Parallel Multipatch Space-Time IGA Solvers for Parabloic Initial-Boundary Value Problems

Transcription

1 Parallel Multipatch Space-Time IGA Solvers for Parabloic Initial-Boundary Value Problems Ulrich Langer Institute of Computational Mathematics (NuMa) Johannes Kepler University Linz (RICAM) Austrian Academy of Sciences (ÖAW) Linz, Austria AANMPDE12, Särkisaari, Finland, August 6 10, 2018

2 Joint work with my collaborators Christoph Hofer (JKU, DK) Martin Neumüller (JKU, NuMa) Ioannis Toulopoulos (RICAM, CM4PDE) Main results have just been published in [1] C. Hofer, U. Langer, M. Neumüller, I. Toulopoulos. Time-multipatch discontinuous Galerkin space-time isogeometric analysis of parabolic evolution problems. ETNA, 2018, v. 49, [2] C. Hofer, U. Langer, M. Neumüller. Robust preconditioning for space-time isogeometric analysis of parabolic evolution problems. [math.na], 2018, arxiv: , arxiv.org. Doctoral Program Computational Mathematics Numerical Analysis and Symbolic Computation

3 Outline 1 Introduction 2 Time Multi-Patch Space-Time IgA 3 Space-Time Solvers 4 Conclusions & Outlooks

4 Parabolic Initial-Boundary Value Model Problem Let us consider the IBVP problem: Find u : Q R such that t u u = f in Q := Ω (0, T ), u = u D := 0 on Σ := Ω (0, T ), u = u 0 on Σ 0 := Ω {0}, (1) as the typical model problem for a linear parabolic evolution equation posed in the space-time cylinder Q = Ω [0, T ]. Our space-time technology can be generalized to more general parabolic equations like div x (A(x, t) x u) + b(x, t) x u + c(x, t) t u + a(x, t)u = f, eddy-current problems, non-linear problems etc.

5 Parabolic Initial-Boundary Value Model Problem Let us consider the IBVP problem: Find u : Q R such that t u u = f in Q := Ω (0, T ), u = u D := 0 on Σ := Ω (0, T ), u = u 0 on Σ 0 := Ω {0}, (1) as the typical model problem for a linear parabolic evolution equation posed in the space-time cylinder Q = Ω [0, T ]. Our space-time technology can be generalized to more general parabolic equations like div x (A(x, t) x u) + b(x, t) x u + c(x, t) t u + a(x, t)u = f, eddy-current problems, non-linear problems etc.

6 Standard Weak Space-Time Variational Formulation Find u H 1,0 0 (Q) such that a(u, v) = l(v) v H 1,1 (Q), (2) 0,0 with the bilinear form a(u, v) = u(x, t) t v(x, t)dxdt + Q Q x u(x, t) x v(x, t)dxdt and the linear form l(v) = f (x, t)v(x, t)dxdt + u 0 (x)v(x, 0)dx, Q Ω For solvability and regularity results, we refer to original papers by the Leningrad School of Mathematical Physics in the 50s, summerized in the monographs by Ladyzhenskaya, Solonnikov & Uralceva (1967) and Ladyzhenskaya (1973)!

7 Parabolic Solvability and Regularity Results If f L 2,1 (Q T ) := {v : T 0 v(, t) L 2 (Ω) dt < } and u 0 L 2 (Ω), then there exists a unique generalized (weak) solution u H 1,0 1,0 0 (Q) of (2) that even belongs to V2,0 (Q T ). If f L 2 (Q T ) and u 0 H0 1 (Ω), then the generalized solution u of (2) belongs to space H,1 0 (Q T ) = W,1 2,0 (Q T ), and u continuously depends on t in the norm of the space H0 1(Ω), where W,1 2,0 (Q T ) = {v H0 1(Q T ) : x v L 2 (Q T )}. Maximal parabolic regularity: t u div x (A(x, t) x u) = f t u X + div x (A(x, t) x u) X C f X, where X = L p ((0, T ), L q (Ω)) = L p,q (Q T ), 1 < p, q <, u 0 = 0, and a = 0, b = 0, c = 1.

8 Parabolic Solvability and Regularity Results If f L 2,1 (Q T ) := {v : T 0 v(, t) L 2 (Ω) dt < } and u 0 L 2 (Ω), then there exists a unique generalized (weak) solution u H 1,0 1,0 0 (Q) of (2) that even belongs to V2,0 (Q T ). If f L 2 (Q T ) and u 0 H0 1 (Ω), then the generalized solution u of (2) belongs to space H,1 0 (Q T ) = W,1 2,0 (Q T ), and u continuously depends on t in the norm of the space H0 1(Ω), where W,1 2,0 (Q T ) = {v H0 1(Q T ) : x v L 2 (Q T )}. Maximal parabolic regularity: t u div x (A(x, t) x u) = f t u X + div x (A(x, t) x u) X C f X, where X = L p ((0, T ), L q (Ω)) = L p,q (Q T ), 1 < p, q <, u 0 = 0, and a = 0, b = 0, c = 1.

9 Some References to Time-parallel methods: Gander (2015): Nice historical overview on 50 years time-parallel method Parareal introduced by Lions, Maday, Turinici (2001) Time-parallel multigrid: Hackbusch (1984),...,Vandewalle (1993),..., Gander & Neumüller (2014): smart time-parallel multigrid,..., Neumüller & Smears (2018),... Space-time methods for parabolic evolution problems: Babuška & Janik (1989,1990), Behr (2008), Schwab & Stevenson (2009), Neumüller & Steinbach (2011), Neumüller (2013), Andreev (2013), Bank & Metti (2013), Mollet (2014), Urban & Patera (2014), Schwab & Stevenson (2014), Bank & Vassilevski (2014), Karabelas & Neumüller (2015), Langer & Moore & Neumüller (2016), Bank & Vassilevski & Zikatanov (2016), Larsson & Molteni (2017), Steinbach & Yang (2017), Langer & Neumüller & Schafelner (2018),...

10 Some References to Time-parallel methods: Gander (2015): Nice historical overview on 50 years time-parallel method Parareal introduced by Lions, Maday, Turinici (2001) Time-parallel multigrid: Hackbusch (1984),...,Vandewalle (1993),..., Gander & Neumüller (2014): smart time-parallel multigrid,..., Neumüller & Smears (2018),... Space-time methods for parabolic evolution problems: Babuška & Janik (1989,1990), Behr (2008), Schwab & Stevenson (2009), Neumüller & Steinbach (2011), Neumüller (2013), Andreev (2013), Bank & Metti (2013), Mollet (2014), Urban & Patera (2014), Schwab & Stevenson (2014), Bank & Vassilevski (2014), Karabelas & Neumüller (2015), Langer & Moore & Neumüller (2016), Bank & Vassilevski & Zikatanov (2016), Larsson & Molteni (2017), Steinbach & Yang (2017), Langer & Neumüller & Schafelner (2018),...

11 References Time-parallel methods: Gander (2015): Nice historical overview on 50 years time-parallel methods Space-time methods for parabolic evolution problems: Steinbach & Yang (2018): Nice overview on space-time methods in the space-time book that will be published in RSCAM by de Gruyter

12 Single-Patch Space-Time IgA: LMN 16 [3] U. Langer, S. Moore, M. Neumüller. Space-time isogeometric analysis of parabolic evolution equations. Comput. Methods Appl. Mech. Engrg., v. 306, pp , t T 0 0 a Q K Ω b x Φ 1 Φ ˆt 1 K Q 0 Ω 1 ˆx t T Q ( ) n x n t Φ 1 1 ˆx2 x2 Ω Φ 0 0 Ω 0 x1 0 1 ˆt ˆQ ˆx1 Space-Time IgA paraphernalia: Q R d+1 ; d = 1 (l) and d = 2 (r). In this talk: Generalization to the multipatch case + Solvers!

13 Outline 1 Introduction 2 Time Multi-Patch Space-Time IgA 3 Space-Time Solvers 4 Conclusions & Outlooks

14 Time Multi-Patch Decomposition of Q We decompose the space-time cylinder Q = Ω (0, T ) into N non-overlapping space-time subcylinder b n = 1, 2,..., N, such that Qn = Ω (tn 1, tn ) = Φn (Q), S Q= N n=1 Q n with the time faces Σn = Q n+1 Q n = Ω {tn }, ΣN = ΣT : T Introduction Time Multi-Patch Space-Time IgA Space-Time Solvers Conclusions & Outlooks

15 Time Multipatch dg IgA spaces V 0h We look for an approximate solution u h to the IBVP (2) in the globally discontinuous, but patch-wise smooth IgA (B-spline, NURBS) spaces V 0h = {v h H 1,0 0 (Q) : v h n := v h Qn B Ξ d+1(q n ), n = 1,..., N} = {v h L 2 (Q) : v n h V n 0h, n = 1,..., N} = span{ϕ i} i I, V0h n = {v h n B Ξ d+1 (Q n n ) : vh n = 0 on Σ} = span{ϕ n,i} i In where B Ξ d+1(q n n ) is the smooth (depending of the polynomial degrees and multiciplicity of the knots) IgA space corresponding to the knot vector Ξ d+1 n = Ξ d+1 n (n n 1,.., n n d+1 ; pn 1,.., p n d+1 ) =... n

16 Stable Time Multipatch dg IgA Scheme Multiplying the PDE t u x u = f by v h + θ n h n t v h, integrating over Q n, integrating by parts, suming over n, and using that the jumps [ u ] across Σ n are 0 at the solution u H,1 0 (Q), we get the consistency identity where a h (u, v h ) = l h (v h ) v h V 0h, N a h (u, v h ) = ( tuv h + θ nh n tu tv h + x u x v h + θ nh n x u x tv h ) dxdt n=1 Q n N + u n 1 v n 1 h,+ dx, n=1 Σ n 1 N l h (v h ) = f [v h + θ nh n tv h ] dxdt + u 0 vh,+ 0 dx. n=1 Q n Σ 0

17 Stable Time Multipatch dg IgA Scheme Multiplying the PDE t u x u = f by v h + θ n h n t v h, integrating over Q n, integrating by parts, suming over n, and using that the jumps [ u ] across Σ n are 0 at the solution u H,1 0 (Q), we get the multi-patch space-time scheme: Find u h V 0h such that where a h (u h, v h ) = l h (v h ) v h V 0h, N a h (u h, v h ) = ( tu h v h + θ nh n tu h tv h + x u h x v h + θ nh n x u h x tv h ) dxdt n=1 Q n N + u n 1 h v n 1 h,+ dx, n=1 Σ n 1 N l h (v h ) = f [v h + θ nh n tv h ] dxdt + u 0 vh,+ 0 dx. n=1 Q n Σ 0

18 Space-Time IgA Scheme and System of IgA Eqns Hence, we look for the solution u h V 0h of the IgA scheme in the form of a h (u h, v h ) = l h (v h ) v h V 0h (3) u h (x, t) = u h (x 1,..., x d, x d+1 ) = i I u iϕ i (x, t) where u h := [u i ] i I R N h= I is the unknown solution vector of control points defined by the solution of the linear system L h u h = f h (4) with huge, non-symmetric, but positive definite system matrix L h.

19 Road Map of the Numerical Analysis v h 2 h = N n=1 ( x v h 2 L 2 (Q + n) θn hn tv h 2 L 2 (Q + 1 ) n) 2 v h n 1 2 L 2 (Σ n 1 ) N v 2 h, = v 2 h + (θ nh n) 1 v 2 L 2 (Q + N n) n=1 n=2 v n 1 2 L 2 (Σ n 1 ) v h 2 L 2 (Σ N ) Coercivity: a h (v h, v h ) µ c v h 2 h, v h V 0h,! u h u h : (4) Boundedness: a h (u, v h ) µ b u h, v h h, u V 0h + H,1 0 (Q), v h V 0h, Consistency: a h (u, v h ) = l h (v h ) v h V 0h, u H 1,0 0 (Q) H,1 (Q): (2) GO: a h (u u h, v h ) = 0 v h V 0h, Cea-like: u u h h (1 + µ b /µ c) inf vh V 0h u v h h,... Convergence rates: u u h h ch p u H p+1 (Q)

20 V 0h Coercivity of the bilinear form a h (, ) We now introduce the mesh-dependent norm v h 2 h = N n=1 ( x v h 2 L 2 (Q + n) θn hn tv h 2 L 2 (Q + 1 ) n) 2 v h n 1 2 L 2 (Σ n 1 ) Lemma (Coercivity / Ellipticity on V 0h ) v h 2 L 2 (Σ N ). The bilinear form a h (, ) : V 0h V 0h R is V 0h coercive wrt the norm h, i.e., there exists a constant µ c = 1/2 such that a h (v h, v h ) µ c v h 2 h, v h V 0h. (5) provided that θ n c 2 inv,0, where v h 2 L 2 ( E) c inv,0hn 1 v h 2 L 2 (E) This lemma immediately yields uniqueness and existence of the solution u h V 0h and u h R N h of (9) and (4), respectively.

21 Uniform Boundedness of a h (, ) on V 0h, V 0h Let us introduce the space V 0h, = V 0h + H,1 0 (Q) equipped with the norm v h, = ( v N 2 h + (θ nh n) 1 v 2 L 2 (Q + N n) v n L 2 (Σ n 1 )). (6) n=1 n=2 Lemma (Boundedness) The bilinear form a h (, ) is uniformly bounded on V 0h, V 0h : a h (u, v h ) µ b u h, v h h, u V 0h,, v h V 0h, (7) with µ b = max(c inv,1 θ max, 2), where θ max = max n {θ n } c 2 inv,0. and c inv,k = c inv,k (p) are constants in the inverse inequalities t xi v h 2 L 2 (E) c inv,1h 2 n x i v h 2 L 2 (E) and v h 2 L 2 ( E) c inv,0h 1 n v h 2 L 2 (E).

22 Consistency and Galerkin orthogonality Lemma (Consitency) If the solution u H 1,0 0 (Q) of the variational problem (2) belongs to H,1 (Q), then it satisfies the consistency identity a h (u, v h ) = l h (v h ) v h V 0h. (8) Lemma (Galerkin orthogonality) If the solution u H 1,0 0 (Q) of the variational problem (2) belongs to H,1 (Q), then the Galerkin orthogonality a h (u u h, v h ) = 0 v h V 0h. (9) holds, where u h V 0h is the space-time dg IgA solution.

23 Cea-like Discretization Error Estimate Theorem (Cea-like Estimate) Let the exact solution u of (2) belong to H 1,0 0 (Q) H,1 (Q), and let u h be the solution of the space-time IgA scheme (9). Then we get ( where v h, = ( N v h = n=1 u u h h (1 + µ b µ c ) inf v h V 0h u v h h,, (10) v 2 h + N n=1 (θnhn) 1 v 2 L 2 (Q + N n) n=2 ( 1 2 x v 2 L 2 (Qn) + θn hn tv 2 L 2 (Qn) v 2 L 2 (Σn 1)) v 2 L 2 (Σ N )). Proof: u u h h u v h h + v h u h h µ c v h u h 2 h a h(v h u h, v h u h ) = a h (v h u, v h u h ) µ b u v h h, v h u h h v n 1 2 L 2 (Σ n 1 )) 1 2 and

24 Approximation Error Estimate Theorem (Approximation Theorem) Let p n + 1 l n 2 and p n + 1 m n 1 be integers, and let u L 2 (Q) such that the restriction u n := u Qn belongs to H ln,mn (Q n) for n = 1,..., N. Then there exists a quasi-interpolant Π h u V 0h such that ( N u Π h u 2 h, = ( x (u Π n h u) 2 L 2 (Q + n) θn hn t(u Πn h u) 2 L 2 (Q n) n=1 + 1 ) 2 (u Π hu) n 1 2 L 2 (Σ n 1 ) + 1 ) 2 u ΠN h u 2 L 2 (Σ N ) N 1 + u Π n h θ n=1 nh u 2 L 2 (Q + N n) n n=2 N ( (C n hn 2(ln 1) + θ nhn 2ln 1 + hn 2mn 1 n=1 N ( ( C n hn 2(ln 1) + h 2(mn 2 1 ) ) n u 2 Hln,mn (Qn) n=1 (u Π n 1 h u) n 1 2 L 2 (Σ n 1 ) ) + θ nhn 2mn 1 u 2 Hln,mn (Qn)

25 A priori Discretization Error Estimate Theorem (A priori Disscretization Error Estimate) u u h h (1 + µ b µ c ) N n=1 ( ( ) C n hn 2(ln 1) + h 2(mn 1 2 ) n u 2 H ln,mn (Q n) We remark that for the case of highly smooth solutions, i.e., p + 1 min(l n, m n), the above estimate takes the form u u h h C N hn p u H p+1,p+1 (Q n) n=1 C h p u H p+1,p+1 (Q) where the last estimate holds if u H p+1,p+1 (Q) and h = max{h n} is assumed.

26 (sp cg, mp dg) IgA for d = 1 and p = 2, 3, error in dg-norm dg meshsize h p=2 p=3 p=4 O(h 2 ) O(h 3 ) O(h 4 )

27 (sp cg, mp dg) IgA for d = 1 and p = 2, 3, 4 Error in the dg-norm and convergence rate for the exact solution u(x, t) = sin(πx) sin( π 2 (t + 1)) and for B-Spline degrees 2, 3 and 4 p = 2 p = 3 p = 4 refinement error eoc error eoc error eoc E E E E E E E E E E E E E E E E E E E E E E E E

28 (mp cg, mp dg) IgA for d = 1 and p = 2, 3 ref Introduction p=2 error e e-05 Time Multi-Patch Space-Time IgA eoc p=3 error e e e e-08 Space-Time Solvers eoc Conclusions & Outlooks

29 Outline 1 Introduction 2 Time Multi-Patch Space-Time IgA 3 Space-Time Solvers 4 Conclusions & Outlooks

30 One huge system of IgA equations Once the basis is chosen, the IgA scheme (9) can be rewritten as a huge system of algebraic equations of the form L h u h = f h (11) for determining the vector u h = ((u 1,i ) i I1,..., (u N,i ) i IN ) R N h of the control points of the IgA solution u h (x, t) = i I n u n,i ϕ n,i (x, t), (x, t) Q n, n = 1,..., N, solving the IgA scheme (9). The system matrix L h is the usual Galerkin (stiffness) matrix, and f h is the rhs (load) vector. The matrix L h is non-symmetric, but positive definite!

31 System Matrix L h The Galerkin matrix L h can be rewritten in the block form A 1 B 2 A 2 L h = B 3 A , B N A N with the matrices A n := M n,x K n,t + K n,x M n,t for n = 1,..., N, B n := M n,x N n,t for n = 2,..., N.

32 Parallel Space-Time Multigrid Solvers Solve A 1 B 2 A 2 L h u h = f h, with L h = B N A N by the time-parallel MGM proposed by Gander & Neumüller ( 16): Ingredients: Time-Restriction and Prolongation Smoother

33 Time-Restriction Two time-slabs are restricted to a single time-slab. Assumption: Same basis on each slab

34 Smoother Inexact damped block Jacobi smoother: u h (k+1) = u h (k) + ω t D 1 h [ f h L h u h (k) ] for k = 1, 2,... with ω t = 1 2 and ω t = 1 2 D h := diag{a n } n=1,...,n Parallel w.r.t. time Replace D 1 1 h by multigrid D h w.r.t space parallel in space

35 Parallel Solver Studies for d = 3 and p = 1 Figure: Computational spatial domain Ω decomposed into 4096 elements (left) and distributed over 32 processors (right). The IgA solution u h (x, t) u(x, t) = sin(πx 1 ) sin(πx 2 ) sin(πx 3 ) sin(πt) is plotted at t = 0.5.

36 Parallel Solver Studies for d = 3 and p = 1 Convergence results in the h - norm for the regular solution u(x, t) = sin(πx 1 ) sin(πx 2 ) sin(πx 3 ) sin(πt) as well as iteration numbers and solving times for the parallel space-time multigrid preconditioned GMRES method on Vulcan N overall dof u u h h eoc c x c t cores iter time [s] E E E E E E E

37 Parallel Solver Studies for d = 3 and p = 1 Convergence results in the norm h for the low regularity solution u(x, t) = cos(βx 1 ) cos(βx 2 ) cos(βx 3 )(1 t) α H s,α+ 1 2 ε (Q), with α = 0.75 and β = 0.3, for an arbitrary s 2 and for an arbitrary small ε > 0, as well as iteration numbers and solving times for the parallel space-time multigrid preconditioned GMRES method on Vulcan BlueGene/Q at LLNL N overall dof u u h h eoc c x c t cores iter time [s] E E E E E E E

38 New Space-Time Multigrid Solver / Preconditioner Utilizing the tensor product structure A = A n = K t M x + M t K x, to construct a cheap approximation  1 to A 1, i.e., D 1 to D 1 : K t K T t, M t M T t K t and M t are small ( 1d - matrices) compared to K x and M x ( 1d, 2d, 3d - SPD-matrices). Idea: Perform decomposition of M 1 t K t 1 Fast Diagonalization: M 1 t K t = XDX 1, X, D C nt nt 2 Complex-Schur: M 1 t K t = QTQ, Q, T C nt nt 3 Real-Schur: M 1 t K t = QTQ T, Q, T R nt nt 4 Ref.: Sangalli & Tani (2016), Tani (2017) for elliptic BVP

39 New Space-Time Multigrid Solver / Preconditioner Utilizing the tensor product structure A = A n = K t M x + M t K x, to construct a cheap approximation  1 to A 1, i.e., D 1 to D 1 : K t K T t, M t M T t K t and M t are small ( 1d - matrices) compared to K x and M x ( 1d, 2d, 3d - SPD-matrices). Idea: Perform decomposition of M 1 t K t 1 Fast Diagonalization: M 1 t K t = XDX 1, X, D C nt nt 2 Complex-Schur: M 1 t K t = QTQ, Q, T C nt nt 3 Real-Schur: M 1 t K t = QTQ T, Q, T R nt nt 4 Ref.: Sangalli & Tani (2016), Tani (2017) for elliptic BVP

40 Fast Diagonalization Eigenvalue decomposition M 1 t K t = XDX 1, i.e. K t X = M t XD D = diag(λ i ), λ i C (Eigenvalues) X C nt nt (Eigenvectors), X 1 X! Defining Y := (M t X) 1 gives M t = Y 1 X 1 K t = Y 1 DX 1 Hence, we obtain A 1 = (K t M x + M t K x ) 1 = ( (Y 1 I) (D M x + I K x ) (X 1 I) ) 1 = (X I) (D M x + I K x ) 1 (Y I) }{{} diag((k x +λ i M x ) 1 )

41 Fast Diagonalization Eigenvalue decomposition M 1 t K t = XDX 1, i.e. K t X = M t XD D = diag(λ i ), λ i C (Eigenvalues) X C nt nt (Eigenvectors), X 1 X! Defining Y := (M t X) 1 gives M t = Y 1 X 1 K t = Y 1 DX 1 Hence, we obtain A 1 = (K t M x + M t K x ) 1 = ( (Y 1 I) (D M x + I K x ) (X 1 I) ) 1 = (X I) (D M x + I K x ) 1 (Y I) }{{} diag((k x +λ i M x ) 1 )

42 Solver for K x + λ i M x Case 1: λ i R + : K x + λ i M x... SPD matrix solvers available, e.g., DD, MG Case 2: λ i = α i + β i i C, α i > 0: (K x + λ i M x ) H K x + λ i M x Rewrite as symmetric, indefinite problem: ( ) Kx + α A i := i M x βm x βm x (K x + α i M x ) Robust preconditioner 1 cond 2 (P 1 i A i ) 2: ( ) Kx + (α P i := i + β i )M x 0 0 K x + (α i + β i )M x K x + (α i + β i )M x is a SPD matrix DD, MG 1 W. Zulehner, SIAM J. Matrix Anal. & Appl., 32(2), , 2011

43 Solver for K x + λ i M x Case 1: λ i R + : K x + λ i M x... SPD matrix solvers available, e.g., DD, MG Case 2: λ i = α i + β i i C, α i > 0: (K x + λ i M x ) H K x + λ i M x Rewrite as symmetric, indefinite problem: ( ) Kx + α A i := i M x βm x βm x (K x + α i M x ) Robust preconditioner 1 cond 2 (P 1 i A i ) 2: ( ) Kx + (α P i := i + β i )M x 0 0 K x + (α i + β i )M x K x + (α i + β i )M x is a SPD matrix DD, MG 1 W. Zulehner, SIAM J. Matrix Anal. & Appl., 32(2), , 2011

44 Solver for K x + λ i M x Case 1: λ i R + : K x + λ i M x... SPD matrix solvers available, e.g., DD, MG Case 2: λ i = α i + β i i C, α i > 0: (K x + λ i M x ) H K x + λ i M x Rewrite as symmetric, indefinite problem: ( ) Kx + α A i := i M x βm x βm x (K x + α i M x ) Robust preconditioner 1 cond 2 (P 1 i A i ) 2: ( ) Kx + (α P i := i + β i )M x 0 0 K x + (α i + β i )M x K x + (α i + β i )M x is a SPD matrix DD, MG 1 W. Zulehner, SIAM J. Matrix Anal. & Appl., 32(2), , 2011

45 Numerical Tests Parallel application for each i = 1,..., n t Unfortunately, cond 2 (X) >> 1 (Eigenvectors) Reliable approximation  only for small n t. n t p p = 2 p = 3 p = 4 p = 5 p = 6 p = 7 p = e e+06 6e+06 3e+07 1e e+06 1e+07 6e+07 4e+08 7e+09 1e e+07 4e+07 3e+08 3e+09 6e+10 3e+11 1e e+08 1e+09 1e+10 3e+11 2e+13 5e+13 4e+14 Table: condition number of X : θ = 0.01 and t i+1 t i = 0.1.

46 Complex Schur decomposition The complex Schur decomposition gives M 1 t K t = QTQ T = uppertriag(t ij ) C, T ii = λ i (Eigenvalues) Q C nt nt with Q 1 = Q cond 2 (Q) = 1. Again, by defining Y := (M t Q ) 1, we obtain M t = Y 1 Q K t = Y 1 TQ. Hence, we have A 1 = (M x K t + K x M t ) 1 = (Q I) (T M x + I K x ) 1 (Y I)

47 Complex Schur decomposition The complex Schur decomposition gives M 1 t K t = QTQ T = uppertriag(t ij ) C, T ii = λ i (Eigenvalues) Q C nt nt with Q 1 = Q cond 2 (Q) = 1. Again, by defining Y := (M t Q ) 1, we obtain M t = Y 1 Q K t = Y 1 TQ. Hence, we have A 1 = (M x K t + K x M t ) 1 = (Q I) (T M x + I K x ) 1 (Y I)

48 Complex Schur decomposition (cont.) T M x + I K x has the following structure K x + λ 1 M x T 12 M x... 0 K x + λ 2 M x T 23 M x Tnt 1n t M x K x + λ nt M x Application is done staggered K x + λ i M x has the same structure as in the diagonal case. Real Schur decomposition works similar only real arithmetic.

49 YETI-footprint: Space mp cg & time mp dg IgA

50 Numerical Tests: Spatial Solver: K x + λ i M x λ i C MinRes method with tolerance ε = 10 8 (K x + (α i + β i )M x ) 1 Direct solver Maximum number of iterations for i = 1,..., n t. ref. x and t p = 2 p = 3 p = 4 p = 5 p =

51 Numerical Tests: Sparse Direct Solver Degree p x, p t = (3, 3); θ = 0.01 and t i t i 1 = 0.1 A i MinRes with ε = 10 4 (It. 10). (K x + (α i + β i )M x ) 1 Direct solver Direct Solver: PARDISO Tolerance Multigrid ε = 10 8 #dofs ref #slabs MG-It Direct Diag x t Setup Solving Setup Solving #dofs ref #slabs MG-It C-Schur R-Schur

52 Numerical Tests: Parallelization in time - Real Schur Degree p x, p t = (3, 3), refinement r x, r t = (2, 3) θ = 0.01 and t i t i 1 = 0.1; (K x + (α i + β i )M x ) 1 Direct solver Direct Solver: PARDISO Tolerance ε = 10 8 #slabs = #Processors. Weak Scaling #dofs #slabs MG-It. Setup Solving 4.5 Setup 4 Solving Procs Time (s)

53 Parallelization in Space & Time Complex Schur Spatial domain YETI-Footprint (84 patches) Degree p x, p t = (3, 3), θ = 0.01 and t [0, 4] Parallel MG as preconditioner/solver for spatial problems #dofs refs #sl. c total c x c t it Setup Solving

54 Parallelization in Space & Time Real Schur Spatial domain YETI-Footprint (84 patches) Degree p x, p t = (3, 3), θ = 0.01 and t [0, 4] Parallel MG as preconditioner/solver for spatial problems #dofs refs #sl. c total c x c t it Setup Solving

55 Parallelization in Space & Time Real Schur Spatial domain YETI-Footprint (84 patches) Degree p x, p t = (3, 3), θ = 0.01 and t [0, 4] Parallel MG as preconditioner/solver for spatial problems #dofs refs #sl. c total c x c t it Setup Solving serial c t =

56 Numerical Tests: Parallelization in Space & Time Alternative Version non-symmetric IETI-DP for A n Spatial domain Ω: 4 4 square grid Degree p x, p t = (3, 3), θ = 0.01 and t i t i 1 = 0.1. GMRes-IETI-DP in the smoother (3 Iterations). Coarsening only in time! #dofs ref x #sl. c total c x c t It. Setup Solving

57 Outline 1 Introduction 2 Time Multi-Patch Space-Time IgA 3 Space-Time Solvers 4 Conclusions & Outlooks

58 Conclusions & Ongoing Work & Outlook Space-time IgA: space singlepatch cg and time-multipatch dg Space-time IgA: space multipatch cg and time-multipatch dg Space-time IgA: Fast generation via tensor techniques Space-time IgA: Fast parallel solvers Functional a posteriori estimates and THB-Spline adaptivity = joint work with S. Matculevich and S. Repin = talk by Svetlana Matculevich on Thursday! space-time multipatch dg IgA + adaptivity + fast generation + efficient parallel solvers??? Adaptive Space-Time FEM = talk by Andreas Schafelner on Tuesday!

59 Conclusions & Ongoing Work & Outlook Space-time IgA: space singlepatch cg and time-multipatch dg Space-time IgA: space multipatch cg and time-multipatch dg Space-time IgA: Fast generation via tensor techniques Space-time IgA: Fast parallel solvers Functional a posteriori estimates and THB-Spline adaptivity = joint work with S. Matculevich and S. Repin = talk by Svetlana Matculevich on Thursday! space-time multipatch dg IgA + adaptivity + fast generation + efficient parallel solvers??? Adaptive Space-Time FEM = talk by Andreas Schafelner on Tuesday!

60 Conclusions & Ongoing Work & Outlook Space-time IgA: space singlepatch cg and time-multipatch dg Space-time IgA: space multipatch cg and time-multipatch dg Space-time IgA: Fast generation via tensor techniques Space-time IgA: Fast parallel solvers Functional a posteriori estimates and THB-Spline adaptivity = joint work with S. Matculevich and S. Repin = talk by Svetlana Matculevich on Thursday! space-time multipatch dg IgA + adaptivity + fast generation + efficient parallel solvers??? Adaptive Space-Time FEM = talk by Andreas Schafelner on Tuesday!

61 Numerical Results: Parallel Solution for p = 1 2d fixed spatial domain Ω = (0, 1) 2 yielding Q = Ω (0, T ) = (0, 1) 3 Exact solution: u(x, t) = sin(πx 1 ) sin(πx 2 ) sin(πt) Figure shows space-time decomposition with 64 subdomains Table shows parallel performance of the parallel AMG hypre preconditioned GMRES stopped after the relative residual error reduction by 10 10! Dofs u u h L2 (Q) Rate iter time [s] cores e e e e e e e e e e e This example was computed on the supercomputer Vulcan BlueGene/Q in Livermore by M. Neumüller.

62 Supercomputing Results on Vulcan: (3+1)D, p = 1 3d fixed spatial domain Ω = (0, 1) 3 yielding Q = Ω (0, T ) = (0, 1) 4 Exact solution: u(x, t) = sin(πx 1 ) sin(πx 2 ) sin(πx 3 ) sin(πt) Table shows parallel performance of the parallel AMG hypre preconditioned GMRES stopped after the relative residual error reduction by 10 6! dofs u u h L2 (Q) rate iter time [s] cores e e e e e e e e e This example was computed on the supercomputer Vulcan BlueGene/Q in Livermore by M. Neumüller.

63 THANK YOU VERY MUCH! Solution 3.32e Introduction Time Multi-Patch Space-Time IgA Space-Time Solvers Conclusions & Outlooks