A New Selection Operator for the Discrete Empirical Interpolation Method---Improved A Priori Error Bound and Extensions

Drmač, Zlatko; Gugercin, Serkan

doi:10.1137/15m1019271

Cited by 271 publications

(280 citation statements)

References 40 publications

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“…In practice, it could be obtained, for example, by applying the standard DEIM point selection algorithm [6,Alg. 1] or by the recent variation [7].…”

Section: Masked Projection Let Y ∈ Rmentioning

confidence: 99%

An Accelerated Greedy Missing Point Estimation Procedure

Zimmermann¹,

Willcox²

2016

SIAM J. Sci. Comput.

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Abstract. Model reduction via Galerkin projection fails to provide considerable computational savings if applied to general nonlinear systems. This is because the reduced representation of the state vector appears as an argument to the nonlinear function, whose evaluation remains as costly as for the full model. Masked projection approaches, such as the missing point estimation and the (discrete) empirical interpolation method, alleviate this effect by evaluating only a small subset of the components of a given nonlinear term; however, the selection of the evaluated components is a combinatorial problem and is computationally intractable even for systems of small size. This has been addressed through greedy point selection algorithms, which minimize an error indicator by sequentially looping over all components. While doable, this is suboptimal and still costly. This paper introduces an approach to accelerate and improve the greedy search. The method is based on the observation that the greedy algorithm requires solving a sequence of symmetric rank-one modifications to an eigenvalue problem. For doing so, we develop fast approximations that sort the set of candidate vectors that induce the rank-one modifications, without requiring solution of the modified eigenvalue problem. Based on theoretical insights into symmetric rank-one eigenvalue modifications, we derive a variation of the greedy method that is faster than the standard approach and yields better results for the cases studied. The proposed approach is illustrated by numerical experiments, where we observe a speed-up by two orders of magnitude when compared to the standard greedy method while arriving at a better quality reduced model.

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“…In practice, it could be obtained, for example, by applying the standard DEIM point selection algorithm [6,Alg. 1] or by the recent variation [7].…”

Section: Masked Projection Let Y ∈ Rmentioning

confidence: 99%

An Accelerated Greedy Missing Point Estimation Procedure

Zimmermann¹,

Willcox²

2016

SIAM J. Sci. Comput.

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“…The off‐line cost of computing DEIM reduced Jacobian arises from the SVD of the nonlinear term snapshots

((), scriptO (n \cdot n_{s}^{2}))

, DEIM algorithm for selecting the interpolation points

((), scriptO (m^{2} \cdot n + m^{3}))

, and matrix operations in

((), scriptO (m^{3} + n \cdot m^{2} + k^{2} \cdot m))

. No additional effort is needed in this stage in the case the DEIM method is employed to approximate the reduced nonlinear term too.…”

Section: Reduced Order Modelingmentioning

confidence: 99%

Efficient approximation of Sparse Jacobians for time‐implicit reduced order models

Ştefanescu

Sandu

2016

Numerical Methods in Fluids

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Summary This paper introduces a sparse matrix discrete interpolation method to effectively compute matrix approximations in the reduced order modeling framework. The sparse algorithm developed herein relies on the discrete empirical interpolation method and uses only samples of the nonzero entries of the matrix series. The proposed approach can approximate very large matrices, unlike the current matrix discrete empirical interpolation method, which is limited by its large computational memory requirements. The empirical interpolation indices obtained by the sparse algorithm slightly differ from the ones computed by the matrix discrete empirical interpolation method as a consequence of the singular vectors round‐off errors introduced by the economy or full singular value decomposition (SVD) algorithms when applied to the full matrix snapshots. When appropriately padded with zeros, the economy SVD factorization of the nonzero elements of the snapshots matrix is a valid economy SVD for the full snapshots matrix. Numerical experiments are performed with the 1D Burgers and 2D shallow water equations test problems where the quadratic reduced nonlinearities are computed via tensorial calculus. The sparse matrix approximation strategy is compared against five existing methods for computing reduced Jacobians: (i) matrix discrete empirical interpolation method, (ii) discrete empirical interpolation method, (iii) tensorial calculus, (iv) full Jacobian projection onto the reduced basis subspace, and (v) directional derivatives of the model along the reduced basis functions. The sparse matrix method outperforms all other algorithms. The use of traditional matrix discrete empirical interpolation method is not possible for very large dimensions because of its excessive memory requirements. Copyright © 2016 John Wiley & Sons, Ltd.

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“…Motivated by recent work by Drmač and Gugercin [11] on a modified DEIM-like algorithm for model reduction, we can improve this bound considerably.…”

Section: Interpretation Of the Bound For Deim-curmentioning

confidence: 99%

“…Lemma 4.4 was inspired by the proof technique developed by Drmač and Gugercin [11] to bound (P T V) −1 , when P is selected by applying a pivoted rank-revealing QR factorization scheme to V. Note that this new bound is on the same order of magnitude as the Drmač-Gugercin scheme. In practice, their scheme seems to give slightly smaller growth that is more consistent over a wide range of examples.…”

Section: −1mentioning

confidence: 99%

A DEIM Induced CUR Factorization

Sorensen

Embree

2015

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We derive a CUR approximate matrix factorization based on the Discrete Empirical Interpolation Method (DEIM). For a given matrix A, such a factorization provides a low rank approximate decomposition of the form A ≈ CUR, where C and R are subsets of the columns and rows of A, and U is constructed to make CUR a good approximation. Given a low-rank singular value decomposition A ≈ VSW T , the DEIM procedure uses V and W to select the columns and rows of A that form C and R. Through an error analysis applicable to a general class of CUR factorizations, we show that the accuracy tracks the optimal approximation error within a factor that depends on the conditioning of submatrices of V and W. For very large problems, V and W can be approximated well using an incremental QR algorithm that makes only one pass through A. Numerical examples illustrate the favorable performance of the DEIM-CUR method compared to CUR approximations based on leverage scores.

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A New Selection Operator for the Discrete Empirical Interpolation Method---Improved A Priori Error Bound and Extensions

Cited by 271 publications

References 40 publications

An Accelerated Greedy Missing Point Estimation Procedure

An Accelerated Greedy Missing Point Estimation Procedure

Efficient approximation of Sparse Jacobians for time‐implicit reduced order models

A DEIM Induced CUR Factorization

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