Induced Dimension Reduction method to solve the Quadratic Eigenvalue Problem

Transcription

1 Induced Dimension Reduction method to solve the Quadratic Eigenvalue Problem R. Astudillo and M. B. van Gijzen Delft Institute of Applied Mathematics Delft University of Technology Delft, the Netherlands Student Krylov Day, February 2nd, 2015 (TU Delft) IDR for QEP 1 / 16

2 Problem Motivation How to solve this problem? Source Millennium Bridge Opening Day (TU Delft) IDR for QEP 2 / 16

3 Problem Motivation Vibration Analysis of Structural Systems The Quadratic eigenvalue problem: Given the matrices M, D, and K of dimension n, find the scalar λ and the vector x, such that: (λ 2 M + λd + K )x = 0 (TU Delft) IDR for QEP 3 / 16

4 Problem Motivation Vibration Analysis of Structural Systems The Quadratic eigenvalue problem: Given the matrices M, D, and K of dimension n, find the scalar λ and the vector x, such that: (λ 2 M + λd + K )x = 0 (TU Delft) IDR for QEP 3 / 16

5 Problem Motivation Vibration Analysis of Structural Systems The Quadratic eigenvalue problem: Given the matrices M, D, and K of dimension n, find the scalar λ and the vector x, such that: (λ 2 M + λd + K )x = 0 (TU Delft) IDR for QEP 3 / 16

6 Table of contents 1 IDR(s) method for solving system of linear equation 2 Approximating eigenpairs with IDR(s) 3 Solving the Quadratic eigenvalue problem with IDR(s) 4 Numerical tests (TU Delft) IDR for QEP 4 / 16

7 IDR(s) method for solving system of linear equation References Sonneveld, and M. B. van Gijzen, SIAM J. Sci. Computing 31, (2008). (TU Delft) IDR for QEP 5 / 16

8 IDR(s) method for solving system of linear equation IDR(s) Theorem G 0 C n G j+1 (A µ j+1 I)(G j S) j = 0, 1, 2..., 1 G j+1 G j, and 2 dimension(g j+1 ) < dimension(g j ) unless G j = {0}. References Sonneveld, and M. B. van Gijzen, SIAM J. Sci. Computing 31, (2008). (TU Delft) IDR for QEP 6 / 16

9 IDR(s) method for solving system of linear equation Source user: Jetzabel. (TU Delft) IDR for QEP 7 / 16

10 IDR(s) method for solving system of linear equation Source user: Jetzabel. (TU Delft) IDR for QEP 7 / 16

11 A Hessenberg decomposition based on IDR(s) Create w k+1 G j+1 (A µ j+1 I)(G j S) Assume that w k, w k 1,..., w k (s+1) are in G j and S P. w k+1 = (A µ j+1 I)(w k with P T (w k s i=1 β iw k i ) = 0, then Aw k = w k+1 + µ j+1 w k + A s β i w k i ) i=1 s β i w k i µ j+1 i=1 s β i w k i i=1 (TU Delft) IDR for QEP 8 / 16

12 A Hessenberg decomposition based on IDR(s) Create w k+1 G j+1 (A µ j+1 I)(G j S) Assume that w k, w k 1,..., w k (s+1) are in G j and S P. w k+1 = (A µ j+1 I)(w k with P T (w k s i=1 β iw k i ) = 0, then Aw k = w k+1 + µ j+1 w k + A s β i w k i ) i=1 s β i w k i µ j+1 i=1 s β i w k i i=1 (TU Delft) IDR for QEP 8 / 16

13 A Hessenberg decomposition based on IDR(s) Create w k+1 G j+1 (A µ j+1 I)(G j S) Assume that w k, w k 1,..., w k (s+1) are in G j and S P. w k+1 = (A µ j+1 I)(w k with P T (w k s i=1 β iw k i ) = 0, then Aw k = w k+1 + µ j+1 w k + A s β i w k i ) i=1 s β i w k i µ j+1 i=1 s β i w k i i=1 (TU Delft) IDR for QEP 8 / 16

14 A Hessenberg decomposition based on IDR(s) Create w k+1 G j+1 (A µ j+1 I)(G j S) Assume that w k, w k 1,..., w k (s+1) are in G j and S P. w k+1 = (A µ j+1 I)(w k with P T (w k s i=1 β iw k i ) = 0, then Aw k = w k+1 + µ j+1 w k + A s β i w k i ) i=1 s β i w k i µ j+1 i=1 s β i w k i i=1 (TU Delft) IDR for QEP 8 / 16

15 A Hessenberg decomposition based on IDR(s) AW m = W m H m + fe T m Λ(H m ) approximated some eigenvalues of A Short-term recurrence method. Implicit filter polynomial. Low-memory requirements. References R. Astudillo and M. B. van Gijzen, AIP Conf. Proc. 1558, 2277 (2013). -, Technical Report 14-04, Delft University of Technology, The Netherlands, (TU Delft) IDR for QEP 9 / 16

16 Solving the Quadratic eigenvalue problem with IDR(s) Linearization and properties The problem (λ 2 M + λd + K )x = 0 can be transformed into the Generalized Eigenvalue Problem Cy = λgy, where C = [ ] D K I 0 and G = [ ] M 0 0 I (TU Delft) IDR for QEP 10 / 16

17 Solving the Quadratic eigenvalue problem with IDR(s) Linearization and properties S = G 1 C = [ M 1 D M 1 ] K I 0 if v = [r 0, 0] T, then [ rj r j 1 ] = S j v References Z. Bai and Y. Su, SIAM J. Matrix Anal. Appl. 26, pp (2005). (TU Delft) IDR for QEP 11 / 16

18 Solving the Quadratic eigenvalue problem with IDR(s) Linearization and properties SW m = W m H m + fe T m if [ ] W (t) m W m = W (b) m Then ( M 1 D)W (t) m + ( M 1 K )W (b) m W (b) m W (t) m = W (u) m T, = W (t) m H m + f (t) e T m = W (b) m H m + f (b) e T m Where T is upper triangular, then we save only W (u) m. (TU Delft) IDR for QEP 12 / 16

19 Solving the Quadratic eigenvalue problem with IDR(s) Linearization and properties SW m = W m H m + fe T m if [ ] W (t) m W m = W (b) m Then ( M 1 D)W (t) m + ( M 1 K )W (b) m W (b) m W (t) m = W (u) m T, = W (t) m H m + f (t) e T m = W (b) m H m + f (b) e T m Where T is upper triangular, then we save only W (u) m. (TU Delft) IDR for QEP 12 / 16

20 A Hessenberg decomposition based on IDR(s) for QEP [ M 1 D M 1 ] K W I 0 m = W m H m + fe T m Λ(H m ) approximated some eigenvalues of QEP: (λ 2 M + λd + K )x = 0 Short-term recurrence method. Implicit filter polynomial. Low-memory requirements. Operation O(n) (TU Delft) IDR for QEP 13 / 16

21 Numerical tests: n = 200; M = rand(n); D = rand(n); K = rand(n); Arnoldi: secs. IDR(s): secs. (TU Delft) IDR for QEP 14 / 16

22 Numerical tests: n = 200; M = rand(n); D = rand(n); K = rand(n); Arnoldi: secs. IDR(s): secs. (TU Delft) IDR for QEP 14 / 16

23 Numerical tests: Wave propagation in a room Dimension: 1681 Convergence to the known eigenvalue λ = i Arnoldi: 0.51 secs. IDR(s): 0.42 secs. References Sleijpen, G. L. G.; van der Vorst, H. A. and van Gijzen, M. B., SIAM News 29, pp 8-19, (1996). (TU Delft) IDR for QEP 15 / 16

24 Thank you for your attention. Questions, comments, suggestions are welcome. (TU Delft) IDR for QEP 16 / 16