摘要(英) |
We numerically investigate the numerical performance of two solution algorithms for the quadratic eigenvalue problems (QEP′s), namely the linearization approach and the polynomial Jacobi-Davidson method. Such eigenvalue computations play an important role and highly-demanded in many computational sciences and engineering applications, such as the noise control in the acoustical design, stability analysis in the structural engineering, and electronic engineering. In the linearization approach, the QEP is linearized as a companion generalized eigenvalue problems (GEVP′s), and then a variety of linear eigensolvers are solved the resulting GEVP′s. On the other hand, the polynomial Jacobi-Davidson method targets the eigenvalue of interests directly without any transformation. The evaluation metrics are the robustness, accuracy, and efficiency. To draw the conclusion for more general situations, we conduct intensive numerical experiments for a large number of test cases generated by a collection of Nonlinear Eigenvalue Problem (NLEPV), with a various problem size and different coefficient matrices properties. |
參考文獻 |
[1] Z. Bai, J. Demmel, J. Dongarra, A. Ruhe, and H.A. van der Vorst. Templates
for the Solution of Algebraic Eigenvalue Problems: A Practical Guide. SIAM,
2000.
[2] F. Tisseur and K. Meerbergen. The quadratic eigenvalue problem. SIAM Rev.,
43:235–286, 2001.
[3] T. Betcke, N.J. Higham, V. Mehrmann, C. Schröder, and F. Tisseur. NLEVP:
A collection of nonlinear eigenvalue problems. ACM T. Math Software, 39:1–28,
2013.
[4] G.L.G. Sleijpen and H.A. van der Vorst. A Jacobi-Davidson iteration method
for linear eigenvalue problems. SIAM J. Matrix Anal. Appl., 17:401–425, 1996.
[5] G.L.G. Sleijpen and H.A. van der Vorst. A Jacobi-Davidson iteration method
for linear eigenvalue problems. SIAM Rev., 42:267–293, 2000.
[6] M. Hochbruck and D. Löchel. A multilevel Jacobi-Davidson method for polynomial
PDE eigenvalue problems arising in plasma physics. SIAM J. Sci. Comput.,
32:3151–3169, 2010.
[7] T.-M. Huang, F.-N. Hwang, S.-H. Lai, W. Wang, and Z.-H. Wei. A parallel
polynomial Jacobi-Davidson approach for dissipative acoustic eigenvalue problems.
Comput. Fluids, 45:207–214, 2011.
[8] F.-N. Hwang, Z.-H. Wei, T.-M. Huang, and W. Wang. A parallel additive
Schwarz preconditioned Jacobi-Davidson algorithm for polynomial eigenvalue
problems in quantum dot simulation. J. Comput. Phys., 229:2932–2947, 2010.
[9] T.-M. Hwang, W.-W. Lin, J.-L. Liu, and W. Wang. Jacobi-Davidson methods
for cubic eigenvalue problems. Numer. Linear Algebra Appl., 12:605–624, 2005.
[10] N.J. Higham, D.S. Mackey, F. Tisseur, and S.D. Garvey. Scaling, sensitivity
and stability in the numerical solution of quadratic eigenvalue problems. Int.
J. Numer. Meth. Engrg., 73:344–360, 2008.
[11] K. Meerbergen. Locking and restarting quadratic eigenvalue solvers. SIAM J.
Sci. Comput., 22(5):1814–1839, 2001.
[12] V. Hernandez, J.E. Roman, and V. Vidal. SLEPc: A scalable and flexible
toolkit for the solution of eigenvalue problems. ACM T. Math Software, 31:351–
362, 2005.
[13] T.-M. Huang, W. Wang, and C.-T. Lee. An efficiency study of polynomial
eigenvalue problem solvers for quantum dot simulations. Taiwanese J. Math.,
14:999–1021, 2010. |