Efficient domain decomposition methods for the time-harmonic Maxwell equations

Transcription

1 Efficient domain decomposition methods for the time-harmonic Maxwell equations Marcella Bonazzoli 1, Victorita Dolean 2, Ivan G. Graham 3, Euan A. Spence 3, Pierre-Henri Tournier 4 1 Inria Saclay (Defi team), CMAP, Palaiseau, France 2 Université Côte d Azur, LJAD, Nice, France 3 University of Bath, UK 4 Sorbonne Université, LJLL, Paris, France Marcella Bonazzoli (Inria, CMAP) Domain Decomposition for Maxwell 03/10/ / 24

2 Outline 1 Time-harmonic wave propagation 2 One-level domain decomposition preconditioners 3 Two-level domain decomposition preconditioners Marcella Bonazzoli (Inria, CMAP) Domain Decomposition for Maxwell 03/10/ / 24

3 Outline 1 Time-harmonic wave propagation 2 One-level domain decomposition preconditioners 3 Two-level domain decomposition preconditioners Marcella Bonazzoli (Inria, CMAP) Domain Decomposition for Maxwell 03/10/ / 24

4 Research topics Wave propagation problems numerical analysis precise and fast simulation implementation on parallel machines [ up to 8000 cores ] medical imaging application (inverse problem) Time-harmonic formulation (Helmholtz - Maxwell) at high frequency Difficulties : non symmetric positive definite matrices very huge matrices in 3D (sparse) : iterative solvers are not robust Marcella Bonazzoli (Inria, CMAP) Domain Decomposition for Maxwell 03/10/ / 24

5 Application: brain imaging for stroke diagnosis (MEDIMAX ANR project) Medical imaging prototype (EMTensor GmbH, Wien): Stroke detection: electric permittivity of brain tissues? inverse problem [IEEE AP 2017], [PAR COMP accepted] Marcella Bonazzoli (Inria, CMAP) Domain Decomposition for Maxwell 03/10/ / 24

6 Research topics Motivation: brain stroke detection time-harmonic Maxwell equations Marcella Bonazzoli (Inria, CMAP) Domain Decomposition for Maxwell 03/10/ / 24

7 Research topics Motivation: brain stroke detection time-harmonic Maxwell equations precise discretization Marcella Bonazzoli (Inria, CMAP) Domain Decomposition for Maxwell 03/10/ / 24

8 Research topics Motivation: brain stroke detection time-harmonic Maxwell equations precise discretization + fast solver Marcella Bonazzoli (Inria, CMAP) Domain Decomposition for Maxwell 03/10/ / 24

9 Research topics Motivation: brain stroke detection time-harmonic Maxwell equations Strategy = precise discretization high order edge finite elements + fast solver Marcella Bonazzoli (Inria, CMAP) Domain Decomposition for Maxwell 03/10/ / 24

10 Research topics Motivation: brain stroke detection time-harmonic Maxwell equations Strategy = precise discretization high order edge finite elements + + fast solver domain decomposition preconditioners Marcella Bonazzoli (Inria, CMAP) Domain Decomposition for Maxwell 03/10/ / 24

11 Time-harmonic Maxwell equations Electric field E(x, t) = Re(E(x)e +iωt ) ( E) γ 2 E = 0, γ := ω µε σ, ε σ := ε i σ ω ω angular frequency µ magnetic permeability, ε(x) electric permittivity, σ(x) electrical conductivity of the medium if σ = 0, γ = ω = ω µε the wavenumber. a(e, v) = Ω ( E v ω 2 µ ε i σ ) E v + B.C. Ω ω E, v H(curl, Ω) = {v L 2 (Ω) 3, v L 2 (Ω) 3 } if σ = 0, a is not positive definite if σ 0, a is not Hermitian Marcella Bonazzoli (Inria, CMAP) Domain Decomposition for Maxwell 03/10/ / 24

12 Outline 1 Time-harmonic wave propagation 2 One-level domain decomposition preconditioners 3 Two-level domain decomposition preconditioners Marcella Bonazzoli (Inria, CMAP) Domain Decomposition for Maxwell 03/10/ / 24

13 Domain decomposition preconditioning Au = b linear systems solvers: robustness memory cost direct (LU, MUMPS...) iterative (CG, GMRES...) preconditioner necessary for iterative solvers M 1 A u = M 1 b choice: Domain Decomposition (DD) preconditioner Domain Ω decomposed into N sub overlapping subdomains Ω i subproblems can be solved with direct solvers concurrently, in parallel Marcella Bonazzoli (Inria, CMAP) Domain Decomposition for Maxwell 03/10/ / 24

14 One-level domain decomposition preconditioners Decomposition of Ω into N sub overlapping subdomains Ω i decomposition of the set of unknowns N into N sub subsets N i Optimized Restricted Additive Schwarz preconditioner (ORAS) M 1 = M 1 Nsub 1,ORAS = Ri T i=1 D i A i 1 R i R i restriction matrix from N to N i A i matrix of the local subproblem on Ω i with impedance conditions as transmission conditions between subdomains: ( E) n + i ω n (E n) D i partition of unity matrix for N i ( N sub i=1 RT i D i R i = I ) extension matrix from N i to N R T i Marcella Bonazzoli (Inria, CMAP) Domain Decomposition for Maxwell 03/10/ / 24

15 Numerical results Reference solution with 114 million (complex-valued) unknowns cores on Curie supercomputer (TGCC-CEA) FreeFem++, HPDDM 1 Degree 1 Degree 2 2.4M 1.5M 8.5M relative error 21M M 43M 74M 13M 27M 46M time to solution (s) GMRES tol. = right-hand sides degree 1: 21 M unks, 130 s degree 2: 5 M unks, 62 s [INT J NUM MOD EL 2018] Marcella Bonazzoli (Inria, CMAP) Domain Decomposition for Maxwell 03/10/ / 24

16 Outline 1 Time-harmonic wave propagation 2 One-level domain decomposition preconditioners 3 Two-level domain decomposition preconditioners Marcella Bonazzoli (Inria, CMAP) Domain Decomposition for Maxwell 03/10/ / 24

17 Two-level domain decomposition preconditioners Goal Independence of the number of iterations of GMRES from the wavenumber ω = ω µε (mesh size h 2π/(g ω) or h ω 3/2 problem size grows with ω) two-level domain decomposition preconditioners: combine a coarse correction with the one-level preconditioner. Two ingredients: rectangular matrix Z (coarse space) reduced size problem E = Z T AZ: the coarse problem algebraic formula to combine (e.g. additive or hybrid) Marcella Bonazzoli (Inria, CMAP) Domain Decomposition for Maxwell 03/10/ / 24

18 Two-level domain decomposition preconditioners General form of 2-level preconditioners M 1 2 = QM 1 1 P + H, H = ZE 1 Z T M 1 1 the 1-level preconditioner Z a rectangular matrix with full column rank (coarse space) E = Z T AZ coarse problem matrix H = ZE 1 Z T coarse correction matrix if P = Q = I : additive if P = I AH, Q = I HA: hybrid or Balancing Neumann Neumann if P = I AH, Q = I : Adapted Deflation Variant 1 (A-DEF1) Marcella Bonazzoli (Inria, CMAP) Domain Decomposition for Maxwell 03/10/ / 24

19 Two-level domain decomposition preconditioners General form of 2-level preconditioners M 1 2 = QM 1 1 P + H, H = ZE 1 Z T M 1 1 the 1-level preconditioner Z a rectangular matrix with full column rank (coarse space) E = Z T AZ coarse problem matrix H = ZE 1 Z T coarse correction matrix if P = Q = I : additive if P = I AH, Q = I HA: hybrid or Balancing Neumann Neumann if P = I AH, Q = I : Adapted Deflation Variant 1 (A-DEF1) Here definition of Z based on a coarser mesh of diameter H cs : the interpolation matrix from V Hcs to V h Marcella Bonazzoli (Inria, CMAP) Domain Decomposition for Maxwell 03/10/ / 24

20 Two-level domain decomposition preconditioners Sign-indefinite time-harmonic Maxwell problem { ( E) ω 2 E = F in Ω E n = 0 on Γ = Ω (PEC BC) Convergence analysis for the dissipative case (σ > 0): Problem with absorption { ( E) ( ω 2 + iξ)e = F in Ω E n = 0 on Γ = Ω (PEC BC) Based on the coercivity of the sesquilinear form a ξ with absorption ξ R \ 0 a ξ (E, v) = E v ( ω 2 + iξ)e v A ξ Ω Ω Marcella Bonazzoli (Inria, CMAP) Domain Decomposition for Maxwell 03/10/ / 24

21 Two-level domain decomposition preconditioners Theory for the 2-level Additive Schwarz preconditioner: Nsub M 1 ξ,as = Ri T (A i ξ ) 1 R i, A i ξ = R ia ξ Ri T, R 0 = Z T i=0 Main theorems: upper bound on the norm of M 1 ξ,as A ξ lower bound on the field of values of M 1 ξ,as A ξ V,M 1 ξ,as A ξv V 2 apply Elman-type estimates for the convergence of GMRES [Bonazzoli, Dolean, Graham, Spence, Tournier. Domain decomposition preconditioning for the high-frequency time-harmonic Maxwell equations with absorption. Submitted <arxiv: >] Marcella Bonazzoli (Inria, CMAP) Domain Decomposition for Maxwell 03/10/ / 24

22 Two-level domain decomposition preconditioners Explicit convergence rates in the parameters: wavenumber ω absorption ξ overlap size δ subdomain diameter H sub coarse-grid diameter H cs Particular cases When ξ ω 2 (maximal absorption) and δ H cs (generous overlap) and H sub H cs ω 1 or δ h (minimal overlap) and h H sub H cs ω 1 then our thery implies convergence of GMRES independent of ω. Marcella Bonazzoli (Inria, CMAP) Domain Decomposition for Maxwell 03/10/ / 24

23 Numerical experiments GMRES with right preconditioning (tolerance 10 6 ) random initial guess all frequencies are present in the error precondition A ξprob by the preconditioner M 1 ξ prec 0 ξ prob ξ prec local problems in each subdomain and coarse space problem solved with a direct solver (MUMPS) FreeFem++ parallelized code (one subdomain one core) Curie (TGCC-CEA) and OCCIGEN (CINES) French supercomputers Marcella Bonazzoli (Inria, CMAP) Domain Decomposition for Maxwell 03/10/ / 24

24 Numerical experiments Setting covered by the theory: PEC BC on Ω, ξ prob = ξ prec 0, AS preconditioner Ω unit cube, h ω 3/2, degree 1 edge elements First optimal particular case: ξ prob = ξ prec = ω 2, δ H sub, H sub H cs ω 1 ω n N sub n cs #AS (57) (63) (67) 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) Marcella Bonazzoli (Inria, CMAP) Domain Decomposition for Maxwell 03/10/ / 24

25 Numerical experiments Setting covered by the theory: PEC BC on Ω, ξ prob = ξ prec 0, AS preconditioner Ω unit cube, h = 1/(20f ) (where f = 2π ω), degree 2 edge elements Second optimal particular case: ξ prob = ξ prec = ω 2, δ h, H sub = 1/(2f ), H cs = 1/(2f ) f ω n n cs N sub #AS (72) (101) (129) (157) (183) (>200) (>200) # : number of iterations 2-level (1-level) Marcella Bonazzoli (Inria, CMAP) Domain Decomposition for Maxwell 03/10/ / 24

26 Numerical experiments Setting NOT covered by the theory: impedance BC on Ω, ξ prob = 0 Ω unit cube, h ω 3/2, degree 1 edge elements ξ prob = 0, ξ prec = ω, impedance BC on Ω H sub ω 0.6, H cs ω 0.9 (non nested meshes), δ h ω n N sub #2-lev n cs Time #1-lev Time (1.6) (2.6) (4.0) (8.9) (9.1) (25.6) (29.5) (88.1) (128.4) (443.1) 2-level: HORAS, 1-level: ORAS, Total Time (GMRES Time) in seconds Marcella Bonazzoli (Inria, CMAP) Domain Decomposition for Maxwell 03/10/ / 24

27 Numerical experiments Setting NOT covered by the theory: impedance BC on Ω, ξ prob = 0 Ω unit cube, degree 2 edge elements h 2π/(20 ω), H cs 2π/(2 ω), δ 2h ξ prob = 0, ξ prec = ω, impedance BC on Ω irregular partitioning (METIS) ω n N sub #2-lev n cs Time #1-lev Time (7.5) (14.8) (18.9) (72.1) (39.8) (174.5) (77.6) > (305.2) 2-level: HORAS, 1-level: ORAS, Total Time (GMRES Time) in seconds Marcella Bonazzoli (Inria, CMAP) Domain Decomposition for Maxwell 03/10/ / 24

28 Numerical experiments Microwave imaging prototype n , n cs subdomains Homogeneous (dissipative) liquid Head model (tissues are dissipative) Non-dissipative cylinder #2-level Time #1-level Time homogeneous liquid (8.6) (6.4) head model (9.2) (6.9) non-dissipative cylinder (9.4) (38.2) 2-level: HORAS, 1-level: ORAS, Total Time (GMRES Time) in seconds Marcella Bonazzoli (Inria, CMAP) Domain Decomposition for Maxwell 03/10/ / 24

29 Conclusion and outlook With high order FEs attain a given accuracy with much fewer unknowns and much less computing time than the degree 1 approximation. Parallel implementation of the domain decomposition preconditioner essential to be able to solve the large scale linear systems arising from the microwave imaging application. Two-level domain decomposition preconditioners improve the dependence of the iteration counts on the wavenumber; robust with respect to heterogeneous and non-dissipative materials. Cast Helmholtz and Maxwell convergence analysis in a common abstract framework. Design a new coarse space for Maxwell (cfr "DtN CS" for Helmholtz). Marcella Bonazzoli (Inria, CMAP) Domain Decomposition for Maxwell 03/10/ / 24