Two-level Domain decomposition preconditioning for the high-frequency time-harmonic Maxwell equations

Transcription

1 Two-level Domain decomposition preconditioning for the high-frequency time-harmonic Maxwell equations Marcella Bonazzoli 2, Victorita Dolean 1,4, Ivan G. Graham 3, Euan A. Spence 3, Pierre-Henri Tournier 2 1 University of Nice Sophia Antipolis, Nice, France 2 Pierre and Marie Curie University, Paris, France 3 University of Bath, UK 4 University of Strathclyde, Glasgow, UK 9th Tutorial and Workshop on FreeFem++ December 14, 2017

2 Goal The convergence analysis for wave propagation problems is challenging Sign-indefinite time-harmonic Maxwell problem { ( E) ω 2 E = F in Ω E n = 0 on Γ = Ω large ω large indefinite, non-hermitian matrix A Pierre-Henri Tournier Two-level domain decomposition preconditioners for Maxwell equations 2/18

3 Goal The convergence analysis for wave propagation problems is challenging Sign-indefinite time-harmonic Maxwell problem { ( E) ω 2 E = F in Ω E n = 0 on Γ = Ω large ω large indefinite, non-hermitian matrix A Goal Find a "good" preconditioner for A: Seek independence of the number of iterations of GMRES from the wavenumber ω = ω µε parallelisable two-level domain decomposition preconditioners: coarse correction combined with the one-level preconditioner Pierre-Henri Tournier Two-level domain decomposition preconditioners for Maxwell equations 2/18

4 Goal The convergence analysis for wave propagation problems is challenging Sign-indefinite time-harmonic Maxwell problem { ( E) ω 2 E = F in Ω E n = 0 on Γ = Ω We developed a theory for the dissipative case (σ > 0): Problem with absorption { ( E) ( ω 2 + i)e = F in Ω E n = 0 on Γ = Ω Work for the Helmholtz equation u ( ω 2 + i)u = f : [Graham, Spence, Vainikko, Domain decomposition preconditioning for high-frequency Helmholtz problems with absorption. Math.Comp. 2017] Pierre-Henri Tournier Two-level domain decomposition preconditioners for Maxwell equations 2/18

5 Strategy Solve with GMRES where B 1 is an approximation of A 1 B 1 Ax = B 1 b, computed using DD Pierre-Henri Tournier Two-level domain decomposition preconditioners for Maxwell equations 3/18

6 Strategy Solve with GMRES where B 1 From is an approximation of A 1 B 1 sufficient conditions for B 1 B 1 Ax = B 1 b, computed using DD A = B 1 A B 1 A (I A 1 A), to be a good preconditioner for A are Pierre-Henri Tournier Two-level domain decomposition preconditioners for Maxwell equations 3/18

7 Strategy Solve with GMRES where B 1 From is an approximation of A 1 B 1 sufficient conditions for B 1 B 1 Ax = B 1 b, computed using DD A = B 1 A B 1 A (I A 1 A), to be a good preconditioner for A are (i) A 1 is a good preconditioner for A in the sense that I A 1 A is small, independently of ω, Pierre-Henri Tournier Two-level domain decomposition preconditioners for Maxwell equations 3/18

8 Strategy Solve with GMRES where B 1 From is an approximation of A 1 B 1 sufficient conditions for B 1 B 1 Ax = B 1 b, computed using DD A = B 1 A B 1 A (I A 1 A), to be a good preconditioner for A are (i) A 1 is a good preconditioner for A in the sense that I A 1 A is small, independently of ω, and (ii) B 1 is a good preconditioner for A in the sense that both the norm and the distance of the field of values from the origin of B 1 A are bounded independently of ω. Pierre-Henri Tournier Two-level domain decomposition preconditioners for Maxwell equations 3/18

9 Remarks (i) A 1 is a good preconditioner for A in the sense that I A 1 A is small, independently of ω (ii) B 1 is a good preconditioner for A in the sense that both the norm and the distance of the field of values from the origin of B 1 A are bounded independently of ω Pierre-Henri Tournier Two-level domain decomposition preconditioners for Maxwell equations 4/18

10 Remarks (i) A 1 is a good preconditioner for A in the sense that I A 1 A is small, independently of ω (ii) B 1 is a good preconditioner for A in the sense that both the norm and the distance of the field of values from the origin of B 1 A are bounded independently of ω we expect that (i) holds when / ω is sufficiently small Helmholtz analogue already proved we proved the PDE-analogue for Maxwell Pierre-Henri Tournier Two-level domain decomposition preconditioners for Maxwell equations 4/18

11 Remarks (i) A 1 is a good preconditioner for A in the sense that I A 1 A is small, independently of ω (ii) B 1 is a good preconditioner for A in the sense that both the norm and the distance of the field of values from the origin of B 1 A are bounded independently of ω we expect that (i) holds when / ω is sufficiently small Helmholtz analogue already proved we proved the PDE-analogue for Maxwell Our theory shows that property (ii) is achieved (GMRES converges in a ω-independent number of iterations when applied to B 1 A ) when ω 2 Pierre-Henri Tournier Two-level domain decomposition preconditioners for Maxwell equations 4/18

12 Remarks (i) A 1 is a good preconditioner for A in the sense that I A 1 A is small, independently of ω (ii) B 1 is a good preconditioner for A in the sense that both the norm and the distance of the field of values from the origin of B 1 A are bounded independently of ω we expect that (i) holds when / ω is sufficiently small Helmholtz analogue already proved we proved the PDE-analogue for Maxwell Our theory shows that property (ii) is achieved (GMRES converges in a ω-independent number of iterations when applied to B 1 A ) when ω 2 "gap" between ω 2 for (ii) and ω for (i) expected to decrease when replacing PEC interface conditions by impedance conditions, as for Helmholtz future work Pierre-Henri Tournier Two-level domain decomposition preconditioners for Maxwell equations 4/18

13 Recap One-level DD preconditioner Consider the linear system: Au = f Cn. Given a decomposition of J1; nk, (N1, N2 ), define: the restriction operator Ri from J1; nk into Ni, RiT as the extension by 0 from Ni into J1; nk. Duplicated unknowns coupled via a partition of unity: N X I= RiT Di Ri i=1 1 MAS := N X 1 MRAS := RiT A 1 i Ri i=1 N X RiT Di A 1 i Ri i=1 N X 1 MORAS := i=1 Pierre-Henri Tournier 1 RiT Di B 1 i Ri Ai = Ri ARiT Optimized transmission conditions [B. Després 1991] for Helmholtz Two-level domain decomposition preconditioners for Maxwell equations 5/18

14 Recap Two-level DD preconditioner General form of two-level preconditioners M 1 2 = QM 1 1 P + H, H = ZE 1 Z ( the conjugate transpose) M 1 1 the one-level preconditioner, Z a rectangular matrix with full column rank (coarse space), E = Z AZ coarse grid matrix, H = ZE 1 Z coarse grid correction matrix, if P = Q = I: additive two-level preconditioner, if P = I AH, Q = I HA: hybrid two-level preconditioner, or balancing Neumann Neumann (BNN). Here definition of Z based on a coarser mesh of element size H cs : the interpolation matrix from V Hcs to V h Pierre-Henri Tournier Two-level domain decomposition preconditioners for Maxwell equations 6/18

15 Theory Main steps Convergence rates for GMRES with 2-level Additive Schwarz preconditioner Nsub M 1,AS = Ri T (A i ) 1 R i, A i = R i A Ri T, R 0 = Z T i=0 explicit in wavenumber ω, absorption, coarse mesh size H cs, subdomain diameter H sub and overlap size δ. Pierre-Henri Tournier Two-level domain decomposition preconditioners for Maxwell equations 7/18

16 Theory Main steps Convergence rates for GMRES with 2-level Additive Schwarz preconditioner Nsub M 1,AS = Ri T (A i ) 1 R i, A i = R i A Ri T, R 0 = Z T i=0 explicit in wavenumber ω, absorption, coarse mesh size H cs, subdomain diameter H sub and overlap size δ. Main theorems: upper bound on the norm of M 1,AS A lower bound on the field of values of M 1,AS A V,M 1,AS A V V 2 apply Elman-type estimates for the convergence of GMRES ω-weighted inner product and norm: (v, w) curl, ω = ( v, w) L2 (Ω) + ω 2 (v, w) L2 (Ω) [Bonazzoli, Dolean, Graham, Spence, Tournier. Domain decomposition preconditioning for the high-frequency time-harmonic Maxwell equations with absorption. Submitted <arxiv: >] Pierre-Henri Tournier Two-level domain decomposition preconditioners for Maxwell equations 7/18

17 Theory Final convergence result Weighted GMRES method: the residual r m is minimized in the norm induced by V, W D ω = (v h, w h ) curl, ω (v h, w h V h with coefficient vectors V, W) Theorem (GMRES convergence for left preconditioning) Ω convex polyhedron. Given ω 0 > 0, there exists C > 0, independent of all parameters, such that, given 0 < a < 1, if (i) ω ω { 0, ( ( (ii) max ( ωh sub ), ( ωh cs) ( ( )) 3 ( ω (iii) m C ( H cs δ then ω 2 ))} ) 2 ) C 1 (1 + ( H cs ) ) 1 2 ( ) δ ω, 2 log ( ) 12 a, r m D ω r 0 D ω a Pierre-Henri Tournier Two-level domain decomposition preconditioners for Maxwell equations 8/18

18 Theory Final convergence result Particular case: when ω 2 (max absorption) and δ H cs (generous overlap), Condition (ii) is satisfied with H sub H cs ω 1, and then bound (iii) implies convergence independent of ω. Theorem (GMRES convergence for left preconditioning) Ω convex polyhedron. Given ω 0 > 0, there exists C > 0, independent of all parameters, such that, given 0 < a < 1, if (i) ω ω { 0, ( ( (ii) max ( ωh sub ), ( ωh cs) ( ( )) 3 ( ω (iii) m C ( H cs δ then ω 2 ))} ) 2 ) C 1 (1 + ( ) ) 1 H cs 2 ( ) δ ω, 2 log ( ) 12 a, r m D ω r 0 D ω a Pierre-Henri Tournier Two-level domain decomposition preconditioners for Maxwell equations 8/18

19 Numerical experiments Ω unit cube, regular or METIS decomposition into subdomains, mesh diameter h ω 3/2, or h 2π/(g ω) if order 2 FEs are used, F = [f, f, f ], f = exp( 400((x 0.5) 2 + (y 0.5) 2 + (z 0.5) 2 )), GMRES with right preconditioning (tolerance 10 6 ), precondition A prob by M 1 prec, 0 prob prec random initial guess all frequencies are present in the error, the local problems in each subdomain and the coarse space problem solved with a direct solver (MUMPS), 1 subdomain 1 processor Pierre-Henri Tournier Two-level domain decomposition preconditioners for Maxwell equations 9/18

20 Numerical experiments Optimal particular case setting H sub H cs ω 1 overlap δ H sub prob = prec = ω 2 PEC B.Cs on Ω order 1 edge FEs h ω 3/2 regular partitioning α = 1 = α ω n N sub n cs #AS #RAS #HRAS (57) 26(37) (64) 28(42) (105) 29(57) 17 n: size of the fine grid problem N sub : number of subdomains n cs : size of the coarse grid problem # : number of iterations 2-level (1-level) Pierre-Henri Tournier Two-level domain decomposition preconditioners for Maxwell equations 10/18

21 Numerical experiments Fewer, larger subdomains H sub ω α H cs ω α overlap δ H sub prob = prec = ω 2 PEC B.Cs on Ω order 1 edge FEs h ω 3/2 regular partitioning α = 0.8, α = 1 ω n N sub n cs #AS #RAS #HRAS # : number of iterations 2-level Pierre-Henri Tournier Two-level domain decomposition preconditioners for Maxwell equations 11/18

22 Numerical experiments Setting not covered by theory H sub ω 0.6 H cs ω 0.9 overlap δ Hsub 2h prob = ω 2 0, prec = ω 2 ω PEC impedance B.Cs on Ω order 1 edge FEs h ω 3/2 regular partitioning prec = ω ω n N sub 2-level n cs Time 1-level Time (1.6) (2.6) (4.0) (8.9) (9.1) (25.6) (29.5) (88.1) (128.4) (443.1) 2-level: # OHRAS, 1-level: # ORAS, Total Time (GMRES Time) Pierre-Henri Tournier Two-level domain decomposition preconditioners for Maxwell equations 12/18

23 Numerical experiments order 2 FEs + fixed number of points per wavelength H sub ω 0.6 (20 ω/(2π)) 0.5 H cs ω 0.9 2π/(2 ω) overlap δ 2h prob = 0, prec = ω impedance B.Cs on Ω order 1 2 edge FEs h ω 3/2 2π/(20 ω) regular METIS partitioning g = 20, α = 0.5, g cs = 2 ω n N sub 2-level n cs Time 1-level Time (7.5) (14.8) (18.9) (72.1) (39.8) (174.5) (77.6) > (305.2) 2-level: # OHRAS, 1-level: # ORAS, Total Time (GMRES Time) Pierre-Henri Tournier Two-level domain decomposition preconditioners for Maxwell equations 13/18

24 Numerical experiments Medimax order 1 edge FEs, 40 points per wavelength n n cs subdomains homogeneous gel brain model plastic cylinder #2-level Time #1-level Time homogeneous gel (8.6) (6.4) brain model (9.2) (6.9) non-conducting cylinder (9.4) (38.2) Pierre-Henri Tournier Two-level domain decomposition preconditioners for Maxwell equations 14/18

25 Numerical experiments Scattering from the COBRA cavity 10 GHz 16 GHz Pierre-Henri Tournier Two-level domain decomposition preconditioners for Maxwell equations 15/18 FigScattering from the COBRA cavity of a plane wave incident upon the cavity aperture at

26 Numerical experiments Scattering from the COBRA cavity order 2 edge FEs, 10 points per wavelength coarse mesh: 3.33 points per wavelength prob = 0, prec = ω f = 10 GHz: n , n cs f = 16 GHz: n , n cs Pierre-Henri Tournier Two-level domain decomposition preconditioners for Maxwell equations 16/18

27 Numerical experiments Scattering from the COBRA cavity order 2 edge FEs, 10 points per wavelength coarse mesh: 3.33 points per wavelength prob = 0, prec = ω f = 10 GHz: n , n cs f = 16 GHz: n , n cs coarse problem too large for a direct solver Pierre-Henri Tournier Two-level domain decomposition preconditioners for Maxwell equations 16/18

28 Numerical experiments Scattering from the COBRA cavity order 2 edge FEs, 10 points per wavelength coarse mesh: 3.33 points per wavelength prob = 0, prec = ω f = 10 GHz: n , n cs f = 16 GHz: n , n cs coarse problem too large for a direct solver inexact coarse solve: use the one-level ORAS preconditioner same decomposition for fine and coarse problems Pierre-Henri Tournier Two-level domain decomposition preconditioners for Maxwell equations 16/18

29 Numerical experiments Scattering from the COBRA cavity order 2 edge FEs, 10 points per wavelength coarse mesh: 3.33 points per wavelength prob = 0, prec = ω f = 10 GHz: n , n cs f = 16 GHz: n , n cs coarse problem too large for a direct solver inexact coarse solve: use the one-level ORAS preconditioner same decomposition for fine and coarse problems Advantages: one-level method already efficient for dissipative pbs (depending on prec ) naturally load-balanced parallel implementation same communication pattern between neighbors for coarse and fine levels R 0 and (R 0 ) T involve no communication GMRES tolerance of 10 2 gives good results Pierre-Henri Tournier Two-level domain decomposition preconditioners for Maxwell equations 16/18

30 Numerical experiments Scattering from the COBRA cavity f = 10 GHz: n , n cs f = 16 GHz: n , n cs g = 10, g cs = 3.33 Total # Total times (seconds) f N sub # it inner it Total Setup GMRES inner 10GHz GHz GHz GHz speedup of 1.7 for f = 10 GHz, 1.45 for f = 16 GHz Pierre-Henri Tournier Two-level domain decomposition preconditioners for Maxwell equations 17/18

31 Perspectives Implementation For now: outer fine problem solved in FreeFem++ inner coarse problem: solved exactly with MUMPS, or inexactly with HPDDM (one-level preconditioner) add support for 2-level preconditioners based on coarse mesh in HPDDM? Recycling Use Krylov-subspace recycling techniques between successive (inexact) solutions of the coarse problem? (recycling techniques are already implemented in HPDDM) Pierre-Henri Tournier Two-level domain decomposition preconditioners for Maxwell equations 18/18

32 Perspectives Implementation For now: outer fine problem solved in FreeFem++ inner coarse problem: solved exactly with MUMPS, or inexactly with HPDDM (one-level preconditioner) add support for 2-level preconditioners based on coarse mesh in HPDDM? Recycling Use Krylov-subspace recycling techniques between successive (inexact) solutions of the coarse problem? (recycling techniques are already implemented in HPDDM) Thank you for your attention! Pierre-Henri Tournier Two-level domain decomposition preconditioners for Maxwell equations 18/18