1,258 research outputs found
Approximation of high-dimensional parametric PDEs
Parametrized families of PDEs arise in various contexts such as inverse
problems, control and optimization, risk assessment, and uncertainty
quantification. In most of these applications, the number of parameters is
large or perhaps even infinite. Thus, the development of numerical methods for
these parametric problems is faced with the possible curse of dimensionality.
This article is directed at (i) identifying and understanding which properties
of parametric equations allow one to avoid this curse and (ii) developing and
analyzing effective numerical methodd which fully exploit these properties and,
in turn, are immune to the growth in dimensionality. The first part of this
article studies the smoothness and approximability of the solution map, that
is, the map where is the parameter value and is the
corresponding solution to the PDE. It is shown that for many relevant
parametric PDEs, the parametric smoothness of this map is typically holomorphic
and also highly anisotropic in that the relevant parameters are of widely
varying importance in describing the solution. These two properties are then
exploited to establish convergence rates of -term approximations to the
solution map for which each term is separable in the parametric and physical
variables. These results reveal that, at least on a theoretical level, the
solution map can be well approximated by discretizations of moderate
complexity, thereby showing how the curse of dimensionality is broken. This
theoretical analysis is carried out through concepts of approximation theory
such as best -term approximation, sparsity, and -widths. These notions
determine a priori the best possible performance of numerical methods and thus
serve as a benchmark for concrete algorithms. The second part of this article
turns to the development of numerical algorithms based on the theoretically
established sparse separable approximations. The numerical methods studied fall
into two general categories. The first uses polynomial expansions in terms of
the parameters to approximate the solution map. The second one searches for
suitable low dimensional spaces for simultaneously approximating all members of
the parametric family. The numerical implementation of these approaches is
carried out through adaptive and greedy algorithms. An a priori analysis of the
performance of these algorithms establishes how well they meet the theoretical
benchmarks
Galerkin approximations of nonlinear optimal control problems in Hilbert spaces
International audienceNonlinear optimal control problems in Hilbert spaces are considered for which we derive approximation theorems for Galerkin approximations. Approximation theorems are available in the literature. The originality of our approachrelies on the identification of a set of natural assumptions that allows us to deal with a broad class of nonlinear evolution equations and cost functionals for which we derive convergence of the value functions associated with the optimal control problem of the Galerkin approximations. This convergence result holds for a broad class of nonlinear control strategies as well. In particular, we show that the framework applies to the optimal control of semilinear heat equations posed on a general compact manifold without boundary. The framework is then shown to apply to geoengineering and mitigation of greenhouse gas emissions formulated for the first time in terms of optimal control of energy balance climate models posed on the sphere
Numerical controllability of the wave equation through primal methods and Carleman estimates
This paper deals with the numerical computation of boundary null controls for
the 1D wave equation with a potential. The goal is to compute an approximation
of controls that drive the solution from a prescribed initial state to zero at
a large enough controllability time. We do not use in this work duality
arguments but explore instead a direct approach in the framework of global
Carleman estimates. More precisely, we consider the control that minimizes over
the class of admissible null controls a functional involving weighted integrals
of the state and of the control. The optimality conditions show that both the
optimal control and the associated state are expressed in terms of a new
variable, the solution of a fourth-order elliptic problem defined in the
space-time domain. We first prove that, for some specific weights determined by
the global Carleman inequalities for the wave equation, this problem is
well-posed. Then, in the framework of the finite element method, we introduce a
family of finite-dimensional approximate control problems and we prove a strong
convergence result. Numerical experiments confirm the analysis. We complete our
study with several comments
Modelling and Inverse Problems of Control for Distributed Parameter Systems; Proceedings of IFIP(W.G. 7.2)-IIASA Conference, July 24-28, 1989
The techniques of solving inverse problems that arise in the estimation and control of distributed parameter systems in the face of uncertainty as well as the applications of these to mathematical modelling for problems of applied system analysis (environmental issues, technological processes, biomathematical models, mathematical economy and other fields) are among the major topics of research at the Dynamic Systems Project of the Systems and Decision Sciences (SDS) Program at IIASA.
In July 1989 the SDS Program was a coorganizer of a regular IFIP (WG 7.2) conference on Modelling and Inverse Problems of Control for Distributed Parameter Systems that was held at IIASA, and was attended by a number of prominent theorists and practitioners. One of the main purpose of this meeting was to review recent developments and perspectives in this field. The proceedings are presented in this volume
Stabilizability for nonautonomous linear parabolic equations with actuators as distributions
The stabilizability of a general class of abstract parabolic-like equations
is investigated, with a finite number of actuators. This class includes the
case of actuators given as delta distributions located at given points in the
spatial domain of concrete parabolic equations. A stabilizing feedback control
operator is constructed and given in explicit form. Then, an associated optimal
control is considered and the corresponding Riccati feedback is investigated.
Results of simulations are presented showing the stabilizing performance of
both explicit and Riccati feedbacks.Comment: 7 figure
Geometric partial differential equations: Surface and bulk processes
The workshop brought together experts representing a wide range of topics in geometric partial differential equations ranging from analyis over numerical simulation to real-life applications. The main themes of the conference were the analysis of curvature energies, new developments in pdes on surfaces and the treatment of coupled bulk/surface problems
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