From the invention of the Schwarz method to the Best Current Methods for Oscillatory Problems: Part 2

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1 and From the invention of the Schwarz method to the Best Current Methods for Oscillatory Problems: Part 2 martin.gander@unige.ch University of Geneva Woudschoten, October 2014

2 Are There Two Equations? ( +k 2 )u = f or ( η)u = f, η > 0? and

3 Are There Two Equations? ( +k 2 )u = f or ( η)u = f, η > 0? Textbook by Zauderer (1989) and

4 Are There Two Equations? ( +k 2 )u = f or ( η)u = f, η > 0? Textbook by Zauderer (1989) Leslie, Bryant, McAveney (1973) Rosmond and Faulkner (1976) and

5 Are There Two Equations? ( +k 2 )u = f or ( η)u = f, η > 0? Textbook by Zauderer (1989) (1859), Hertz... Leslie, Bryant, McAveney (1973) Rosmond and Faulkner (1976) and

6 Are There Two Equations? ( +k 2 )u = f or ( η)u = f, η > 0? Textbook by Zauderer (1989) (1859), Hertz... Leslie, Bryant, McAveney (1973) Rosmond and Faulkner (1976) and

7 Are There Two Equations? ( +k 2 )u = f or ( η)u = f, η > 0? Textbook by Zauderer (1989) (1859), Hertz... Leslie, Bryant, McAveney (1973) Rosmond and Faulkner (1976) and

8 Are There Two Equations? ( +k 2 )u = f or ( η)u = f, η > 0? Textbook by Zauderer (1989) (1859), Hertz... Leslie, Bryant, McAveney (1973) Rosmond and Faulkner (1976) and

9 Are There Two Equations? ( +k 2 )u = f or ( η)u = f, η > 0? Textbook by Zauderer (1989) (1859), Hertz... Leslie, Bryant, McAveney (1973) Rosmond and Faulkner (1976) x Solution of the equation 0.5 y x y x Solution of Laplaces equation x and

10 with ILU, Multigrid or Schwarz Example: Open cavity with point source at the center QMR ILU( 0 ) ILU(1e-2) k it Mflops it Mflops it Mflops > 2000 > > 2000 > and

11 with ILU, Multigrid or Schwarz Example: Open cavity with point source at the center QMR ILU( 0 ) ILU(1e-2) k it Mflops it Mflops it Mflops > 2000 > > 2000 > k Smoothing steps 2.5π 5π 10π 20π Iterative ν = 2 7 div div div Preconditioner ν = Iterative ν = 5 7 stag div div Preconditioner ν = and

12 with ILU, Multigrid or Schwarz Example: Open cavity with point source at the center QMR ILU( 0 ) ILU(1e-2) k it Mflops it Mflops it Mflops > 2000 > > 2000 > k Smoothing steps 2.5π 5π 10π 20π Iterative ν = 2 7 div div div Preconditioner ν = Iterative ν = 5 7 stag div div Preconditioner ν = k Overlap 10π 20π 40π 80π 160π Iterative h div div div div div Preconditioner h Iterative fixed div div div div div Preconditioner fixed and

13 with ILU, Multigrid or Schwarz Example: Open cavity with point source at the center QMR ILU( 0 ) ILU(1e-2) k it Mflops it Mflops it Mflops > 2000 > > 2000 > k Smoothing steps 2.5π 5π 10π 20π Iterative ν = 2 7 div div div Preconditioner ν = Iterative ν = 5 7 stag div div Preconditioner ν = k Overlap 10π 20π 40π 80π 160π Iterative h div div div div div Preconditioner h Iterative fixed div div div div div Preconditioner fixed and Why it is Difficult to Solve Problems with Classical Iterative Methods, Ernst and MJG 2012

14 New Method 1: Preconditioner Enquist, Ying 2010: Preconditioner for the Equation The paper introduces the sweeping preconditioner, which is highly efficient for iterative solutions of the variable coefficient equation including very high frequency problems. The first central idea of this novel approach is to construct an approximate factorization of the discretized equation by sweeping the domain layer by layer, starting from an absorbing layer or boundary condition. and

15 Numerical Experiment (Enquist, Ying 2010) and

16 New Method 2: DDM Chen, Xiang 2012: A Domain Decomposition Method for Equations in Unbounded Domain The method is based on the decomposition of the domain into non-overlapping layers and the idea of source transfer which transfers the sources equivalently layer by layer so that the solution in the final layer can be solved using a PML method defined locally outside the last two layers. and

17 Numerical Experiment (Chen, Xiang 2012) and

18 New Method 3: Stolk 2013: A rapidly converging domain decomposition method for the equation A new domain decomposition method is introduced for the heterogeneous 2-D and 3-D equations. Transmission conditions based on the perfectly matched layer (PML) are derived that avoid artificial reflections and match incoming and outgoing waves at the subdomain interfaces. Our most remarkable finding concerns the situation where the domain is split into many thin layers along one of the axes, say J subdomains numbered from 1 to J. Following [3] we will also call these quasi 2-D subdomains. Generally, an increase in the number of subdomains leads to an increase in the number of iterations required for convergence. Here we propose and study a method where the number of iterations is essentially independent of the number of subdomains. and

19 Numerical Experiments (Stolk 2013) and

20 Numerical Experiments (Stolk 2013) and

21 Underlying Fundamental Algorithms 1 An optimal discrete algorithm based on matrix factorization and Au = y ( +k 2 )u = f u 1m u 11 D 1 L 1,2 L 2,1 D Ln 1,n L n,n 1 in Ω = (0,1) (0,π) D n u 1 u 2. u n u nm u n1 x = f 1 f 2. f n

22 Block Factorization The block LU factorization of the block tridiagional matrix is D 1 L 1,2 L 2,1 D 2 L 2,3 A = = LU L n,n 1 D n T 1 L 2,1 T 2 = L n,n 1 T n I T 1 1 L 1,2 I T 1 2 L 2, where the diagonal blocks T i satisfy the recurrence { D1 i = 1, T i = D i L i,i 1 T 1 i 1 L i 1,i 1 < i n. I and

23 Forward and Backward Substitution Au = LUu = f Lv = f, Uu = v Forward substitution: T 1 L 2,1 T L n,n 1 T n v 1 =T 1 1 f 1 v 1 v 2. v n f 1 = f 2. f n v 2 =T 1 2 (f 2 L 2,1 v 1 )=T 1 2 (f 2 L 2,1 T 1 1 f 1 )=T 1 2 f 2 v 3 =T 1 3 (f 3 L 3,2 v 2 )=T 1 3 (f 3 L 3,2 T 1 2 f 2 )=T 1 3 f 3... and

24 Forward and Backward Substitution Au = LUu = f Lv = f, Uu = v Forward substitution: T 1 L 2,1 T L n,n 1 T n v 1 =T 1 1 f 1 v 1 v 2. v n f 1 = f 2. f n v 2 =T 1 2 (f 2 L 2,1 v 1 )=T 1 2 (f 2 L 2,1 T 1 1 f 1 )=T 1 2 f 2 v 3 =T 1 3 (f 3 L 3,2 v 2 )=T 1 3 (f 3 L 3,2 T 1 2 f 2 )=T 1 3 f 3.. = Source transfer: f j = f j L j,j 1 T 1 j 1 f j 1 and v n = u n. and

25 Forward and Backward Substitution Au = LUu = f Lv = f, Uu = v Forward substitution: T 1 L 2,1 T L n,n 1 T n v 1 =T 1 1 f 1 v 1 v 2. v n f 1 = f 2. f n v 2 =T 1 2 (f 2 L 2,1 v 1 )=T 1 2 (f 2 L 2,1 T 1 1 f 1 )=T 1 2 f 2 v 3 =T 1 3 (f 3 L 3,2 v 2 )=T 1 3 (f 3 L 3,2 T 1 2 f 2 )=T 1 3 f 3.. = Source transfer: f j = f j L j,j 1 T 1 j 1 f j 1 and v n = u n = preconditioner: forward and backward sweep. and

26 Underlying Fundamental Algorithms 2 π y Ω 1 Γ Ω Nataf et al (1994): algorithm ( +k 2 )u1 n = f in Ω 1 x u1 n +DtN 2(u1 n) = xu2 n 1 +DtN 2 (u2 n 1 ) on Γ ( +k 2 )u2 n = f in Ω 2 x u2 n DtN 1(u2 n) = xu1 n 1 DtN 1 (u1 n 1 ) on Γ This algorithm converges in two iterations, x and

27 Underlying Fundamental Algorithms 2 π y Ω 1 Γ 21 Γ 12 Ω Nataf et al (1994): algorithm ( +k 2 )u1 n = f in Ω 1 x u1 n +DtN 2(u1 n) = xu2 n 1 +DtN 2 (u2 n 1 ) on Γ 12 ( +k 2 )u2 n = f in Ω 2 x u2 n DtN 1(u2 n) = xu1 n 1 DtN 1 (u1 n 1 ) on Γ 21 This algorithm converges in two iterations, independantly of the overlap! x and

28 Underlying Fundamental Algorithms 2 π y Γ 12 Γ 23 Γ 34 Γ 45 Ω 1 Ω 2 Ω 3 Ω 4 Ω Nataf et al (1994): algorithm ( +k 2 )ui n = f in Ω i x ui n +DtN i,i+1 (ui n) = xu n 1 i+1 +DtN i,i+1(u n 1 i+1 ) on Γ i,i+1 x ui n DtN i,i 1 (ui n) = xu n 1 i 1 DtN i,i 1(u n 1 i 1 ) on Γ i 1,i With N subdomains, it converges in N iterations, x and

29 Underlying Fundamental Algorithms 2 π y Γ 12 Γ 23 Γ 34 Γ 45 Ω 1 Ω 2 Ω 3 Ω 4 Ω Nataf et al (1994): algorithm ( +k 2 )ui n = f in Ω i x ui n +DtN i,i+1 (ui n) = xu n 1 i+1 +DtN i,i+1(u n 1 i+1 ) on Γ i,i+1 x ui n DtN i,i 1 (ui n) = xu n 1 i 1 DtN i,i 1(u n 1 i 1 ) on Γ i 1,i With N subdomains, it converges in N iterations, or in two when sweeping back and forth once, independently of N = Stolk DDM. x and

30 Underlying Fundamental Algorithms 2 π y Γ 12 Γ 23 Γ 34 Γ 45 Ω 1 Ω 2 Ω 3 Ω 4 Ω Nataf et al (1994): algorithm ( +k 2 )ui n = f in Ω i x ui n +DtN i,i+1 (ui n) = xu n 1 i+1 +DtN i,i+1(u n 1 i+1 ) on Γ i,i+1 x ui n DtN i,i 1 (ui n) = xu n 1 i 1 DtN i,i 1(u n 1 i 1 ) on Γ i 1,i With N subdomains, it converges in N iterations, or in two when sweeping back and forth once, independently of N = Stolk DDM. x and It is the continuous formulation of block LU!

31 1 DtN operator is non local. Fourier symbol is DtN(u) = k 2 ξ 2 Use approximations for practical algorithms: absorbing boundary conditions (ABCs) k 2 ξ 2 p +qξ 2. and perfectly matched layers (PMLs) k 2 ξ 2 P(ξ) Q(ξ). and

32 1 DtN operator is non local. Fourier symbol is DtN(u) = k 2 ξ 2 Use approximations for practical algorithms: absorbing boundary conditions (ABCs) k 2 ξ 2 p +qξ 2. and perfectly matched layers (PMLs) k 2 ξ 2 P(ξ) Q(ξ). This leads to Optimized (OSM): based on subdomain decomposition can run in parallel or sweeping and

33 1 DtN operator is non local. Fourier symbol is DtN(u) = k 2 ξ 2 Use approximations for practical algorithms: absorbing boundary conditions (ABCs) k 2 ξ 2 p +qξ 2. and perfectly matched layers (PMLs) k 2 ξ 2 P(ξ) Q(ξ). This leads to Optimized (OSM): based on subdomain decomposition can run in parallel or sweeping Examples: OO0 and OO2 (Japhet, Nataf 1999), OSM (many in , MJG, Kwok 2010), (Chen, Xiang 2012), DDM (Stolk 2013) and

34 2 Block factorization: need to approximate dense matrices T i { D1 i = 1, T i = D i L i,i 1 T 1 i 1 L i 1,i 1 < i n. = Use algebraic (compute and approximate the limit by a sparse matrix) or continuous techniques (as in OSM) and

35 2 Block factorization: need to approximate dense matrices T i { D1 i = 1, T i = D i L i,i 1 T 1 i 1 L i 1,i 1 < i n. = Use algebraic (compute and approximate the limit by a sparse matrix) or continuous techniques (as in OSM) This leads to Approximate Factorization Preconditioners: based on grid layers instead of subdomains sweep forward and backward Examples: Frequency Filtering (Wittum 1992), AILU (G, Nataf 2000), preconditioner (Enquist, Ying 2010) and

36 2 Block factorization: need to approximate dense matrices T i { D1 i = 1, T i = D i L i,i 1 T 1 i 1 L i 1,i 1 < i n. = Use algebraic (compute and approximate the limit by a sparse matrix) or continuous techniques (as in OSM) This leads to Approximate Factorization Preconditioners: based on grid layers instead of subdomains sweep forward and backward Examples: Frequency Filtering (Wittum 1992), AILU (G, Nataf 2000), preconditioner (Enquist, Ying 2010) Disadvantage: can not use adaptive mesh refinement! and MJG, Zhang 2014: Iterative Solvers for the Equation: Factorizations, Preconditioners and

37 s Equations James C., in Faradays Lines of Force (1856): All the mathematical sciences are founded on relations between physical laws and laws of numbers, so that the aim of exact science is to reduce the problems of nature to the determination of quantities by operations with numbers. The original s equations (20 equations in 20 unknowns) are nowadays written in the form ε E t curl H+σE = J, µ H Related form using only the electric field: t +curl E = 0 ε 2 E t 2 +σ E t +curl 1 curl E = J µ t and

38 Time Harmonic and Time Discretized Variants Time harmonic s equations: E(x,t) = Re(E(x)exp(iωt)) H(x,t) = Re(H(x)exp(iωt)) = (σ +iωε)e curl H = J, iωµh+curl E = 0. Time discretized s equations: ( ε En+1 E n t curl H n+1 ) ( +H n 2 +σ E n+1 ) +E n 2 = J, ( µ Hn+1 H n t +curl E n+1 ) +E n 2 = 0. = (σ +ε η)e curl H = J, µ ηh+curl E = g, where (E,H) := (E n+1,h n+1 ), η := 2 t, J := J ηεe n +2σE n curl H n, g = ηµh n curl E n. and

39 Are There Two s Equations? Eliminating H, we obtain in the time harmonic case ( ω 2 ε+iωσ)e+curl 1 curl E = iωj. µ and in the time discretized case (ε η +σ)e+curl 1 µ curl E = J curl η Many solvers for the time discretized case: 1 µ η g. Domain decomposition: Toselli 1997, Toselli, Widlund, Wohlmuth 1998, Zou 2011,... Multigrid: Hiptmair 1999, Hiptmair, Xu 2007,... Very few solvers for the time harmonic case with large wave number! and

40 for s Equation x z Γ 21 Γ 12 Ω 2 Ω 1 iωεe 1,n +curl H 1,n σe 1,n =J in Ω 1 iωµh 1,n +curl E 1,n =0 in Ω 1 (Bn 1 +S 1 Bn 2 )(E 1,n,H 1,n )=(Bn 1 +S 1 Bn 2 )(E 2,n 1,H 2,n 1 )on Γ 12 iωεe 2,n +curl H 2,n σe 2,n =J in Ω 2 iωµh 2,n +curl E 2,n =0 in Ω 2 (Bn 2 +S 2 Bn 1 )(E 2,n,H 2,n )=(Bn 2 +S 2 Bn 1 )(E 1,n 1,H 1,n 1 )on Γ 21 where Bn is the impedance operator Ω Bn(E,H) := n E +n (H n) = s. Z y and

41 Result for One New Variant Theorem (Dolean, G, Gerardo-Giorda (2009)) For σ = 0, if S l have the Fourier symbol [ k 2 σ l := F(S l ) = γ y kz 2 2k y k z l 2k y k z kz 2 ky 2 ], γ l C(k z,k y ), then the convergence factor of the new algorithm is ρ= ( k 2 ω 2 i ω) 2 1 γ 1 ( k ( k 2 ω 2 +i ω) 2 1 γ 2 ( 2 ω 2 +i ω) 2 1 γ 1 ( k 2 ω 2 i ω) 2 1 γ 2 ( Theorem ( Method (2009)) 1 k 2 ω 2 +i ω) 2 k 2 ω 2 2 L k 2 ω 2 i ω) 2e 2 The choice γ l = 1/( k 2 ω 2 +i ω) 2 is optimal, since then ρ 0, for all Fourier modes k; the method converges in two iterations. Similar results for two other variants. and

42 Optimized and 1. taking γ 1 = γ 2 = 0: classical algorithm 2. taking γ 1 = γ 2 = 1 s iω εµ k 2 s+iω εµ ( ρ 2 (ω,ε,µ,l, k,s) = with s C: k 2 ω 2 εµ s k 2 ω 2 εµ+s 1 3. taking γ 1 = γ 2 = k 2 2ω 2 εµ+2iω εµs ρ 3 (ω,ε,µ,l, k,s)= 4. taking γ l = 1 s l iω εµ k 2 k 2 ω 2 εµ iω εµ k 2 ω 2 εµ+iω εµ ) 2 e 2 s l +iω εµ, l = 1,2, s 1 s 2. with s C: k 2 ω 2 εµl 1 2 ρ 2(ω,ε,µ,L, k,s) 5. taking γ l = 1 k 2 2ω 2 εµ+2iω εµs l, l = 1,2, s 1 s 2.

43 Optimal Choice Based on Asymptotic Analysis Based on equioscillation, we find for the choice s l = p l (1+i) the optimized results: with overlap, L = h without overlap, L = 0 Case ρ parameters ρ parameters 1 1 k + ω 2 h none 1 none 2 1 2C 1 6 ωh 1 3 p = C 1 3ω 2 h (k+ 2 ω 2 ) h 3 p = (k2 + ω2 ) h p 5 C 10 ω h 1 1 = C 5ω h p 2 = C 5ω h 5 3 p (k 2 + ω 2 ) = (k2 + ω2 ) 2 5 h h 1 5 p 2= (k2 + ω2 ) h C 1 4ω C h p = CC 1 4ω 2 h 2(k ω 2 ) 4 1 C(k 2 C h p = + ω 2 ) h 3 p 1 = C 8ω C C 8ω h C (k2 + ω2 ) 1 8 where ω = ω εµ and C ω = min ( k 2 + ω2, ω 2 k 2 ). C 1 4 h h p 2 = C 8ω C 3 4 h 4 3 p 1= (k2 + ω2 ) 8 3 C h 1 4 p 2= (k2 + ω2 ) 8 1 C 3 4 h 3 4 and

44 Transverse Electric Waves Joint work with Dolean and Gerardo-Giorda (2009) We consider the transverse electric waves problem (TE) in Ω = (0,1) 2 : Subdomains Ω 1 = (0,β) (0,1) and Ω 2 = (α,1) (0,1) Overlap L = β α pulsation ω = 2π relative residual reduction of 10 6 with overlap, L = h without overlap, L = 0 h 1/32 1/64 1/128 1/256 1/32 1/64 1/128 1/256 Case 1 27(21) 47(27) 72(33) 118(45) -(73) -(100) -(138) -(181) Case 2 19(14) 19(15) 22(17) 26(19) 41(26) 57(34) 79(40) 111(47) Case 3 13(13) 14(14) 17(17) 21(18) 41(23) 56(25) 80(28) 115(35) Case 4 19(14) 21(16) 24(18) 27(19) 31(24) 35(28) 41(30) 47(33) Case 5 13(13) 15(15) 17(18) 19(19) 56(26) 64(30) 76(32) 76(35) and

45 Comparison of the Asymptotics and iterations 10 2 case 1 case 2 case 3 case 4 case5 O(h 1 ) O(h 1/3 ) O(h 1/3 ) O(h 1/5 ) O(h 1/5 ) iterations 10 2 case 2 case 3 case 4 case5 O(h 1/2 ) O(h 1/2 ) O(h 1/4 ) O(h 1/4 ) h overlapping case 10 2 non-overlapping case h

46 The Chicken Problem Joint work with Dolean (2009) and Heating a chicken in our Whirlpool Talent Combi 4 microwave oven

47 The Chicken Problem The electric field intensity in the chicken and

48 Scattering of a plane wave by a dielectric cylinder Joint Work with Dolean, El Bouajaji and Lanteri (2010) Method L 2 err. E z L 2 err. E z N s # iter # iter Classical Optimized Classical Optimized DGTH-P ( 6.1) ( 4.7) DGTH-P (10.7) ( 6.7) DGTH-P (15.0) ( 8.2) DGTH-P (19.5) (10.3) DGTH-P pk ( 7.2) ( 5.1) EZ and

49 Exposure of head tissue to plane wave 1800 MHz Joint with Dolean, El Bouajaji, Lanteri, Perrussel (2010) Z Y and REX X # d.o.f Ns RAM LU time LU iter Elapsed time 26,854, /3.1 GB 496 sec sec /1.2 GB 132 sec sec 44,491, /3.2 GB 528 sec sec /1.3 GB 142 sec sec DGTH-P1, single precision MUMPS LU, contour lines of Ex Skin (εr = and σ = 1.23 S/m), skull (εr = and σ = 0.43 S/m), CSF (Cerebro Spinal Fluid) (εr = and σ = 2.92 S/m), brain (εr = and σ = 1.15 S/m)

COBRA Cavity Joint work with Dolean, Lanteri, Lee and Peng (2014): 33 subdomains, h = λ 0 /4, 21 562 026 DOFs plane wave incident opon the cavity aperture f =

50 COBRA Cavity Joint work with Dolean, Lanteri, Lee and Peng (2014): 33 subdomains, h = λ 0 /4, DOFs plane wave incident opon the cavity aperture f = 17.6GHz Orthogonal incidence: 18 iterations to reach 1e-3 (29 without optimization) Oblique incidence: 23 iterations to reach 1e-3 (51 without optimization) and

51 COBRA Cavity Results 1 and

52 COBRA Cavity Results 2 and

53 s New preconditioners based on one fundamental idea: 1. Block factorizations 2. Schwarz methods for strip decompositions Both are direct solvers, if DtN or Schur complements are used: = methods Approximations lead to practical algorithms: = methods Examples: OO0, OO2, Generalized Schwarz, Robin Schwarz,, Frequency filtering, AILU, Preconditioner, and many more Preprints are available at gander and MJG, Zhang 2014: Iterative Solvers for the Equation: Factorizations, Preconditioners and

Comparison of a One and Two Parameter Family of Transmission Conditions for Maxwell s Equations with Damping

Comparison of a One and Two Parameter Family of Transmission Conditions for Maxwell s Equations with Damping M. El Bouajaji 1, V. Dolean 2, M. J. Gander 3 and S. Lanteri 1 1 Introduction Transmission conditions