Harrachov 2007

1. -Computational Methods with Applications- Aug. 19-25 Harrachov 2007 On a weighted quasi-residual minimization strategy of QMR for solving complex symmetric shifted linear systems Tomohiro Sogabe (Joint work with S.-L. Zhang) Dept. of Computational Science & Engineering, Nagoya University, Japan

2. Outline ・ Complex symmetric shifted linear systems - Shifted linear systems - Shifted COCG, Shifted COCR, Shifted QMR_SYM ・ Shifted QMR_SYM -Advantages and shortcomings over shifted COCG(R) ・Improving the speed of convergence of shifted QMR_SYM (On a weighted quasi-residual minimization strategy) - A proposal of a weight - Comparison of computational cost ・Numerical examples - Large scale electronic structure calculation（Si & Cu atoms） ・Conclusion

3. List of main symbols ・Krylov subspace : ・W-dot product on : ・W-norm on : (W : N-by-N h.p.d. matrix) Note Matrix A is complex symmetric if A is not Hermitian but symmetric, i.e.

4. Shifted linear systems Linear systems Shifted linear systems ・Lattice QCD Numerical computation of the strong interaction between quarks mediated by gluons ・Large scale electronic structure calculation Dynamics computation of nanostructures based on quantum mechanics

5. Our main interest Complex symmetric shifted linear systems Krylov subspace (KS) methods for solving complex symmetric linear systems

6. KS methods KS methods Generate Krylov basis KS methods j i Reuse of the basis No need for matrix-vector and dot products. KS methods for shifted linear systems For more efficient computaion,

7. KS solvers for non-Hermitian shifted linear systems Non-Hermitian GMRES ( Datta & Saad, 1991 ) ( Freund, 1993 ) QMR for SLS ( Jegerlehner, 1996 ) BiCG-M ( Jegerlehner, 1996 ) BiCGSTAB-M .. ( Frommer & Grassner, 2003 ) Shifted GMRES(k) BiCGSTAB(ℓ) Shifted ( Frommer, 2003 )

8. KS methods for complex symmetric shifted linear systems Complex symmetric linear systems (van der Vorst & Melissen, 1990) COCG (S. & Zhang, 2007) COCR (Freund, 1992) QMR_SYM Complex symmetric shifted linear systems Shifted COCG ( Takayama et al., 2006 ) ( S. & Zhang, manuscript 2007 ) Shifted COCR Shifted QMR_SYM ← Readily obtained from two papers by Freund, 1992, 1993.

9. Property of each method COCG: COCR: QMR_SYM: Shifted COCG: Shifted COCR: Shifted QMR_SYM:

10. Choice for weight, e.g. s.t. Comp. symm. linear systems QMR_SYM (Freund, 1992) Approximate solutions The comp. symm. Lanczos process

11. 4-term recurrences relation Comp. symm. linear systems QMR_SYM (Freund, 1992) For n=1,2,… １．Run nth step of the complex symmetric Lanczos process ２．Solve ３．Update End

12. Shifted comp. symm. linear systems Shifted QMR_SYM Approximate solutions The comp. symm. Lanczos process

13. Shifted comp. symm. linear systems Shifted QMR_SYM For n=1,2,… １．Run nth step of the complex symmetric Lanczos process (i=1,2,…,m) ２．Solve ３．Update (i=1,2,…,m) End 4-term recurrences relation

14. Advantage and shortcoming of the shifted QMR_SYM Cost per iteration (1<<m) low high ・Shifted COCG ・Shifted COCR ・Shifted QMR_SYM Need for the choice of a suitable seed system Required Not required ・ShiftedCOCG ・Shifted QMR_SYM Possible to avoid (Cf. S. et al. 2007) ・ShiftedCOCR

15. Need for the choice of a suitable seed system Number of systems: 1001, seed system: σ=0.900+0.001i 4 Shifted COCG 0 -4 -8 -12 0.4 0.6 0.8 1.0 1.2 1.4 Re(σ)

16. If 1<<m, update solutions require the dominant computational cost Comparison of costs per iteration step Shifted QMR_SYM Shifted COCG(R) COCG(R) matrix・vector multiplication 1 1 m O(m) O(m) Polynomial comp. O(m) Update solutions 6Nm 4Nm 4Nm m: number of shifted linear systems N: order of matrices

17. s.t. s.t. （　：upper bidiagonal） A proposal of a weight for least squares problems to reduce the computational cost per iteration Main idea

18. Main idea (cont.)

19. Comparison of costs per iteration step Shifted QMR_SYM Shifted COCG(R) COCG(R) 1 1 m O(m) Polynomial comp. O(m) O(m) Update solutions 6Nm 4Nm 4Nm How ischosen? 4Nm

21. For convenience shifted QMR_SYM with the weight is referred to as shifted QMR_SYM （B） .

22. On property of each method Cost per iteration (1<<m) low high ・Shifted COCG ・Shifted QMR_SYM ・Shifted COCR ・Shifted QMR_SYM(B) Need for the choice of a suitable seed system Required Not required ・ShiftedCOCG ・Shifted QMR_SYM Possible to avoid (Cf. S. et al. 2007) ・Shifted QMR_SYM(B) ・ShiftedCOCR

23. Some properties of shifted QMR_SYM Theorem 1.Let A be real symmetric, If shifted QMR_SYM is applied to the systems of the form then, shifted QMR_SYM has the following properties: 1.All matrix-vector multiplicationscan be done in real arithmetic. For each system i, each approximate solution holds minimal residual norm. 2. ⇒ In terms of number of iterations, shifted QMR_SYM always converges at fewer iterations than shifted COCG(R) . The QMR_SYM holds the above two properties. (Cf. Freund, 1992)

24. Some properties of shifted QMR_SYM(B) Theorem 2.Let A be real symmetric, If shifted QMR_SYM(B) is applied to the systems of the form then, shifted QMR_SYM(B) has the following properties: 1.All matrix-vector multiplicationscan be done in real arithmetic. 2. ⇒ Shifted QMR_SYM(B) and shifted COCG generate approximate solutions with the same residual 2-norm .

25. Comparison of costs per iteration step Shifted QMR_SYM(B) Shifted QMR_SYM Shifted COCG(R) matrix・vector multiplication (real・real) O(m) O(m) O(m) Polynomial comp. Update solutions 4Nm 6Nm 4Nm m: number of shifted linear systems, N: order of matrices

26. Numerical examples Code Stopping criterion

27. （large scale electronic structure calculation） Example 1 Si512 atoms surface reconstructuring simulation (Cf. Takayama et al., Phys. Rev. B, 2006)

28. Shifted COCG Shifted COCG 0 -4 Shifted QMR_SYM （B） -8 Shifted QMR_SYM Shifted QMR_SYM -12 Example 1 σ=1.000+0.001i (log10 || rn ||/ || b ||) Relative residual 2-rnom 0 50 100 150 200 250 Number of iterations

29. 7.2[s] 5.2[s] 5.1[s] Reσ∈ [0.4, 1.4] Example 1 (Ratio of computation time) （The computational time for shifted COCG is scaled to 1.） Shifted QMR_SYM ◆ Shifted QMR_SYM(B) ■ Ratio of computation time Number of shifted linear systems

30. （large scale electronic structure calculation） Example 2 Cu1568 atoms surface reconstructuring simulation (Cf. Takayama et al., Phys. Rev. B, 2006)

31. 118[s] 103[s] 87[s] Reσ∈ [-0.5, 1.0] Example 2 (Ratio of computation time) （The computational time for shifted COCG is scaled to 1.） Shifted QMR_SYM ◆ Shifted QMR_SYM(B) ■ Ratio of computation time Number of shifted linear systems

32. Conclusion 1．Shifted QMR_SYM(B) is proposed. 2．In terms of number of iteration steps Shifted QMR_SYM(B) required almost the same iteration steps as shifted COCG and shifted QMR_SYM, 3．In terms of computational time For small m, it converged faster than the others. For large m, it converged in the almost the same time as shifted COCG and converged about 20% faster than shifted QMR_SYM . 4．No need for the choice of a suitable seed system.

33. Appendix 1 Proof of Theorem 2

34. Appendix 3 ShiftedCOCR COCR

35. Appendix 4 ShiftedCOCG COCG

36. Appendix ５ LS-electronic structure calculation １．Computation of Green’s function ２．Computation of density matrices ３．Physical quantity

37. Appendix 6 LS-electronic structure calculation １． Computation of Green’s function Numerical integration The number of mesh ⇒The number of shifted linear systems

38. 前処理について　 補足資料 ８ 前処理がシフト方程式に対して適用できない例 多項式前処理は適用可能（Cf. Jegerlehner, 1996 ）