A parallel additive schwarz preconditioned jacobi-davidson algorithm for polynomial eigenvalue problems in quantum dot simulation

Journal of Computational Physics

Volumn 229, Issue 8, 2010, Pages 2932-2947

  Hwang, Feng Nan ^a   Wei, Zih Hao ^a   Huang, Tsung Ming ^b   Wang, Weichung ^c  

^a NATIONAL CENTRAL UNIVERSITY   (Taiwan)

^b NATIONAL TAIWAN NORMAL UNIVERSITY   (Taiwan)

^c NATIONAL TAIWAN UNIVERSITY   (Taiwan)

Author keywords

Jacobi Davidson methods; Parallel computing; Polynomial eigenvalue problems; Quantum dot simulation; Restricted additive Schwarz preconditioning; Schr dinger's equation

Indexed keywords

EIGENVALUES AND EIGENFUNCTIONS; NANOCRYSTALS; POLYNOMIALS; SEMICONDUCTOR QUANTUM DOTS;

ADDITIVE SCHWARZ PRECONDITIONING; EIGENPAIRS; JACOBI- DAVIDSON; JACOBI-DAVIDSON ALGORITHM; JACOBI-DAVIDSON METHODS; PARALLEL COM- PUTING; POLYNOMIAL EIGENVALUE PROBLEMS; QUANTUM DOT SIMULATION; RESTRICTED ADDITIVE SCHWARZ PRECONDITIONING; SCHRÖDINGE EQUATION;

PARALLEL PROCESSING SYSTEMS;

EID: 78649447470     PISSN: 00219991     EISSN: 10902716     Source Type: Journal    
DOI: 10.1016/j.jcp.2009.12.024     Document Type: Article

References (49)

1. P. Arbenz, M. Bečka, R. Geus, U. Hetmaniuk, T. Mengotti, On a parallel multilevel preconditioned Maxwell eigensolver, Parallel Comput. 32 (2006) 157-165.
2. S. Balay, K. Buschelman, W.D. Gropp, D. Kaushik, M.G. Knepley, L.C. McInnes, B.F. Smith, H. Zhang, PETSc Web page, 2009..
3. G. Bastard, Wave Mechanics Applied to Semiconductor Heterostructures, John Wiley and Sons, New York, 1991.
4. L. Bergamaschi, A. Martinez, G. Pini, F. Sartoretto, Eigenanalysis of finite element 3D flow models by parallel Jacobi-Davidson, Lect. Notes Comput. Sci. 2840 (2003) 565-569.
5. L. Bergamaschi, G. Pini, F. Sartoretto, Computational experience with sequential and parallel, preconditioned Jacobi-Davidson for large, sparse symmetric matrices, J. Comput. Phys. 188 (2003) 318-331.
6. M.M. Betcke, H. Voss, Stationary Schrödinger equations governing electronic states of quantum dots in the presence of spin-orbit splitting, Appl. Math. 52 (2007) 267-284.
7. M.M. Betcke, H. Voss, Numerical simulation of electronic properties of coupled quantum dots on wetting layers, Nanotechnology 19 (2008) 165204.
8. T. Betcke, Optimal scaling of generalized and polynomial eigenvalue problems, SIAM J. Matrix Anal. Appl. 30 (2008) 1320-1338.
9. X.-C. Cai, M. Sarkis, A restricted additive Schwarz preconditioner for general sparse linear systems, SIAM J. Sci. Comput. 21 (1999) 792-797.
10. S.L. Chuang, N. Peyghambarian, S. Koch, Physics of Optoelectronic Devices, Wiley, New York, 1996.
11. E.A. de Andrada e Silva, G.C. La Rocca, F. Bassani, Spin-orbit splitting of electronic states in semiconductor asymmetric quantum wells, Phys. Rev. B 55 (1997) 16293.
12. D. El-Moghraby, R.G. Johnson, P. Harrison, Calculating modes of quantum wire and dot systems using a finite differencing technique, Comput. Phys. Commun. 150 (2003) 235-246.
13. K. Elssel, H. Voss, Reducing sparse nonlinear eigenproblems by automated multi-level substructuring, Adv. Eng. Software 39 (2008) 828-838.
14. E. Frisch, M. Frisch, G.W. Trucks, Gaussian 03, Gaussian, Pittsburgh, PA, 2003.
15. F. Gelbard, K.J. Malloy, Modeling quantum structures with the boundary element method, J. Comput. Phys. 172 (2001) 19-39.
16. M.B. Van Gijzen, The parallel computation of the smallest eigenpair of an acoustic problem with damping, Int. J. Numer. Meth. Eng. 45 (1999) 765-777.
17. V. Hernandez, J.E. Roman, V. Vidal, SLEPc: a scalable and flexible toolkit for the solution of eigenvalue problems, ACM Trans. Math. Software 31 (2005) 351-362.
18. T.-M. Huang, W. Wang, C.-T. Lee, An efficiency study of polynomial eigenvalue problem solvers for quantum dot simulations, Technical Report, NCTS Preprints in Mathematics 2009-11-002, National Tsing Hua University, Hsinchu, Taiwan, 2009.
19. T.-M. Hwang, W.-W. Lin, W.-C. Wang, W. Wang, Numerical simulation of three dimensional pyramid quantum dot, J. Comput. Phys. 196 (2004) 208-232.
20. T.-M. Hwang, W. Wang, Energy states of vertically aligned quantum dot array with nonparabolic effective mass, Comput. Math. Appl. 49 (2005) 39-51.
21. T.-M. Hwang, W.-C. Wang, W. Wang, Numerical schemes for three-dimensional irregular shape quantum dots over curvilinear coordinate systems, J. Comput. Phys. 226 (2007) 754-773.
22. T.-M. Hwang, W.-H. Wang, W. Wang, Efficient numerical schemes for electronic states in coupled quantum dots, J. Nanosci. Nanotechnol. 8 (2008) 3695-3709.
23. A.V. Knyazev, Toward the optimal preconditioned eigensolver: locally optimal block preconditioned conjugate gradient method, SIAM J. Sci. Comput. 23 (2001) 517-541.
24. R.B. Lehoucq, J.A. Scott, An evaluation of software for computing eigenvalues of sparse nonsymmetric matrices, Preprint MCS-P, 1195, 1996.
25. Y. Li, J.-L. Liu, O. Voskoboynikov, C.P. Lee, S.M. Sze, Electron energy level calculations for cylindrical narrow gap semiconductor quantum dot, Comput. Phys. Commun. 140 (2001) 399-404.
26. Y. Li, O. Voskoboynikov, C.P. Lee, S.M. Sze, Computer simulation of electron energy levels for different shape InAs/GaAs semiconductor quantum dots, Comput. Phys. Commun. 141 (2001) 66-72.
27. M. Markiewicz, H. Voss, Electronic states in three dimensional quantum dot/wetting layer structures, Lect. Notes Comput. Sci. 3980 (2006) 684-693.
28. R.V.N. Melnik, K.N. Zotsenko, Finite element analysis of electronic states, Modell. Simul. Mater. Sci. Eng. 12 (2004) 465-477.
29. M. Nool, A. van der Ploeg, Parallel Jacobi-Davidson for solving generalized eigenvalue problems, Lect. Notes Comput. Sci. 1573 (1999) 58-70.
30. M. Nool, A. van der Ploeg, A parallel Jacobi-Davidson-type method for solving large generalized eigenvalue problems in magnetohydrodynamics, SIAM J. Sci. Comput. 22 (2000) 95-112.
31. M. Parrinello, From silicon to RNA: the coming of age of ab initio molecular dynamics, Solid State Commun. 102 (1997) 107-120.
32. C. Pryor, Eight-band calculations of strained InAs/GaAs quantum dots compared with one-, four-, and six-band approximations, Phys. Rev. B 57 (1998) 7190-7195.
33. Y. Saad, Iterative Methods for Sparse Linear Systems, SIAM, Philadelphia, 2003.
34. G.L.G. Sleijpen, A.G.L. Booten, D.R. Fokkema, H.A. van der Vorst, Jacobi-Davidson type methods for generalized eigenproblems and polynomial eigenproblems, BIT 36 (1996) 595-633.
35. G.L.G. Sleijpen, H.A. van der Vorst, A Jacobi-Davidson iteration method for linear eigenvalue problems, SIAM Rev. 42 (2000) 267-293.
36. B.F. Smith, P.E. Bjørstad, W. Gropp, Domain Decomposition: Parallel Multilevel Methods for Elliptic Partial Differential Equations, Cambridge University Press, Cambridge, 1996.
37. A. Stathopoulos, Nearly optimal preconditioned methods for Hermitian eigenproblems under limited memory. Part I: Seeking one eigenvalue, SIAM J. Sci. Comput. 29 (2007) 481-514.
38. A. Stathopoulos, J.R. McCombs, Nearly optimal preconditioned methods for Hermitian eigenproblems under limited memory. Part II: Seeking many eigenvalues, SIAM J. Sci. Comput. 29 (2007) 2162-2188.
39. F. Tisseur, Backward error and condition of polynomial eigenvalue problems, Linear Algebra Appl. 309 (2000) 339-361.
40. S. Tomov, J. Langou, A. Canning, L.W. Wang, J. Dongarra, Comparison of nonlinear conjugate-gradient methods for computing the electronic properties of nanostructure architectures?, Lect Notes Comput. Sci. 3516 (2005) 317-325.
41. S. Tomov, J. Langou, J. Dongarra, A. Canning, L. Wang, Conjugate-gradient eigenvalue solvers in computing electronic properties of nanostructure architectures, Int. J. Comput. Sci. Eng. 2 (2006) 205-212.
42. A. Toselli, O.B. Widlund, Domain Decomposition Methods-Algorithms and Theory, Springer, Berlin, 2005.
43. C. Vömel, S.Z. Tomov, O.A. Marques, A. Canning, L.W. Wang, J.J. Dongarra, State-of-the-art eigensolvers for electronic structure calculations of large scale nano-systems, J. Comput. Phys. 227 (2008) 7113-7124.
44. C. Vömel, S.Z. Tomov, L.W. Wang, O.A. Marques, J.J. Dongarra, The use of bulk states to accelerate the band edge state calculation of a semiconductor quantum dot, J. Comput. Phys. 223 (2007) 774-782.
45. H. Voss, Iterative projection methods for computing relevant energy states of a quantum dot, J. Comput. Phys. 217 (2006) 824-833.
46. H. Voss, Numerical calculation of the electronic structure for three-dimensional quantum dots, Comput. Phys. Commun. 174 (2006) 441-446.
47. H. Voss, A Jacobi-Davidson method for nonlinear and nonsymmetric eigenproblems, Comput. Struct. 85 (2007) 1284-1292.
48. W. Wang, T.-M. Hwang, J.-C. Jang, A second-order finite volume scheme for three dimensional truncated pyramidal quantum dot, Comput. Phys. Commun. 174 (2006) 371-385.
49. W. Wang, T.-M. Hwang, W.-W. Lin, J.-L. Liu, Numerical methods for semiconductor heterostructures with band nonparabolicity, J. Comput. Phys. 190 (2003) 141-158.