-preconditioned for large - A case study Matthias Bollhöfer, André Eppler TU Braunschweig Institute Computational Mathematics Syrene-MOR Workshop, TU Hamburg October 30, 2008
2 / 20 Outline 1 2 Overview CF- 3 CF- with Krylov-subspace 4
3 / 20 Outline CF- with 1 2 Overview CF- 3 CF- with Krylov-subspace 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 / 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 / 20 Outline CF- with 1 2 Overview CF- 3 CF- with Krylov-subspace 4
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 / 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 / 20 Outline CF- with 1 2 Overview CF- 3 CF- with Krylov-subspace 4
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 / 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 / 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 / 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 / 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 / 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 / 20 Outline CF- with 1 2 Overview CF- 3 CF- with Krylov-subspace 4
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. 2 1. T = 1 2........ 1, b = 1 2 TX + XT T = bb T (2) 1 1. 1 Rn
18 / 20 Preconditioning vs None CF- with With Without n rank steps time steps time 20 9 5 1.919e 02 38 1.827e 01 30 11 6 2.601e 02 106 1.364e + 00 40 12 5 3.059e 02 201 5.593e + 00 50 13 5 5.600e 02 408 2.152e + 01...... 80 15 6 7.152e 02 1029 2.938e + 02 90 15 6 1.036e 01 1296 5.426e + 02 100 15 6 8.857e 02 1531 8.489e + 02 Remark without determined the ranks too large.
19 / 20 Example with jumping coefficients n = 500 CF- with step CF &CF 0 5.0000e + 02 5.0000e + 02 1 1.4350e + 02 1.3831e + 02 2 4.8951e + 01 2.8610e + 01 3 2.0432e + 01 1.2106e + 01... 11 1.8789e + 00 1.4227e 02 12 1.8724e + 00 7.3986e 03 13 1.8659e + 00 3.0673e 03 Remark CF is sensitive to wrong shift parameters
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 / 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 / 20 Discussion Thank you for your attention!
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