×

A new fast method of solving the high dimensional elliptic eigenvalue problem. (English) Zbl 1429.65271

Summary: In this paper, we develop a novel method to solve the elliptic eigenvalue problem. The univariate multi-wavelet approach provides a simple diagonal preconditioner for second order elliptic problems, which gives an almost constant condition number for efficiently solving the corresponding linear system. Here, we shall consider a new fast numerical approach for approximating the smallest elliptic eigenvalue by using the multi-wavelet basis in the multi-grid discretization scheme. Moreover, we develop a new numerical scheme coupled with sparse grids method in the calculation. This new approach saves storage in degrees of freedom and thus is more efficient in the computation. Several numerical experiments are provided for validating the proposed numerical scheme, which show that our method retains the optimal convergence rate for the smallest eigenvalue approximation with much less computational cost comparing with “eigs” in full grids.

MSC:

65N25 Numerical methods for eigenvalue problems for boundary value problems involving PDEs
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs

Software:

LOBPCG; JDQZ; JDQR
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Bai, Z.; Demmel, J.; Dongarra, J.; Ruhe, A., Templates for the solution of algebraic eigenvalue problems: a practical guide, (van der Vorst, H., Society for Industrial and Applied Mathematics (2000))
[2] Chegini, N.; Dahlke, S.; Friedrich, U.; Stevenson, R., “Piecewise tensor product wavelet bases by extensions and approximation rates.”, Math. Comput., 82, 284, 2157-2190 (2013) · Zbl 1277.65093
[3] Cohen, A.; Dahmen, W.; DeVore, R., “adaptive wavelet methods for elliptic operator equations: convergence rates.”, Math. Comput., 70, 233, 27-75 (2001) · Zbl 0980.65130
[4] Cohen, A.; Masson, R., “wavelet methods for second-order elliptic problems, preconditioning, and adaptivity.”, SIAM J. Sci. Comput., 21, 3, 1006-1026 (1999) · Zbl 0981.65132
[5] Daubechies, I., “Ten lectures on wavelets.”, Proceedings of the CBMS-NSF Regional Conference Series in Applied Mathematics, 61 (1991) · Zbl 0776.42018
[6] Dijkema; Jan, T.; Schwab, C.; Stevenson, R., “An adaptive wavelet method for solving high-dimensional elliptic PDEs.”, Constr. Approx., 30, 3, 423 (2009) · Zbl 1205.65313
[7] Donovan; George, C.; Jeffrey, S.; Geronimo; Hardin, D. P., “Orthogonal polynomials and the construction of piecewise polynomial smooth wavelets.”, SIAM J. Math. Anal., 30, 5, 1029-1056 (1999) · Zbl 0932.42021
[8] Garcke, J., “Sparse grids in a nutshell.”, Sparse Grids and Applications, 57-80 (2012), Springer: Springer Berlin, Heidelberg
[9] Grebenkov; Denis, S.; Nguyen, B.-T., “geometrical structure of laplacian eigenfunctions.”, SIAM Rev., 55, 4, 601-667 (2013) · Zbl 1290.35157
[10] Griebel, M., “Adaptive sparse grid multilevel methods for elliptic PDEs based on finite differences.”, Computing, 61, 2, 151-179 (1998) · Zbl 0918.65078
[11] Hackbusch; Wolfgang; Khoromskij, B. N.; Sauter, S.; Tyrtyshnikov, E. E., “Use of tensor formats in elliptic eigenvalue problems.”, Numer. Linear Algebra Appl., 19, 1, 133-151 (2012) · Zbl 1274.35252
[12] Hu, X.; Cheng, X., “Acceleration of a two-grid method for eigenvalue problems.”, Math. Comput., 80, 275, 1287-1301 (2011) · Zbl 1232.65141
[13] Huang, R.; Sun, J.; Yang, C., “Recursive integral method with Cayley transformation.”, Numer. Linear Algebra Appl., 25, 6, e2199 (2018) · Zbl 1513.65099
[14] Kressner, D.; Tobler, C., “Preconditioned low-rank methods for high-dimensional elliptic PDE eigenvalue problems.”, Comput. Methods Appl. Math. Comput. Methods Appl. Math., 11, 3, 363-381 (2011) · Zbl 1283.65025
[15] Li, H.; Yang, Y., “The adaptive finite element method based on multi-scale discretizations for eigenvalue problems.”, Comput. Math. Appl., 65, 7, 1086-1102 (2013) · Zbl 1266.65196
[16] Lin, Q.; Xie, H., “A multi-level correction scheme for eigenvalue problems.”, Math. Comput., 84, 291, 71-88 (2015) · Zbl 1307.65159
[17] Pabel, R., Adaptive Wavelet Methods for Variational Formulations of Nonlinear Elliptic PDEs on Tensor-product Domains (2015), Logos Verlag Berlin GmbH · Zbl 1329.65276
[18] Sun, J.; Zhou, A., Finite Element Methods for Eigenvalue Problems (2016), Chapman and Hall/CRC
[19] Temlyakov; Nikolaevich, V., “approximations of functions with bounded mixed derivative.”, Trudy Matematicheskogo Instituta imeni VA Steklova, 178, 3-113 (1986)
[20] Wang, Z.; Tang, Q.; Guo, W.; Cheng, Y., “Sparse grid discontinuous Galerkin methods for high-dimensional elliptic equations.”, J. Comput. Phys., 314, 244-263 (2016) · Zbl 1349.65636
[21] Xu, J.; Zhou, A., “A two-grid discretization scheme for eigenvalue problems.”, Math. Comput., 70, 233, 17-25 (2001) · Zbl 0959.65119
[22] Yang, Y.; Bi, H., “Two-grid finite element discretization schemes based on shifted-inverse power method for elliptic eigenvalue problems.”, SIAM J. Numer. Anal., 49, 4, 1602-1624 (2011) · Zbl 1236.65143
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.