)j1\sigma = (\sigma_j)_{j\ge 1} of the random inputs, and prove convergence rates of best NN-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 NN-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

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    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 n1/2n^{-1/2} 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

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    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

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    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

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    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

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    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 (CkC^{k}, k1k \geq 1) 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

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    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 L2L^{2} 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

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    Let \Omega \subset \RR^d, d1d \geqslant 1, be a bounded domain with piecewise smooth boundary Ω\partial \Omega and let UU be an open subset of a Banach space YY. Motivated by questions in "Uncertainty Quantification," we consider a parametric family P=(Py)yUP = (P_y)_{y \in U} of uniformly strongly elliptic, second order partial differential operators PyP_y on Ω\Omega. 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 Pyuy=fyP_y u_y = f_y, yUy \in U, with mixed Dirichlet-Neumann boundary conditions in the case when the boundary and the interface are smooth and in the general case for d=2d=2. 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 Ω\Omega, possibly augmented by some locally constant functions. This implies that the parametric, elliptic PDEs (Py)yU(P_y)_{y \in U} admit a shift theorem that is uniform in the parameter yUy\in U. In turn, this then leads to hmh^m-quasi-optimal rates of convergence (i.e. algebraic orders of convergence) for the Galerkin approximations of the solution uu, where the approximation spaces are defined using the "polynomial chaos expansion" of uu with respect to a suitable family of tensorized Lagrange polynomials, following the method developed by Cohen, Devore, and Schwab (2010)
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