Efficient approximation of Sparse Jacobians for time‐implicit reduced order models. (23rd June 2016)

• Record Type: Journal Article
• Authors: Ştefănescu, Răzvan; Sandu, Adrian
• Is Part Of: International journal for numerical methods in fluids. Volume 83:Number 2(2017)
• Page Start: 175
• Page End: 204
• Publication Date: 2016-06-23
• Subjects: POD -- DEIM -- implicit reduced order modeling -- shallow water equations -- finite difference; Fluid dynamics -- Mathematics -- Periodicals; 532
• DOI: 10.1002/fld.4260
• Languages: English
• ISSNs: 0271-2091

Abstract

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. Abstract : This paper introduces a sparse matrix discrete interpolation method to effectively compute matrix approximations in the reduced order modeling framework. The method uses only samples of the nonzero entries of the matrix series. The sparse matrix approximation strategy is compared against various existing methods for computing reduced Jacobians in the case of the 1D Burgers and 2D shallow water equations models.