On domain decomposition preconditioners for finite element approximations of the Helmholtz equation using absorption
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1 On domain decomposition preconditioners for finite element approximations of the Helmholtz equation using absorption Ivan Graham and Euan Spence (Bath, UK) Collaborations with: Paul Childs (Emerson Roxar, Oxford), Martin Gander (Geneva) Douglas Shanks (Bath) Eero Vainikko (Tartu, Estonia) CUHK Lecture 3, January 2016
2 Outline of talk: Seismic inversion, HF Helmholtz equation FE discretization, preconditioned GMRES solvers sharp analysis of preconditioners based on absorption new theory for Domain Decomposition for Helmholtz almost optimal (scalable) solvers (2D implementation) some open theoretical questions
3 Motivation
4 Seismic inversion Inverse problem: reconstruct material properties of subsurface (characterised by wave speed c(x)) from observed echos. Regularised iterative method: repeated solution of the (forward problem): the wave equation Frequency domain: u + 1 c 2 2 u t 2 = f or its elastic variant ( ) ωl 2 u u = f, ω = frequency c solve for u with approximate c. Large domain of characteristic length L. effectively high frequency
5 Seismic inversion Inverse problem: reconstruct material properties of subsurface (wave speed c(x)) from observed echos. Regularised iterative method: repeated solution of the (forward problem): the wave equation Frequency domain: u + 2 u t 2 = f or its elastic variant ( ) ωl 2 u u = f, ω = frequency c solve for u with approximate c. Large domain of characteristic length L. effectively high frequency - time domain vs freqency domain
6 Marmousi Model Problem Schlumberger 2007: Solver of choice based on principle of limited absorption (Erlangga, Osterlee, Vuik, 2004) This work: Analysis of this approach and use it to build better methods...
7 Analysis for: interior impedance problem u k 2 u = f in bounded domain Ω u iku n = g on Γ := Ω...Also truncated sound-soft scattering problems in Ω B R Ω Ω Γ
8 Linear algebra problem weak form with absorption k 2 k 2 + i, η = η(k, ) ( a (u, v) := u. v (k 2 +k 2 )uv ) ik uv Ω Γ = fv + gv ShiftedLaplacian Ω Γ (Fixed order) finite element discretization A u := (S (k 2 +k 2 )M Ω ikm Γ )u = f Often: h k 1 but pollution effect: for quasioptimality need h k 2??, h k 3/2?? Melenk and Sauter 2011, Zhu and Wu 2013
9 Linear algebra problem weak form with absorption k 2 k 2 + i, ( a (u, v) := u. v (k 2 + i)uv ) ik uv Ω Γ = fv + gv Shifted Laplacian Ω Γ Finite element discretization A u := (S (k 2 + i)m Ω ikm Γ )u = f Blackboard Often: h k 1 but pollution effect: for quasioptimality need h k 2??, h k 3/2?? Melenk and Sauter 2011, Zhu and Wu 2013
10 Preconditioning with A 1 and its approximations A 1 Au = A 1 f. Elman theory for GMRES requires: A 1 A 1, and dist(0, fov(a 1 A)) 1 any norm Sufficient condition: I A 1 A 2 C < 1. Blackboard In practice use Writing B 1 Au = B 1 f, where B 1 I B 1 A = I B 1 a sufficient condition is: A + B 1 A 1. A (I A 1 A), I A 1 A 2 and I B 1 A 2 small, i.e. A 1 to be a good preconditioner for A and B 1 to be a good preconditioner for A.
11 Preconditioning with A 1 and its approximations A 1 Au = A 1 f. Elman theory for GMRES requires: A 1 A 1, and dist(0, fov(a 1 A)) 1 Sufficient condition: I A 1 A 2 C < 1. In practice use B 1 B 1 Au = B 1 f, easily computed approximation of A 1. Writing I B 1 so we require A = I B 1 A + B 1 A (I A 1 A), I A 1 A 2 and I B 1 A 2 small, i.e. A 1 to be a good preconditioner for A and B 1 to be a good preconditioner for A. Part 1
12 Preconditioning with A 1 and its approximations A 1 Au = A 1 f. Elman theory for GMRES requires: A 1 A 1, and dist(0, fov(a 1 A)) 1 Sufficient condition: I A 1 A 2 C < 1. In practice use B 1 B 1 Au = B 1 f, easily computed approximation of A 1. Writing I B 1 so we require A = I B 1 A + B 1 A (I A 1 A), I A 1 A 2 and I B 1 A 2 small, i.e. A 1 to be a good preconditioner for A and B 1 to be a good preconditioner for A. Part 2
13 A very short history Bayliss et al 1983, Laird & Giles Erlangga, Vuik & Oosterlee 04 and subsequent papers: Precondition A with MG approximation of A 1 ɛ ɛ k 2 (simplified Fourier eigenvalue analysis) Kimn & Sarkis 13 used k 2 to enhance domain decomposition methods Engquist and Ying, 11 Used k to stabilise their sweeping preconditioner...others...
14 Part 1 Theorem 1 (with Martin Gander and Euan Spence) For star-shaped domains Smooth (or convex) domains, quasiuniform meshes: I A 1 ɛ A ɛ k. Corner singularities, locally refined meshes: I D 1/2 A 1 ɛ AD 1/2 ɛ k. D = diag(m Ω ). So ɛ/k sufficiently small = convergence. k independent GMRES
15 Shifted Laplacian preconditioner = k Solving A 1 Ax = A 1 1 on unit square h k 3/2 k # GMRES
16 Shifted Laplacian preconditioner = k 3/2 Solving A 1 Ax = A 1 1 on unit square h k 3/2 k # GMRES
17 Shifted Laplacian preconditioner = k 2 Solving A 1 Ax = A 1 1 on unit square h k 3/2 k # GMRES
18 Proof of Theorem 1: via continuous problem a ɛ (u, v) = Ω fv + gv, v H 1 (Ω) ( ) Γ Theorem (Stability) Assume Ω is Lipschitz and star-shaped. Then, if ɛ/k sufficiently small, u 2 L 2 (Ω) + k2 u 2 L 2 (Ω) }{{} =: u 2 1,k f 2 L 2 (Ω) + g 2 L 2 (Γ), k indept of k and ɛ cf. Melenk 95, Cummings & Feng 06 More absorption: k ɛ k 2 general Lipschitz domain OK. Key technique in proof: Rellich/Morawetz Identities More detail of proof: Lecture 4
19 Proof continued Exact solution estimate : u L 2 (Ω) k 1 f L 2 (Ω) (*) Finite element solution: A u = f Estimate: u 2 k 1 h d f 2 (**) proof of (**) uses (*) and FE quasioptimality (h small enough) Lecture 4 Hence I A 1 ɛ A A 1 A A k 1 h d iɛm Ω ɛ k. Locally refined meshes: I D 1/2 A 1 ɛ AD 1/2 ɛ k.
20 Exterior scattering problem with refinement h k 1, Solving A 1 Ax = A 1 1 on unit square # GMRES 1 with diagonal scaling k = k = k 3/
21 A trapping domain k = k = k 3/2 10π/ π/ π/ π/ Stability result fails... quasimodes Lecture 4 Betcke, Chandler-Wilde, IGG, Langdon, Lindner, 2010
22 Part 2: How to approximate A 1? Erlangga, Osterlee, Vuik (2004): Geometric multigrid: problem elliptic Engquist & Ying (2012): Since the shifted Laplacian operator is elliptic, standard algorithms such as multigrid can be used for its inversion Domain Decomposition (DD): Many non-overlapping methods ( = 0) Benamou & Després Gander, Magoules, Nataf, Halpern, Dolean... General issue: coarse grids, scalability? Conjecture If large enough, classical overlapping DD methods with coarse grids will work (giving scalable solvers). However Classical analysis for = 0 (Cai & Widlund, 1992) leads to coarse grid size H k 2
23 Classical additive Schwarz To solve a problem on a fine grid FE space S h Coarse space S H (here linear FE) on a coarse grid Subdomain spaces S i on subdomains Ω i, overlap δ H sub H in this case
24 Classical additive Schwarz p/c for matrix C Approximation of C 1 : R T i C 1 i R i + R T HC 1 H R H i R i = restriction to S i, C i = R i CR T i Dirichlet BCs R H = restriction to S H C H = R H CR T H Apply to A to get B 1
25 Non-standard DD theory - applied to A Coercivity Lemma There exisits Θ = 1, with Im [Θa (v, v)] k 2 v 2 1,k. ( ) Projections onto subpaces: Lecture 4 a (Q i v h, w i ) = a (v h, w i ), v h S h, w i S i. Guaranteed well-defined by ( ). Analysis of B 1 A equivalent to analysing Q := i Q i + Q H operator in FE space S h.
26 Non-standard DD theory - applied to A Coercivity Lemma There exisits Θ = 1, with Im [Θa (v, v)] k 2 v 2 1,k }{{}. ( ) u 2 Ω +k2 u 2 Ω Projections onto subpaces: Lecture 4 a (Q H v h, w H ) = a (v h, w H ), v h S h, w H S H. Guaranteed well-defined by ( ). Analysis of B 1 A equivalent to analysing Q := i Q i + Q H operator in FE space S h.
27 Convergence results Assume overlap δ H and k 2 Theorem (with Euan Spence and Eero Vainikko) (i) For all coarse grid sizes H, B 1 A Dk 1. (ii) Provided Hk 1 (no pollution!). dist(0, fov(b 1 A ) Dk ) 1, Note: D k = stiffness matrix for Helmholtz energy: (u, v) H 1 + k 2 (u, v) L 2 Hence k independent (weighted) GMRES convergence when k 2 and Hk 1
28 Convergence results - general Assume overlap δ H Theorem (with Euan Spence and Eero Vainikko) (i) For all coarse grid sizes H, (ii) Provided Hk (/k 2 ) 3 B 1 A Dk k 2 /. dist(0, fov(b 1 A ) Dk ) (/k 2 ) 2, Same results for right preconditioning (duality) Extension to general overlap, and one-level Schwarz Hence k independent GMRES convergence when k 2 and Hk 1
29 Some steps in proof k 2 (v h, Qv h ) 1,k = j (v h, Q j v h ) 1,k + (v h, Q H v h ) 1,k (v h, Q H v h ) 1,k = Q H v h 2 1,k + ((I Q H)v h, Q H v h ) 1,k Second term is small (condition on kh) [ Galerkin Orthogonality, duality, regularity] (v h, Qv h ) 1,k Q j v h 2 1,k + Q Hv h 2 1,k j ( ) k 2 v h 2 1,k
30 Some steps in proof k 2 (v h, Qv h ) 1,k = j (v h, Q j v h ) 1,k + (v h, Q H v h ) 1,k (v h, Q H v h ) 1,k = Q H v h 2 1,k + ((I Q H)v h, Q H v h ) 1,k Second term is small (condition on kh) [ Galerkin Orthogonality, duality, regularity] (v h, Qv h ) 1,k j Q j v h 2 1,k + Q Hv h 2 1,k ( k 2 ) 2 vh 2 1,k Lecture 4
31 Useful Variants HRAS: Multiplicative between coarse and local solves only add up once on regions of overlap ImpHRAS impedance boundary conditions on local solves All experiments: unit square, h k 3/2, n k 3, δ H. Standard GMRES - minimise residual in Euclidean norm (Theory has weights)
32 B 1 as preconditioner for A = k 2 # GMRES iterates with HRAS: k H k 1 H k 0.9 H k Scope for increasing H when = k 2 Is there scope for reducing?
33 B 1 as preconditioner for A = k # GMRES iterates with HRAS: k H k 1 H k 0.9 H k * * Method still works when = k provided Hk 1
34 The real problem: B 1 as preconditioner for A H k 1 # GMRES iterates with HRAS: k = k = k 2 cf. Shifted Laplace Local problems of size k k Coarse grid problem of k 2 k 2 (dominates) approximation (FGMRES) Hierarchical
35 The coarse grid problem: inner iteration problem of size k 2 k 2, with k (Hierarchical) subdomains of size k k no inner coarse grid # GMRES iterates with ImpHRAS k H inner k k 0.3
36 The real problem: Inner outer FGMRES = k # FGMRES iterates with HRAS (Inner iterations ImpHRAS) k time (s) (1) (2) (3) (5) (5) (7) 3417 k 4 n 4/3 O(k 2 ) independent solves of size k
37 A more challenging application 3D SEG Salt model Childs, IGG, Shanks, 2016 Hybrid Sweeping preconditioner with one level RAS inner solve Boundary condition chosen as optimised Robin condition
38 ω = 3π ω = 6π ω = 9π Nsub Iterations Solve time (s) Iterations Solve time (s) Iterations Solve time (s) 2x2x e e e+01 4x4x e e e+01 8x8x e e e+01 PPW Shifted problem (ω/c(x)) 2 ((ω i)/c(x)) 2 cf. k
39 Summary k and ɛ explicit analysis allows rigorous explanation of some empirical observations and formulation of new methods. When ɛ k, When ɛ k 2, A 1 ɛ B 1 is optimal preconditioner for A is optimal preconditioner for A When preconditioning A with B 1, empirical best choice is k New framework for DD analysis - Helmholtz energy and sesquilinear form. Open questions in analysis when k 2 1
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