Projected Schur Complement Method for Solving Non-Symmetric Saddle-Point Systems (Arising from Fictitious Domain Approaches)

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1 Projected Schur Complement Method for Solving Non-Symmetric Saddle-Point Systems (Arising from Fictitious Domain Approaches) Jaroslav Haslinger, Charles University, Prague Tomáš Kozubek, VŠB TU Ostrava Radek Kučera, VŠB TU Ostrava SNA 07, Januar 22, Ostrava

2 OUTLINE Preliminaries on saddle-point systems Projected Schur complement method Projected BiCGSTAB algorithm Hierarchical multigrid Poisson-like solver for singular matrices Numerical experiments

3 OUTLINE Preliminaries on saddle-point systems Projected Schur complement method Projected BiCGSTAB algorithm Hierarchical multigrid Poisson-like solver for singular matrices Numerical experiments

4 Formulation: Non-symmetric sadle-point system ( A B 1 B 2 0 ) ( u λ ) = ( f g ) Assumptions A... non-symmetric (n n) matrix... singular with p = dim Ker A as V h H 1 per(ω)... actions of A can be computed by Poisson-like solver based on FFT B 1, B 2... full rank (m n) matrices, m n... sparse, their actions are cheap... B 1 B 2 as Γ γ or as mixed boudary condions are prescribed Algorithms based on Schur complement reductions Case 1: symmetric non-singular Case 2: non-symmetric non-singular Case 3: symmetric singular Case 4: non-symmetric singular

5 Case 1: symmetric non-singular (positive definite A) ( A B ) ( u ) ( f ) = u = A 1 (f B λ) B 0 λ = g = BA 1 B λ = BA 1 f g Algorithm Step 1: Assemble d := BA 1 f g. Step 2: Solve Sλ = d by the CGM with S := BA 1 B. Step 3: Assemble u = A 1 (f B λ). Matrix-vector products Sµ are computed by: Sµ := ( B ( A 1 ( B µ ))) Algorithm requires O((m + 2)n A ) flops in the worst case.

6 Case 2: non-symmetric non-singular ( A B 1 B 2 0 λ = g = B 2 A 1 B 1 λ = B 2A 1 f g ) ( u ) ( f ) = u = A 1 (f B 1 λ) Algorithm is analogous. the Schur complement S := B 2 A 1 B 1 is non-symmetric a Krylov method for non-symmetric matrices is required (GMRES, BiCG,...) A := ( A B 1 B 2 0 = B 2 A 1 I ) ( A B 1 0 S ) ( I 0 ) Theorem. Let A be non-singular. Then A is invertible iff S is invertible. First Prev Next Last Go Back Full Screen Close Quit

7 Case 3: symmetric singular (positive semidefinite A, FETI) a generalized inverse A satisfying A = AA A an (n p) matrix N whose columns span Ker A Au + B λ = f f B λ Im A Ker A u = A (f B λ) + Nα N (f B λ) = 0 & Bu = g BA B λ BNα = BA f g Reduced system: ( BA B BN N B 0 )( λ α ) ( BA = f g N f an orthogonal projector + the projected CG )

8 Case 4: non-symmetric and singular a generalized inverse A satisfying A = AA A an (n p) matrices N, M whose columns span Ker A, Ker A, respectively Au + B 1 λ = f f B 1 λ Im A Ker A u = A (f B 1 λ) + Nα M (f B 1 λ) = 0 & B 2 u = g B 2 A B 1 λ B 2Nα = B 2 A f g Reduced system: ( B2 A B 1 B 2 N M B 1 0 )( λ α ) ( B2 A = f g M f two orthogonal projectors + the projected BiCG )

9 ( A B ) Theorem 1. The saddle-point matrix A := 1 is invertible iff B 2 0 B 1, B 2 have full ranks Ker A Ker B 2 = {0} (NSC) A Ker B 2 Im B 1 = {0} The generalized Schur complement: ( B2 A B 1 B 2 N S := M B 1 0 ) Theorem 2. The following three statements are equivalent: The necessary and sufficient condition (NSC) holds. A is invertible. S is invertible.

10 OUTLINE Preliminaries on saddle-point systems Projected Schur complement method Projected BiCGSTAB algorithm Hierarchical multigrid Poisson-like solver for singular matrices Numerical experiments

11 New notation in the reduced system: F = B 2 A B 1, G 1 = N B 2, G 2 = M B 1, d = B 2 A f g, e = M f ( F G 1 G 2 0 ) ( λ α ) = ( d e ) Two orthogonal projectors: P 1 : R m Ker G 1, P 1 = I G 1 (G 1 G 1 ) 1 G 1 P 1 splits the saddle-point structure of the system: P 2 : R m Ker G 2, P 2 = I G 2 (G 2 G 2 ) 1 G 2 P 1 Fλ = P 1 d, G 2 λ = e P 2 decomposes λ = λ Im + λ Ker, λ Im Im G 2, λ Ker Ker G 2. At first: λ Im = G 2 (G 2 G 2 ) 1 e At second: P 1 Fλ Ker = P 1 (d Fλ Im ) on Ker G 2

12 Theorem 3. The linear operator P 1 F: Ker G 2 Ker G 1 is invertible. Proof. As both null-spaces Ker G 1 and Ker G 2 have the same dimension, it is enough to prove that P 1 F is injective. Let µ Ker G 2 be such that P 1 Fµ = 0. Then Fµ Ker P 1 = Im G 1 and, therefore, there is β R l so that Fµ = G 1 β and G 2µ = 0. Letting y = (µ, β ), we get Sy = 0 implying µ = 0 as S is invertible.

13 Algorithm PSCM Step 1.a: Assemble G 1 := N B 2, G 2 := M B 1. Step 1.b: Assemble d := B 2 A f g, e := M f. Step 1.c: Assemble H 1 := (G 1 G 1 ) 1, H 2 := (G 2 G 2 ) 1. Step 1.d: Assemble λ Im := G 2 H 2e, d := P1 (d Fλ Im ). Step 1.e: Solve P 1 Fλ Ker = d on Ker G2. Step 1.f: Assemble λ := λ Im + λ Ker. Step 2: Assemble α := H 1 (d Fλ). Step 3: Assemble u = A (f B 1 λ) + Nα. An iterative projected Krylov subspace method for non-symmetric operators can be used in Step 1.e. Matrix-vector products Fµ and P k µ, k = 1, 2 are computed by: Fµ := ( B 2 ( A ( B 1 µ))), P k µ := µ ( G k (H k (G k µ)) ).

14 OUTLINE Preliminaries on saddle-point systems Projected Schur complement method Projected BiCGSTAB algorithm Hierarchical multigrid Poisson-like solver for singular matrices Numerical experiments

15 Fλ = d on R m Algorithm BiCGSTAB [ɛ, λ 0, F, d ] λ Initialize: r 0 := d Fλ 0, p 0 := r 0, r 0 arbitrary, k := 0 While r k > ɛ 1 p k := Fp k 2 α k := (r k ) r 0 /( p k ) r 0 3 s k := r k α k p k 4 s k := Fs k 5 ω k := ( s k ) s k /( s k ) s k 6 λ k+1 := λ k + α k p k + ω k s k 7 r k+1 := s k ω k s k 8 β k+1 := (α k /ω k )(r k+1 ) r 0 /(r k ) r 0 9 p k+1 := r k+1 + β k+1 (p k ω k p k ) 10 k := k + 1 end

16 P 1 Fλ = d on Ker G2 Algorithm ProjBiCGSTAB [ɛ, λ 0, F, P 1, P 2, d ] λ Initialize: λ 0 Ker G 2, r 0 := d P1 Fλ 0, p 0 := r 0, r 0 arbitrary, k := 0 While r k > ɛ 1 p k := P 1 Fp k 2 α k := (r k ) r 0 /( p k ) r 0 3 s k := r k α k p k 4 s k := P 1 Fs k 5 ω k := ( s k ) s k /( s k ) s k 6 λ k+1 := λ k + α k P 2 p k + ω k P 2 s k 7 r k+1 := s k ω k s k 8 β k+1 := (α k /ω k )(r k+1 ) r 0 /(r k ) r 0 9 p k+1 := r k+1 + β k+1 (p k ω k p k ) 10 k := k + 1 end

17 P 1 Fλ = d on Ker G2 Algorithm ProjBiCGSTAB [ɛ, λ 0, F, P 1, P 2, d ] λ Initialize: λ 0 Ker G 2, r 0 := P 2 ( d P1 Fλ 0 ), p 0 := r 0, r 0, k := 0 While r k > ɛ 1 p k := P 2 P 1 Fp k 2 α k := (r k ) r 0 /( p k ) r 0 3 s k := r k α k p k 4 s k := P 2 P 1 Fs k 5 ω k := ( s k ) s k /( s k ) s k 6 λ k+1 := λ k + α k p k + ω k s k 7 r k+1 := s k ω k s k 8 β k+1 := (α k /ω k )(r k+1 ) r 0 /(r k ) r 0 9 p k+1 := r k+1 + β k+1 (p k ω k p k ) 10 k := k + 1 end

18 OUTLINE Preliminaries on saddle-point systems Projected Schur complement method Projected BiCGSTAB algorithm Hierarchical multigrid Poisson-like solver for singular matrices Numerical experiments

19 Consider a family of nested partitions of the fictitious domain Ω with stepsizes: h j, 0 j J the first iterate is determined by the result from the nearest lower level the terminating tolerance ɛ on each level is ɛ := νh p j Algorithm: Hierarchical Multigrid Scheme Initialize: Let λ 0,(0) Ker Ker (G (0) 2 ) be given. ProjBiCGSTAB[νh p 0, λ 0,(0) Ker, F (0), P (0) 1, P (0) 2, d(0) ] λ Ker (0). For j = 1,..., J, 1 prolongate λ (j 1) Ker 2 project λ0,(j) Ker λ 0,(j) Ker λ0,(j) Ker := P (j) 0,(j) 2 λ Ker 3 ProjBiCGSTAB[νh p j, λ 0,(j) Ker, F (j), P (j) 1, P (j) 2, d(j) ] λ (j) Ker end Return: λ Ker := λ (J) Ker.

20 OUTLINE Preliminaries on saddle-point systems Projected Schur complement method Projected BiCGSTAB algorithm Hierarchical multigrid Poisson-like solver for singular matrices Numerical experiments

21 Circulant matrices and Fourier transform A = a 1 a n... a 2 a 2 a 1... a 3 a 3 a 2... a a n a n 1... a 1 = ( a, Ta, T 2 a,, T n 1 a ) T k f(ω) = R f(x k)e ixω dx = e ikω f(ω) XA = (Dx 0, Dx 1, Dx 2,, Dx n 1 ) = DX Let A be circulant. Then A = X 1 DX, where X is the DFT matrix and D = diag(â), â = Xa, a = A(:, 1).

22 Multiplying procedure: A v = X 1 ( D (Xv) )... Moore-Penrose 0 d := fft(a) 1 v := fft(v) 2 v := v. d 1 3 A v := ifft(v) O(2n log 2 n) Multiplying procedures: Nα, N v and Mα, M v I DD = diag(1, 1, 1, 0,..., 0) = X 1 = (N, Y) Therefore we can define the operation: ind(α) = 1 v α := ind(α) 1 v := ifft(v) ( α 0 2 Nα := ifft(v α ) 2 N v := ind 1 (v) ) } R n O(n log 2 n)

23 Kronecker product of matrices: A x A y = A x R n x n x, A y R n y n y a y 11A x... a y 1n y A x..... a y n y 1A x... a y n y n y A x Lemma 1: (A x A y )(B x B y ) = A x B x A y B y (A x A y ) = A x A y N = N x N y Lemma 2: (A x A y )v = vec(a x VA y ), where V = vec 1 (v). V = (v 1,..., v ny ) R n x n y vec(v) = v 1. R n xn y v ny

24 Kronecker product and circulant matrices: Let A x, A y be circulant, then: A = A x I y + I x A y = X 1 x D xx x X 1 X y + X 1 X x X 1 D yx y y = (X 1 x X 1 y )(D x I y + I x D y )(X x X y ) = X 1 DX x y with X = X x X y (DFT matrix in 2D) D = D x I y + I x D y (diagonal matrix) where X x, X y are the DFT matrices, D x = diag(x x a x ), D y = diag(x y a y ) and a x = A x (:, 1), a y = A y (:, 1), respectively.

25 Multiplying procedure: A v = X 1 ( D (Xv) ) 0 d x := fft(a x ), d y := fft(a y ) V := vec 1 (v) 1 V := fft(v) 2 V := fft(v ) 3 V := vec 1 (D vec(v)) 4 V := ifft(v) 5 V := ifft(v ) A v := vec(v) Number of arithmetic operations : O(2n(log 2 n x + log 2 n y ) + n) O(n log 2 n), n = n x n y Multiplying procedures: Nα, N v, Mα, M v... analogous

26 OUTLINE Preliminaries on saddle-point systems Projected Schur complement method Projected BiCGSTAB algorithm Hierarchical multigrid Poisson-like solver for singular matrices Numerical experiments

27 Ω = (0, 1) (0, 1) ω = {(x, y) R 2 : (x 0.5) 2 + (y 0.5) 2 = } { u = f in ω u = g on γ where the right hand-sides f, g are chosen appropriately to the exact solution û ex (x, y) = 100((x 0.5) 3 (y 0.5) 3 ) x 2. The auxiliary boundary Γ is obtained by shifting γ in the normal direction with δ = 8h. 1 Ω ν ω γ Γ Figure 1: Geometry of ω. Figure 2: Right hand side f. Figure 3: Ex. solution û ex ω.

28 Classical FDM Step h n/m Iters. C.time[s] Err L2 (ω) Err H1 (ω) Err L2 ( ω) 1/ / e e e-2 1/ / e e e-2 1/ / e e e-2 1/ / e e e-3 1/ / e e e-3 Convergence rates: New FDM, PSCM Step h n/m Iters. C.time[s] Err L2 (ω) Err H1 (ω) Err L2 ( ω) 1/ / e e e-3 1/ / e e e-4 1/ / e e e-4 1/ / e e e-5 1/ / e e e-5 Convergence rates: New FDM, PSCM+Multigrid Step h n/m Iters. C.time[s] Err L2 (ω) Err H1 (ω) Err L2 ( ω) 1/ / e e e-3 1/ / e e e-4 1/ / e e e-5 1/ / e e e-5 1/ / e e e-5 Convergence rates:

29 New FDM: BiCGM - iterations, discretization error Classical FDM New FDM New FDM+Multigrid Step h Iters. Err H 1 (ω) Iters. Err H 1 (ω) Iters. Err H 1 (ω) 1/ e e e-2 1/ e e e-3 1/ e e e-3 1/ e e e-3 1/ e e e-4 Conv. rates: H 1 error 10 1 cond(p 1 F N(G 2 )) δ δ Figure 1: H 1 (ω)-error sensitivity on δ. Figure 2: cond(p 1 F N(G 2 )) sensitivity on δ.

30 Conclusions The efficient implementation of the FDM with an auxiliary boundary. The saddle-point system is solved by the PSCM (non-symmetric analogy of FETI). The fast Poisson-like solver for singular matrices are used. The fast implementation is matrix free.

31 References R. Glowinski, T. Pan, J. Periaux (1994): A fictitious domain method for Dirichlet problem and applications. Comput. Meth. Appl. Mech. Engrg. 111, Farhat, C., Mandel, J., Roux, F., X. (1994): Optimal convergence properties of the FETI domain decomposition method, Comput. Methods Appl. Mech. Engrg., 115, R. K. (2005): Complexity of an algorithm for solving saddle-point systems with singular blocks arising in wavelet-galerkin discretizations, Appl. Math. 50, J. Haslinger, T. Kozubek, R.K., G. Peichel (2007): Projected Schur complement method for solving non-symmetric systems arising from a smooth fictitious domain approach, in preparing.