Inexact Data-Sparse BETI Methods by Ulrich Langer. (joint talk with G. Of, O. Steinbach and W. Zulehner)

Transcription

1 Inexact Data-Sparse BETI Methods by Ulrich Langer (joint talk with G. Of, O. Steinbach and W. Zulehner) Radon Institute for Computational and Applied Mathematics Austrian Academy of Sciences and Institute for Computational Mathematics Johannes Kepler University Linz, Austria This paper was supported by the Austrian Sciene Fund (FWF) and by the German Research Foundation (DFG)

2 Contents 1. Introduction 2. Data-Sparse Boundary Element Domain Decompostion Methods 3. Inexact BETI Methods 4. Preconditioning and Solution of Two-fold Saddle Point Problems 5. Data-Sparse Boundary Element Preconditioners 6. Numerical Results 7. Summary and Work in Progress

3 1. Introduction A lot of FETI papers have been published since Farhat and Roux proposed this DD technique in 1991: new applications: structural mechanics, contact problems, Helmholtz,... new versions: FETI-2, FETI-DP, Algebraic versions etc. FETI Analysis History: Mandel and Tezaur (1996): Dirichlet FETI preconditioner (PC) C: Klawonn and Widlund (2000,2001): New FETI PCs; Inexact versions: Brenner (2002, 2003): Schwarz analysis! Mandel and Tezaur (2001), Klawonn, Widlund, Dryja (2002),...: FETI-DP κ(c 1 F ) O((1 + log(h/h)) 2 ) and robust!

4 BETI = Boundary Element counterpart of FETI: Langer and Steinbach (2002/2003): BETI Langer and Steinbach (2003 / DD15): BETI-FETI-Coupling Idea: Start from the skeleton formulation of elliptic BVP: p p (S i u)(x)v(x)ds x = [(N i f)(x) (S i g)(x)]v(x)ds x i=1 Γ i=1 i Γ i (1) q p q p Ri Si,hR FEM i u + Ri Si,hR BEM i u = Ri f FEM R i i Si,hR BEM i g (2) i=1 i=q+1 i=1 i=q+1

5 In this talk: = Inexact (no elimination of any unkowns!) = Data-sparse approximations of all BE matrices (V, K, K T, D): Ṽ V, K K, D D: X X = O(discretization error) and property preserving! = Two-fold saddle point problem for determining t, u, λ = Subspace iteration for n-fold saddle point problems (n = 2) = Data-sparse preconditioners for Ṽi, S i, F = B T i S + i B i Ultimate Goal for Inexact Data-Sparse BETI = asymptotically almost optimal complexity + robustness! Complexity = O(N S ln q (N S )) = O(h (d 1) ln q (h 1 ))!

6 2. Data-Sparse Boundary Element DDM Model boundary value problem (BVP): d = 2, 3 div(α û) = 0 in Ω R d and û = g on Γ = Ω = Γ D (3) Non-overlapping quasi-regular domain decomposition (DD): Ω Ω Ω 7 Ω 8 9 Ω α Ω Ω Ω Ω n 1 α Ω = Ω 1 Ω q Ω q+1 Ω p Ω i Ω j = for i j, Γ ij = Γ i Γ j, Γ i = Ω i, Γ S = p i=1 Γ i Γ C = Γ S \ Γ D = Γ S \ Γ diam(ω i ) = O(H), α(x) = α i > 0, x Ω i, i = 1,..., p (4)

7 Therefore, the weak form of our model BVP (3): Find û H 1 (Ω) with given û Γ = g H 1/2 (Γ) such that: p α i i=1 Ω i û(x) v(x)dx = 0 v H 1 0 (Ω), (5) can be rewritten in the form p α i û(x)v(x)ds x = 0 for all v H n 0 1 (Ω) (6) i i=1 Γ i by the use of Green s first formula and the fact that û(x) = 0 for x Ω i. (7)

8 The solutions of the local subproblems (7) are given by the representation formula that holds for x Ω i : û(x) = E(x, y) û(y)ds y E(x, y)û(y)ds y (8) n i n i Γ i where E(x, y) = (1/(2π)) log x y resp. E(x, y) = (1/(4π)) x y 1 is the fundamental solution of in 2D resp. 3D. Taking the trace on Γ i and the normal derivative gives Caldaron s representation of the Cauchy date on Γ i : 1 2 u i(x) := 1 1 2û Γ i (x) =... 2 t i(x) := 1 u i (x) =..., x Γ 2 n i (9) i Γ i

9 The complete Cauchy data (u i, t i = u i / n i ) on Γ i satisfy the Calderon equations ( ) ( 1 ui = 2 I K ) ( ) i V i ui 1 t i D i 2 I + (10) K i t i where (V i t i )(x) := E(x, y)t i (y)ds y, x Γ i, Γ i (K i u i )(x) := E(x, y)u i (y)ds y, x Γ i, n y Γ i (K it i )(x) := E(x, y)t i (y)ds y, x Γ i, and n x Γ i (D i u i )(x) := E(x, y)u i (y)ds y, x Γ i, n x n y Γ i

10 Inserting the second boundary integral equation of (10) into (6) we have to find û H 1/2 (Γ S ) with û Γ = g such that p [ α i D i û Γi, v Γi Γi + ( 1 ] 2 I + K i )t i, v Γi Γi = 0 (11) i=1 for all test functions v H 1/2 0 (Γ S ) = {v H 1/2 (Γ S ) : v Γ = 0}, where t i H 1/2 (Γ i ) are the unique solutions of the local variational problems [ α i V i t i, τ i Γi ( 1 ] 2 I + K i)û Γi, τ i Γi = 0 (12) for all test functions τ i H 1/2 (Γ i ). REF.: Hsiao and Wendland (1991), Costabel (1987)

11 After homogenization of the Dirichlet boundary condition via the ansatz û = ĝ + u with ĝ Γ = g and u Γ = 0, we can rewrite (11) and (12) as mixed variational problem to find t = (t 1, t 2,..., t p ) T = T 1 T 2... T p = H 1/2 (Γ 1 ) H 1/2 (Γ 2 )... H 1/2 (Γ p ) and u U = H 1/2 0 (Γ S ) such that [ α i τ i, V i t i Γi τ i, ( 1 ] 2 I + K i)u Γi Γi = τ i, g i for all τ i T i (13) for i = 1, 2,..., p and p [ α i ( 1 ] 2 I + K i)t i, v Γi Γi D i u Γi, v Γi Γi i=1 where the right-hand sides are given by the relations = F, v v U, (14) τ i, g i = α i τ i, ( 1 2 I + K i)ĝ Γi Γi i = 1, 2,..., p, (15) F, v = p α i f i, v Γi Γi = i=1 p α i D i ĝ Γi, v Γi Γi. (16) i=1

12 The Galerkin discretization of the symmetric mixed variational formulation (13) and (14) now gives immediately the mixed boundary element equations to find t h = (t 1,h, t 2,h,..., t p,h ) T h = T 1,h T 2,h... T p,h (e.g. piecewise constant) and u h U h (e.g. continuous piecewise linear) such that [ α i τ i,h, V i t i,h Γi τ i,h, ( 1 ] 2 I + K i)u h Γi Γi = τ i,h, g i (17) for all τ i,h T i,h, i = 1, 2,..., p, and p α i [ ( 1 ] 2 I + K i)t i,h, v h Γi Γi D i u h Γi, v h Γi Γi + i=1 = F, v h (18) for all v h U h. { } u u h 2 U + t t h 2 T c inf u v h 2 U + inf t τ h 2 T. (19) v h U h τ h T h

13 The symmetric mixed boundary element equations (17)-(18) now immediately gives the symmetric, but indefinite Galerkin system with dense block matrix entries: α 1 V 1,h α 1 K 1,h R 1,h t 1. g α p V p,h α p K p,h R p,h t pu =. g p, (20) α 1 R1,h K 1,h... α prp,h K p,h D h f where K i,h = [ ψl, i ( 1 2 I + K i)ϕ i m L2 (Γ i )], (21) p D h = α i Ri,hD i,h R i,h, (22) f = i=1 p Ri,hf i. (23) i=1

14 Data-Sparse Approximation via symmetric FMM: α 1 Ṽ 1,h α 1 K 1,h R 1,h.... α p Ṽ p,h α 1 R1,h K 1,h... α prp,h K p,h Results: Data-sparse and property preserving! α p Kp,h R p,h D h t 1. t pũ = g 1. g p f. (24) Quasi-optimal error estimate: { } u ũ h 2 U + t t h 2 T c inf u v h 2 U + inf t τ h 2 T. (25) v h U h τ h T h

15 3. Inexact BETI In order to avoid assambled matrices and vectors, we tear the global potential (displacements) vector u on the subdomain boundaries Γ i by introducing the individual local unknows u i = R i,h u. (26) Conversely, the global continuity of the potentials is enforced by the constraints p B i u i = 0 (27) i=1 again interconnecting the local potential vectors across the subdomain boundaries as usual in FETI.

16 By introducing Lagrange multipliers λ R L, we easily see that the one-fold saddle point problem (24) is equivalent to the twofold saddle point problem: Find t = (t 1,..., t p ) R M, u = (u 1,..., u p ) R N, and λ = (λ 1,..., λ L ) R L such that V K 0 K D B 0 B 0 t u λ = g f 0, (28) where V = blockdiag(α i Ṽ i,h ), D = blockdiag(α i Di,h ), K = blockdiag( α i Ki,h ), B = (B 1,..., B p ), g = ( g 1,..., g p ) R M and f = ( f 1,..., f p ) R N are given..

17 For the floating subdomains Ω j, j = 1,..., q, we have D j,h e j = 0, Kj,h e j = 0, Sj,h e j = 0, (29) where S j,h e j = α j ( D j,h + K j,hṽ j,h 1 K j,h ) are the FMM BE SC. If Dj,h s psd is replaced by D j,h + β j e j e j spd then there exists a unique solution v j of the regularized equations α j ( D j,h + β j e j e j )v j = f j + B j λ α j K j,h t j (30) If the right-hand side of (30) fulfils the solvability conditions for the original equations without the regularization term e j ( f j + B j λ α j K i,h t j ) = e j ( f j + B j λ) = 0, (31) then the solution v j of (30) is orthogonal to ker D j,h, i.e. e j v j = 0.

18 Now the solution u = (u 1,..., u p ) of the original system (28) can be recovered by the formulas u i = v i + γ i e i, (32) with γ i = 0 for i = q + 1,..., p and appropriatly choosen γ i for i = 1,..., q. Taking into account (29) - (32), we easily observe that the unique solution t R M, v R N, λ R L and γ R q of the three-fold saddle point problem V K 0 0 K D B 0 0 B 0 G 0 0 G 0 t v λ γ = g f 0 e (33) immediately yields the solution (t, u, λ) of the two-fold saddle point problem (28), where now D is SPD, since we replaced the singular blocks D j,h by the regularized ones for the floating subdomains. The L q matrix G and the vector e R q are defined by the relations G = (B 1 e 1,..., B q e 1 ) and e = (e 1 f 1,..., e q f q ). (34)

19 Let us now introduce the orthogonal projection P = I G(G G) 1 G : Λ := R M Λ 0 = ker G = (range G) Λ (35) Since P G = 0, the application of P to the third equation of (33) gives P Bv = 0 that excludes γ from the first three equations of (33). Let us now represent λ as λ = T 0 λ 0 + λ e (36) with known λ e = G(G G) 1 e (ker G ) = range G, fulfilling the constraints G λ e = e, and unknown T 0 λ 0 ker G, i.e. G T 0 λ 0 = 0, where λ 0 R L 0 and L 0 = dim Λ 0. The columns of T 0 span a basis of Λ 0. Now we can define t, v and λ 0 from the two-fold saddle point problem V K 0 K D B P T 0 0 T0 P B 0 t v λ 0 = g d 0, (37) where here d = f B λ e. Once t, v, λ 0 are defined from (37), we get λ from (36), γ = (G G) 1 G Bv from the third equation of (33), and, finally, u from (32).

20 4. Preconditioning and Solution of Two-fold SPP Let us rewrite the two-fold saddle point problem (37) in the form A 1 B1 0 B 1 A 2 B2 =, (38) 0 B 2 0 x 1 x 2 x 3 where A 1 = V, A 2 = D, B 1 = K and B 2 = T 0 P B. The following matrices are SPD: S 1 = A 1 = V = blockdiag(α i Ṽ i,h ) A 2 = D = blockdiag(α i Di,h + β j e j e j ) S 2 = A 2 + B 1 A 1 1 B 1 = D + K V 1 K S 3 = B 2 S 1 2 B 2 = T 0 P B(D + K V 1 K) 1 B P T 0 = F b 1 b 2 b 3

21 It is easy to see that the system matrix A 1 B1 0 S B 1 A 2 B2 1 I S 1 = B 1 S 2 I 0 B 2 0 B 2 S 3 1 BT 1 S 1 2 BT 2 I of (38) possesses a block LU-decomposition K 3 = L 3 U 3 (39) that motivates the preconditioner (denoted by hats ˆX) ( ) τ τ 2 τ 1 Ŝ 1 ˆL 3 = 2 ˆL2 = τ B 2 E 2 Ŝ 2 B 1 τ 2 Ŝ 2 (40) 3 B 2 Ŝ 3

22 Theorem 4.1 If the parameter τ 1 is chosen such that 0 < τ 1 < λ min (Ŝ 1 1 A 1) then the matrix ˆL 1 2 K 2 is symmetric and positive definite in the scalar product (u, v) 2 = (D 2 u, v) l2 with ( ) ( τ ˆL 2 = 1 Ŝ 1 A1 B, K 2 = T ) ) 1 (A, D B 1 Ŝ2 B 1 A 2 = 1 τ 1 Ŝ 1. 2 Ŝ2 ˆL 1 If the parameter τ 2 is chosen such that 0 < τ 2 < λ min ( 2 K 2) then the preconditioned matrix ˆL 1 3 K 3 is SPD in the scalar product (u, v) 3 = (D 3 u, v) l2 with ( ) D D 3 = 2 ( ˆL 1 2 K 2 τ 2 I) 0. 0 Ŝ 3

23 Theorem 4.2 If σ 1 Ŝ 1 S 1 σ 1 Ŝ 1, σ 2 Ŝ 2 S 2 σ 2 Ŝ 2, σ 3 Ŝ 3 S 3 σ 3 Ŝ 3 and the parameters τ 1 and τ 2 are chosen such that τ 1 < λ 1, τ 2 < λ 2, then the matrices ˆL 1 2 K 2 and ˆL 1 3 K 3 are symmetric and positive definite with rerspect to the scalar products (.,.) 2 and (.,.) 3 and λ min ( ˆL 1 3 K 3) λ 3, λ max ( ˆL 1 3 K 3) λ 3, κ( with explicit formulas for λ 1, λ 2, λ 3, λ 3 = λ(σ)! ˆL 1 3 K 3) λ 3 /λ 3 = Use PCG-Solver with PC ˆL 3 and SP (D 3 u, v) l2!

24 5. Data-Sparse BE Preconditioners ˆX X Ŝ1 = ˆV V, i.e. ˆVi,h Ṽi,h (local!) = Artificial Multilevel Single Layer Preconditioner proposed by Steinbach (2003): σ 1 /σ 1 = O((1 + log(h/h)) 2 ) Ŝ2 S 2 = D + K V 1 K = S BE, i.e. Ŝ BE,i S BE,i (local!) = Opposite order preconditioner Ŝ 1 BE,i = M 1 i,hṽ i,hm 1 i,h proposed by Steinbach & Wendland (1998) (dense): σ 2 /σ 2 = O(1) Ŝ3 = ˆF S 3 = F = T 0 P B(D + K V 1 K) 1 B P T 0 = Data-sparse scaled hypersingular BETI preconditioner proposed by Langer and Steinbach (2002) in our BETI paper: ˆF 1 = Q DQ: σ 3 /σ 3 = O((1 + log(h/h)) 2 )

25 Theorem 5.1 (Final Result) If σ 1 /σ 1 = O(1) (e.g. Data-sparse GMG- or AMG-PC), σ 2 /σ 2 = O(1) (e.g. opposite order preconditioner), σ 3 /σ 3 = O((1 + log(h/h)) 2 ) (e.g. scaled data-sparse hypersingular BETI preconditioner), then the PCG acceleration gives the complexity estimates: = Iter(ε) = O((1 + log H h ) log 1 ε ) = Ops parallel (ε) = O(( H h )d 1 (1 + log H h )3 log 1 ε ) Robust with respect to coefficient jumps!

26 6. Numerical Results We solve the Dirichlet potential problem div(α û) = 0 in Ω with given Dirichlet data g on Γ = Ω:

27 L N M M c N i t 1 t 2 It. D-Error e e e e e e e-6 Table 6.1 BETI two-fold saddle point formulation for 8 cubes.

28 BETI Schur BETI spp BETI 2spp L t 2 it t 2 it t 2 it 0 0 3(2) (10) (15) (17) (19) (21) Table 6.2 Comparison of different BETI versions for 8 cubes.

29 L N M M c N i t 1 t 2 It. D-error e e e e e e e-12 Table 6.3 BETI two-fold SP formulation for 8 cubes with α = 10 5.

30 7. Summary and Work in Progress We propose Inexact Data-Sparse BETI methods the complexity (w.r.t. arithmetical costs and memory) of which is up to some polylogarithmic factor proportional to the nubmber of skeleton unknowns!!! Parallel Implementation: Ops parallel (ε) = O(( H h )d 1 (1 + log H h )3 log 1 ε ) Inexact Data-Sparse BETI-FETI, especially, for non-linear problems in the FEM subdomains Application to Elasticity, Helmholtz, Maxwell etc. equations

31 I am hoping very much to see you at the DD17 held in St. Wolfgang (Austria), July 3-7, More information can be found at the preliminary DD17 homepage Homepages of our university institutes: Institute of Computational Mathematics, JKU Linz Institute of Mathematics D, TU Graz IANS, University of Stuttgart Thank you very much for your attention!