Space-time adaptive wavelet methods for parabolic evolution problems

Schwab, Christoph; Stevenson, Rob

doi:10.1090/s0025-5718-08-02205-9

Cited by 169 publications

(191 citation statements)

References 41 publications

(46 reference statements)

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“…In view of the above result, we are therefore interested in choosing univariate sequences T = (t k ) k≥0 such that the Lebesgue constant λ k or the quantity δ k have moderate algebraic growth with k. It is well known that for particular sets of points such as the Chebychev points 81) or the Gauss-Lobatto (or Clemshaw-Curtis) points…”

Section: Proofmentioning

confidence: 99%

Approximation of high-dimensional parametric PDEs

Cohen¹,

DeVore²

2015

Acta Numerica

230

266

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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 a → u(a) where a is the parameter value and u(a) 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 n-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 n-term approximation, sparsity, and n-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.

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Section: Proofmentioning

confidence: 99%

Approximation of high-dimensional parametric PDEs

Cohen¹,

DeVore²

2015

Acta Numerica

230

266

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“…In order to overcome the instability of the classical L 2 -in-time error-bound formulation, we follow the space-time approach recently devised by Urban and Patera, 13,12 ; we consider a space-time variational and corresponding finite element formulation that produces a favorable inf-sup stability constant and then incorporate the space-time truth within a space-time reduced basis approach. The approach is inspired by the recent work on the space-time Petrov-Galerkin formulation by Schwab and Stevenson 11 . The main contribution of this work is twofold.…”

Section: Introductionmentioning

confidence: 99%

A space-time hp-interpolation-based certified reduced basis method for Burgers' equation

Yano

Patera

Urban

2014

Math. Models Methods Appl. Sci.

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Received (Day Month Year) Revised (Day Month Year) Communicated by (xxxxxxxxxx)We present a space-time certified reduced basis method for Burgers' equation over the spatial interval (0, 1) and the temporal interval (0, T ] parametrized with respect to the Peclet number. We first introduce a Petrov-Galerkin space-time finite element discretization, which enjoys a favorable inf-sup constant that decreases slowly with Peclet number and final time T . We then consider an hp interpolation-based space-time reduced basis approximation and associated Brezzi-Rappaz-Raviart a posteriori error bounds. We detail computational procedures that permit offline-online decomposition for the three key ingredients of the error bounds: the dual norm of the residual, a lower bound for the inf-sup constant, and the space-time Sobolev embedding constant. Numerical results demonstrate that our space-time formulation provides improved stability constants compared to classical L 2 -error estimates; the error bounds remain sharp over a wide range of Peclet numbers and long integration times T , unlike the exponentially growing estimate of the classical formulation for high Peclet number cases.

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“…On montre (Proposition 1 basée sur [6]) que pour des problèmes coercifs, la constante inf-sup β de b est positive et bornée inférieurement.…”

Section: Introductionunclassified

A New Error Bound for Reduced Basis Approximation of Parabolic Partial Differential Equations

Urban¹,

Patera²

2012

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We consider a space-time variational formulation for linear parabolic partial differential equations. We introduce an associated Petrov-Galerkin truth finite element discretization with favorable discrete inf-sup constant β δ : β δ is unity for the heat equation; β δ grows only linearly in time for non-coercive (but asymptotically stable) convection operators. The latter in turn permits effective long-time a posteriori error bounds for reduced basis approximations, in sharp contrast to classical (pessimistic) exponentially growing energy estimates. RésuméNous considérons une formulation variationnelle espace-temps pour leséquations différentielles paraboliques linéaires. Nous y associons une discrétisation paréléments finis de Petrov-Galerkin pour laquelle la constante de stabilité inf-sup β δ possède des propriétés agréables : β δ est unité pour l'équation de la chaleur; β δ a une croissance seulement linéaire en temps pour des opérateurs de convection non-coercifs (mais asymptotiquement stables). Dans le cadre des approximations par bases réduites, cette dernière propriété permet d'obtenir des bornes efficaces pour l'erreur a posteriori en temps long, en net contraste avec les estimateurs d'erreur enénergie classiques (pessimistes) qui présentent une croissance exponentielle.

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Space-time adaptive wavelet methods for parabolic evolution problems

Cited by 169 publications

References 41 publications

Approximation of high-dimensional parametric PDEs

Approximation of high-dimensional parametric PDEs

A space-time hp-interpolation-based certified reduced basis method for Burgers' equation

A New Error Bound for Reduced Basis Approximation of Parabolic Partial Differential Equations

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