ADI-preconditioned FGMRES for solving large generalized Lyapunov equations - A case study

Transcription

1 -preconditioned for large - A case study Matthias Bollhöfer, André Eppler TU Braunschweig Institute Computational Mathematics Syrene-MOR Workshop, TU Hamburg October 30, 2008

2 2 / 20 Outline 1 2 Overview CF- 3 CF- with Krylov-subspace 4

3 3 / 20 Outline CF- with 1 2 Overview CF- 3 CF- with Krylov-subspace 4

4 4 / 20 Linear Descriptor system CF- with Eẋ(t) = Ax(t) + Bu(t) y(t) = Cx(t) + Du(t) with x state variables, u input variables, y output variables

5 5 / 20 Model Reduction by Balanced Truncation CF- with When applying BT one has to solve of type AXE T +EXA T = BB T, A T YE +E T YA = C T C. (1) Properties pair (E,A) stable large scale E,A sparse

6 6 / 20 Outline CF- with 1 2 Overview CF- 3 CF- with Krylov-subspace 4

7 7 / 20 Different solvers CF- with Direct methods Bartels-Stewart Hammarlings method... Iterative methods Sign function method type methods (CF-, Cyclic LR-Smith..) (Preconditioned) Krylov-subspace methods...

8 8 / 20 CF- method [Li,White 04] CF- with Cholesky Factor Alternating Direct Iimplicit Iteration computes the Cholesky factor Z of desired solution X = ZZ T Algorithm 1 compute shift params. p 1,...p j e.g. [Wachspress 95] 2 z 1 = 2p 1 (A + p 1 E) 1 B Z = [z] 3 For i=2..j z i = P i 1 z i 1,where 2pi+1 P i = [I (p i+1 + p i )(A + p i+1 E) 1 ] 2pi Z = [Z z i ]

9 9 / 20 Outline CF- with 1 2 Overview CF- 3 CF- with Krylov-subspace 4

10 10 / 20 (m) [Saad 95] CF- with Flexible GMRES, m steps per restart to solve Px = b Algorithm 1 given initial solution x 0 2 Arnoldi process: r 0 = b Px 0 β = r 0 v 1 = r 0 /β For j = 1..m l j := M 1 j v j q = Pl j For i = 1..j h i,j := (q, v i ) q := q h ij v i h j+1,j = q v j+1 = q/h j+1,j X m = x 0 + m i=1 l ic i, where c = (c 1,..., c m ) T solves H m c βe 1 3 Restart if necessary, goto 2

11 11 / 20 Changes in for equation 1 CF- with b BB T x 0 X 0 Y 0 X T 0 r 0 BB T AX 0 Y 0 X T 0 E T EX 0 Y 0 X T 0 AT R 0 S 0 R T 0 K m (r 0, P) = span{r 0, Pr 0,..., P m 1 r 0 } K m (R 0 S 0 R0 T, A, E) = span{r 0 S 0 R T 0, AR 0S 0 R T 0 E T + ER 0 S 0 R T 0 AT,...} span{r 0 S 0 R T 0, R 1S 1 R T 1,...}

12 12 / 20 Changes in for equation 2 CF- with Main observation If x 0 and b are symmetric, low rank matrices then r 0 and all elements of the Krylov-subspace methods are symmetric as well. Explicit structure preservation in Krylov-subspace methods!

13 13 / 20 (m) for CF- with Algorithm 1 given initial solution X 0 Y 0 X T 0 2 R 0 S 0 R0 T = BBT AX 0 Y 0 X0 T E T EX 0 Y 0 X0 T AT β = R 0 S 0 R0 T V 1 = R 0 W 1 = S 0 /β For j = 1..m L j D j L T j := M 1 j V j W j Vj T QRQ T AL j D j L T j E T + EL j D j L T j A T For i = 1..j h i,j := (QRQ T, V i W i V T i ) QRQ T QRQ T h ij V i W i V T i h j+1,j = QRQ T, V j+1 W j+1 V j + 1 = QRQ T /h j+1,j X m Y m X T m X 0 Y 0 X T 0 + m i=1 L id i L T i c i, where c = (c 1,..., c m ) T solves H m c βe 1

14 14 / 20 CF- with Remarks 2 is replaced by F standard scalar product (u, v) is replaced by trace(u T V ) usually the ranks can be reduced on the fly using full rank decompositions (SVD,QR...) backward-error can be used as stopping criterion

15 15 / 20 CF- with Remarks 2 is replaced by F standard scalar product (u, v) is replaced by trace(u T V ) usually the ranks can be reduced on the fly using full rank decompositions (SVD,QR...) backward-error can be used as stopping criterion in general preconditioning does not preserve symmetric low-rank matrices CF- used as preconditioner (symmetry is preserved by construction)

16 16 / 20 Outline CF- with 1 2 Overview CF- 3 CF- with Krylov-subspace 4

17 17 / 20 Test example CF- with Consider the discretized 1D Laplacian denoted by T. We want to solve the equation (E = I) of dimension n T = , b = 1 2 TX + XT T = bb T (2) Rn

18 18 / 20 Preconditioning vs None CF- with With Without n rank steps time steps time e e e e e e e e e e e e e e + 02 Remark without determined the ranks too large.

19 19 / 20 Example with jumping coefficients n = 500 CF- with step CF &CF e e e e e e e e e e e e e e 03 Remark CF is sensitive to wrong shift parameters

20 20 / 20 Benefits CF- with With the combination of both algorithms we were able get benefits of both: fast convergence of structure preserving (symmetry) property of GMRES as Krylov subspace method robustness of GMRES low rank truncation only as accurate as the desired approximate solution (10 8 ) preconditioning allows even coarser truncation (e.g.10 4 ) The combined approach generalizes earlier work in [Damm 08].

21 21 / 20 Future work investigate further parameter influences on algorithm apply this to arising from circuit use further Krylov-subspace methods e.g. BICGStab improve low rank truncation implementation in C

22 22 / 20 Discussion Thank you for your attention!

23 Prof. Dr. M.Sc. H. Faßbender J. Amorocho D. TU Braunschweig TU Braunschweig Prof. Dr. Peter Benner Chemnitz UT Dr. P. Lang ITWM Kaiserslautern Dipl.-Math. techn. A. Schneider Dipl.-Math. techn. Chemnitz UT T. Mach Chemnitz UT Pro M. Bo TU Brau System Reduction for Nanoscale IC Design Dipl.-Math. O. Schmidt ITWM Kaiserslautern Dr. Tatjana Stykel TU Berlin Dipl.-Ma A. E TU Brau Prof. Dr. M. Hinze Dipl.-Math. techn. M.Sc.??? University of Hamburg M. Kunkel H. M. Sahadet M. Vierling University of Hamburg TU Berlin University of Hamburg