Reduced basis methods with adaptive snapshot computations

Ali, Mazen; Steih, Kristina; Urban, Karsten

doi:10.1007/s10444-016-9485-9

Cited by 31 publications

(49 citation statements)

References 42 publications

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“…For more details on the choices for W we refer to [1] and [9], for example. Concerning existence, uniqueness and regularity of a solution to (5), we refer to [9]. In order to derive a variational form of type (3), we write (5) as a single fourth-order parabolic equation for c by…”

Section: Example 24 (Cahn-hilliard Equations)mentioning

confidence: 99%

“…For the fully discrete POD setting, the spatial discretization points are chosen appropriately for the numerical integration of the polynomials. In the context of reduced basis methods, adaptive wavelet discretizations are used in [5] in the offline snapshot computation phase. In [54], a reduced basis method is developed which is certified by a residual bound relative to the infinite-dimensional weak solution.…”

Section: Introductionmentioning

confidence: 99%

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POD reduced-order modeling for evolution equations utilizing arbitrary finite element discretizations

Gräßle

Hinze

2018

Adv Comput Math

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The main focus of the present work is the inclusion of spatial adaptivity for the snapshot computation in the offline phase of model order reduction utilizing Proper Orthogonal Decomposition (POD-MOR) for nonlinear parabolic evolution problems. We consider snapshots which live in different finite element spaces, which means in a fully discrete setting that the snapshots are vectors of different length. From a numerical point of view, this leads to the problem that the usual POD procedure which utilizes a singular value decomposition of the snapshot matrix, cannot be carried out. In order to overcome this problem, we here construct the POD model / basis using the eigensystem of the correlation matrix (snapshot gramian), which is motivated from a continuous perspective and is set up explicitly e.g. without the necessity of interpolating snapshots into a common finite element space. It is an advantage of this approach that the assembling of the matrix only requires the evaluation of inner products of snapshots in a common Hilbert space. This allows a great flexibility concerning the spatial discretization of the snapshots. The analysis for the error between the resulting POD solution and the true solution reveals that the accuracy of the reduced order solution can be estimated by the spatial and temporal discretization error as well as the POD error. Finally, to illustrate the feasibility our approach, we present a test case of the Cahn-Hilliard system utilizing h-adapted hierarchical meshes and two settings of a linear heat equation using nested and non-nested grids.

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Section: Example 24 (Cahn-hilliard Equations)mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

POD reduced-order modeling for evolution equations utilizing arbitrary finite element discretizations

Gräßle

Hinze

2018

Adv Comput Math

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“…A posteriori error estimates play an important role within the RBM, at least for the following reasons: (1) The error estimator is used in a weak Greedy algorithm to construct the reduced model. This is e.g.…”

Section: Introductionmentioning

confidence: 99%

“…Effectivity index over online CPU-time for Helmholtz problem on (1) , (2) ; strong greedy sampling. Circles: Hierarchical error estimator for different ; crosses: Standard error estimator.…”

mentioning

confidence: 99%

A hierarchical a posteriori error estimator for the Reduced Basis Method

Hain¹,

Ohlberger

Radic³

et al. 2019

Adv Comput Math

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In this contribution we are concerned with tight a posteriori error estimation for projection based model order reduction of inf -sup stable parameterized variational problems. In particular, we consider the Reduced Basis Method in a Petrov-Galerkin framework, where the reduced approximation spaces are constructed by the (weak) Greedy algorithm. We propose and analyze a hierarchical a posteriori error estimator which evaluates the difference of two reduced approximations of different accuracy. Based on the a priori error analysis of the (weak) Greedy algorithm, it is expected that the hierarchical error estimator is sharp with efficiency index close to one, if the Kolmogorov N-with decays fast for the underlying problem and if a suitable saturation assumption for the reduced approximation is satisfied. We investigate the tightness of the hierarchical a posteriori estimator both from a theoretical and numerical perspective. For the respective approximation with higher accuracy we study and compare basis enrichment of Lagrange-and Taylor-type reduced bases. Numerical experiments indicate the efficiency for both, the construction of a reduced basis using the hierarchical error estimator in a weak Greedy algorithm, and for tight online certification of reduced approximations. This is particularly relevant in cases where the inf -sup constant may become small depending on the parameter. In such cases a standard residual-based error estimator -complemented by the successive constrained method to compute a lower bound of the parameter dependent inf -sup constant -may become infeasible.

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“…¶ Institute for Computational and Applied Mathematics, University of Münster, Einsteinstrasse 62, 48149 Münster, Germany, {mario.ohlberger,stephan.rave,felix.schindler}@uni-muenster.de interested in approximating (1) for many parameters, efficiency is related to an overall computational cost that is minimal compared to the combined cost of separate approximations for each parameter. To this end one employs model reduction with reduced basis (RB) methods, where one usually considers a common approximation space Q h for all parameters (with the notable exceptions [2,18]) and where one iteratively builds a reduced approximation space Q red ⊂ Q h by an adaptive greedy search, the purpose of which is to capture the manifold of solutions of (1):…”

mentioning

confidence: 99%

True Error Control for the Localized Reduced Basis Method for Parabolic Problems

Ohlberger¹,

Rave²,

Schindler³

2017

Model Reduction of Parametrized Systems

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We present an abstract framework for a posteriori error estimation for approximations of scalar parabolic evolution equations, based on elliptic reconstruction techniques [10,9,3,5]. In addition to its original application (to derive error estimates on the discretization error), we extend the scope of this framework to derive offline/online decomposable a posteriori estimates on the model reduction error in the context of Reduced Basis (RB) methods. In addition, we present offline/online decomposable a posteriori error estimates on the full approximation error (including discretization as well as model reduction error) in the context of the localized RB method [14]. Hence, this work generalizes the localized RB method with true error certification to parabolic problems. Numerical experiments are given to demonstrate the applicability of the approach.

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Reduced basis methods with adaptive snapshot computations

Cited by 31 publications

References 42 publications

POD reduced-order modeling for evolution equations utilizing arbitrary finite element discretizations

POD reduced-order modeling for evolution equations utilizing arbitrary finite element discretizations

A hierarchical a posteriori error estimator for the Reduced Basis Method

True Error Control for the Localized Reduced Basis Method for Parabolic Problems

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