44,728 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]
Space-Time Isogeometric Analysis of Parabolic Evolution Equations
We present and analyze a new stable space-time Isogeometric Analysis (IgA)
method for the numerical solution of parabolic evolution equations in fixed and
moving spatial computational domains. The discrete bilinear form is elliptic on
the IgA space with respect to a discrete energy norm. This property together
with a corresponding boundedness property, consistency and approximation
results for the IgA spaces yields an a priori discretization error estimate
with respect to the discrete norm. The theoretical results are confirmed by
several numerical experiments with low- and high-order IgA spaces
Numerical methods for time-fractional evolution equations with nonsmooth data: a concise overview
Over the past few decades, there has been substantial interest in evolution
equations that involving a fractional-order derivative of order
in time, due to their many successful applications in
engineering, physics, biology and finance. Thus, it is of paramount importance
to develop and to analyze efficient and accurate numerical methods for reliably
simulating such models, and the literature on the topic is vast and fast
growing. The present paper gives a concise overview on numerical schemes for
the subdiffusion model with nonsmooth problem data, which are important for the
numerical analysis of many problems arising in optimal control, inverse
problems and stochastic analysis. We focus on the following aspects of the
subdiffusion model: regularity theory, Galerkin finite element discretization
in space, time-stepping schemes (including convolution quadrature and L1 type
schemes), and space-time variational formulations, and compare the results with
that for standard parabolic problems. Further, these aspects are showcased with
illustrative numerical experiments and complemented with perspectives and
pointers to relevant literature.Comment: 24 pages, 3 figure
Weak order for the discretization of the stochastic heat equation
In this paper we study the approximation of the distribution of
Hilbert--valued stochastic process solution of a linear parabolic stochastic
partial differential equation written in an abstract form as driven by a Gaussian
space time noise whose covariance operator is given. We assume that
is a finite trace operator for some and that is
bounded from into for some . It is not required
to be nuclear or to commute with . The discretization is achieved thanks to
finite element methods in space (parameter ) and implicit Euler schemes in
time (parameter ). We define a discrete solution and for
suitable functions defined on , we show that |\E \phi(X^N_h) - \E
\phi(X_T) | = O(h^{2\gamma} + \Delta t^\gamma) \noindent where . Let us note that as in the finite dimensional case the rate of
convergence is twice the one for pathwise approximations
Multiscale differential Riccati equations for linear quadratic regulator problems
We consider approximations to the solutions of differential Riccati equations
in the context of linear quadratic regulator problems, where the state equation
is governed by a multiscale operator. Similarly to elliptic and parabolic
problems, standard finite element discretizations perform poorly in this
setting unless the grid resolves the fine-scale features of the problem. This
results in unfeasible amounts of computation and high memory requirements. In
this paper, we demonstrate how the localized orthogonal decomposition method
may be used to acquire accurate results also for coarse discretizations, at the
low cost of solving a series of small, localized elliptic problems. We prove
second-order convergence (except for a logarithmic factor) in the
operator norm, and first-order convergence in the corresponding energy norm.
These results are both independent of the multiscale variations in the state
equation. In addition, we provide a detailed derivation of the fully discrete
matrix-valued equations, and show how they can be handled in a low-rank setting
for large-scale computations. In connection to this, we also show how to
efficiently compute the relevant operator-norm errors. Finally, our theoretical
results are validated by several numerical experiments.Comment: Accepted for publication in SIAM J. Sci. Comput. This version differs
from the previous one only by the addition of Remark 7.2 and minor changes in
formatting. 21 pages, 12 figure
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