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

Urban, Karsten; Patera, Anthony T.

doi:10.21236/ada557547

Cited by 31 publications

(54 citation statements)

References 6 publications

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“…In addition, these methods can remove spurious temporal modes (e.g., unstable growth, artificial dissipation) from the state space, which can in principle lead to more accurate long-time responses. Further, space-time reduced-basis ROMs [45,46,50,49] are equipped with error bounds that are observed to grow linearly (rather than exponentially) in the final time. While these approaches are quite promising, they exhibit several drawbacks in terms of applicability to general large-scale nonlinear dynamical systems.…”

mentioning

confidence: 99%

Space--Time Least-Squares Petrov--Galerkin Projection for Nonlinear Model Reduction

Choi¹,

Carlberg²

2019

SIAM J. Sci. Comput.

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This work proposes a space-time least-squares Petrov-Galerkin (ST-LSPG) projection method for model reduction of nonlinear dynamical systems. In contrast to typical nonlinear model-reduction methods that first apply (Petrov-)Galerkin projection in the spatial dimension and subsequently apply time integration to numerically resolve the resulting low-dimensional dynamical system, the proposed method applies projection in space and time simultaneously. To accomplish this, the method first introduces a low-dimensional space-time trial subspace, which can be obtained by computing tensor decompositions of state-snapshot data. The method then computes discrete-optimal approximations in this space-time trial subspace by minimizing the residual arising after time discretization over all space and time in a weighted 2 -norm. This norm can be defined to enable complexity reduction (i.e., hyper-reduction) in time, which leads to space-time collocation and space-time Gauss-Newton with Approximated Tensors (GNAT) variants of the ST-LSPG method. Advantages of the approach relative to typical spatial-projection-based nonlinear model reduction methods such as Galerkin projection and least-squares Petrov-Galerkin projection include a reduction of both the spatial and temporal dimensions of the dynamical system, and a priori error bounds that bound the solution error by the best space-time approximation error and whose stability constants exhibit slower growth in time. Numerical examples performed on model problems in fluid dynamics demonstrate the ability of the method to generate orders-of-magnitude computational savings relative to spatial-projection-based reduced-order models without sacrificing accuracy for a fixed spatio-temporal discretization.

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confidence: 99%

Space--Time Least-Squares Petrov--Galerkin Projection for Nonlinear Model Reduction

Choi¹,

Carlberg²

2019

SIAM J. Sci. Comput.

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“…for all 1 ≤ p ≤ P , where D k is defined in (34) and E (p) k in (36). As a consequence, there is only one term composing the matrices B and B fdPG that differs, namely the time matrix in the double summation over p, p ′ where this matrix is M…”

Section: Methods 2: Fully Discrete Petrov-galerkin Methodsmentioning

confidence: 99%

“…Remark 2.4 (Heat equation). Sharp estimates of the inf-sup constant β for the heat equation (with µ(t) ≡ 1 on I so that α = M = 1 and κ = 0) can be found in [36,10] using the above norms.…”

Section: Moreover Using Again the Coercivity Of A(t) And The Boundedmentioning

confidence: 99%

Low-rank approximation of linear parabolic equations by space-time tensor Galerkin methods

et al. 2019

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We devise a space-time tensor method for the low-rank approximation of linear parabolic evolution equations. The proposed method is a stable Galerkin method, uniformly in the discretization parameters, based on a Minimal Residual formulation of the evolution problem in Hilbert-Bochner spaces. The discrete solution is sought in a linear trial space composed of tensors of discrete functions in space and in time and is characterized as the unique minimizer of a discrete functional where the dual norm of the residual is evaluated in a space semi-discrete test space. The resulting global space-time linear system is solved iteratively by a greedy algorithm. Numerical results are presented to illustrate the performance of the proposed method on test cases including non-selfadjoint and time-dependent differential operators in space. The results are also compared to those obtained using a fully discrete Petrov-Galerkin setting to evaluate the dual residual norm.

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“…Nevertheless, uniform stability was not proven and in [2] it was even shown that indeed the Crank-Nicolson method is generally not a stable space-time method. The interpretation of essentially time-stepping methods via the spacetime formulation was also exploited for reduced basis approximations [38,39] as well as for (quantized) tensor train (QTT) low rank tensor approximations [23] just to mention a few. In that regard, we would like to mention [38,39] in the context of the Crank-Nicolson method and [31] in the context of the discontinuous Galerkin method for the heat equation.…”

Section: Introductionmentioning

confidence: 99%

“…The interpretation of essentially time-stepping methods via the spacetime formulation was also exploited for reduced basis approximations [38,39] as well as for (quantized) tensor train (QTT) low rank tensor approximations [23] just to mention a few. In that regard, we would like to mention [38,39] in the context of the Crank-Nicolson method and [31] in the context of the discontinuous Galerkin method for the heat equation. These works give stability results by introducing subspace-dependent norms but not directly with respect to the natural space-time norms.…”

Section: Introductionmentioning

confidence: 99%

Stability of Petrov–Galerkin Discretizations: Application to the Space-Time Weak Formulation for Parabolic Evolution Problems

Mollet

2014

Computational Methods in Applied Mathematics

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This paper is concerned with the stability of Petrov-Galerkin discretizations with application to parabolic evolution problems in space-time weak form. We will prove that the discrete inf-sup condition for an a priori fixed Petrov-Galerkin discretization is satisfied uniformly under standard approximation and smoothness conditions without any further coupling between the discrete trial and test spaces for sufficiently regular operators. It turns out that one needs to choose different discretization levels for the trial and test spaces in order to obtain a positive lower bound for the discrete inf-sup condition which is independent of the discretization levels. In particular, we state the required number of extra layers in order to guarantee uniform boundedness of the discrete inf-sup constants explicitly. This general result will be applied to the space-time weak formulation of parabolic evolution problems as an important model example. In this regard, we consider suitable hierarchical families of discrete spaces. The results apply, e.g., for finite element discretizations as well as for wavelet discretizations. Due to the Riesz basis property, wavelet discretizations allow for optimal preconditioning independently of the grid spacing. Moreover, our predictions on the stability, especially in view of the dependence on the refinement levels w.r.t. the test and trial spaces, are underlined by numerical results. Furthermore, it can be observed that choosing the same discretization levels would, indeed, lead to stability problems.2010 Mathematical subject classification: 35A15, 35K90, 41A17, 49K40.

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

Cited by 31 publications

References 6 publications

Space--Time Least-Squares Petrov--Galerkin Projection for Nonlinear Model Reduction

Space--Time Least-Squares Petrov--Galerkin Projection for Nonlinear Model Reduction

Low-rank approximation of linear parabolic equations by space-time tensor Galerkin methods

Stability of Petrov–Galerkin Discretizations: Application to the Space-Time Weak Formulation for Parabolic Evolution Problems

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