The Nullspace free eigenvalue problem and the inexact Shift and invert Lanczos method. V. Simoncini. Dipartimento di Matematica, Università di Bologna
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1 The Nullspace free eigenvalue problem and the inexact Shift and invert Lanczos method V. Simoncini Dipartimento di Matematica, Università di Bologna and CIRSA, Ravenna, Italy 1
2 The generalized eigenvalue problem Given Ax = λmx A sym. pos. semidef. M sym. positive def. C sparse basis for large null space of A Find min λ i 0 λ i Difficulty: zero eigenvalues pollute approximation 2
3 Equivalent formulation for smallest eigenvalue min C T Mx=0 0 x R n xt Ax x T Mx C T Mx = 0 constraint Nullspace free eigenvectors 3
4 Outline General Spectral Transformation Inexact Shift-and-Invert Lanczos Method Inexactness vs. Constraint Alternative Problem Formulations Augmented Formulation Modified Formulation A numerical example Computational enhancements 4
5 Original Problem: Solve General Spectral Transformation Ax = λmx Transformed Problem: Fix σ R and rewrite as (A σm) 1 Mx = ηx η = (λ σ) 1 σ close to eigenvalues of interest η large λ = σ + 1 η Fast convergence of Lanczos method is expected Other methods: Jacobi-Davidson, Subspace iteration, Preconditioned Inverse Iteration, etc. 5
6 Given v 1 R, General Shift and Invert Lanczos (SI(σ) Lanczos) for j = 1, 2,... ṽ = (A σm) 1 Mv j v j+1 t j+1,j = ṽ V j T :,j T :,j = V T j Mṽ V j = [v 1,..., v j ] Yielding (A σm) 1 MV j = V j T j + [0,..., 0, r j+1 ] 6
7 (A σm) 1 MV j = V j T j + [0,..., 0, r j+1 ] If T j s (i) j = η (i) j s (i) j i = 1,..., j then ( ) σ + 1, V η (i) j s (i) j j, i = 1,..., j are approximate eigenpairs of Ax = λmx Warning: In our setting A is singular. If σ is close to zero, then many zero eigenvalues may be detected Take v 1 s.t. C T Mv 1 = 0 C T MV j = 0 j (exact arithmetic) 7
8 General Inexact Shift and Invert Lanczos Take v 1 s.t. C T Mv 1 = 0, for j = 1, 2,... w Mv j ˆv approx. solves (A σm)ˆv = w M-orthogonalize ˆv w.r. to {v 1,..., v j } v j+1 V j = [v 1, v 2,..., v j ] C T MV j? = 0 Krylov subspace inner solvers 8
9 Natural fixes [1 ] Null space is available. Explicitly deflate (purge) null space Starting vector v 1 M N (A) span(v j ) M N (A) for j = 1, 2,... w Mv j ˆv approx. solves (A σm)ˆv = w ˆv (I π)ˆv M-orthogonalize ˆv w.r. to {v 1,..., v j } v j+1 π projection onto N (A) π = C(C T MC) 1 C T M see e.g. Golub, Zhang, Zha
10 [2 ] Use σ = 0 and generate approximation space in Range(A) A MV j = V j T j + r j+1 e T j At each iteration i, solve consistent system Ay = Mv i Arbenz, Drmac 2000 Preconditioning techniques for singular A: Notay , Hiptmair, Neymeyr 2001,... 10
11 Enforcing the constraint Given v s.t. C T Mv = 0, approximately solve (A σm)x = Mv (1) with the constraint C T Mx = 0 This problem is equivalent to A σm MC (MC) T 0 x y = Saddle point linear system Exploit work on preconditioning Mv 0 (A σm)z = Mb (2) (A σm)p 1 ẑ = Mb ( ) ẑ m approx. solution to (*) z m = P 1 ẑ m approx. solution to (2) 11
12 Structured preconditioning Given the linear system K B B T 0 x y = f g Effective preconditioners include P = K 1 0 Q = K 1 B 0 S B T 0 R = K 1 B 0 S S B T K 1 B Large bibliography. See Benzi, Golub, Liesen, Acta Numerica
13 Definite preconditioning (A σm)x = Mv (1) If A σm MC (MC) T 0 x y = Mv 0 (2) is preconditioned by P = K (MC) T K1 1 (MC) then MINRES soln of (2) is given by z m = MINRES soln. of (1) preconditioned by K 1, K 1 = A 1 τm A 1 C = 0 x m 0, where x m is 13
14 Indefinite preconditioning (A σm)x = Mv (1) If A σm MC (MC) T 0 x y = Mv 0 (2) is preconditioned by Q = K 1 MC (MC) T 0, K 1 = A 1 τm A 1 C = 0 then GMRES soln of (2) is given by z m = x m, where x m is 0 GMRES soln. of (1) preconditioned by K 1 14
15 Solve Preconditioned system Important remark (A σm)k 1 1 ˆx = Mv K 1 = A 1 τm so that Then it holds x m = K 1 1 ˆx m C T Mx m = 0 15
16 Alternative Problem Formulations Ax = λmx C T Mx = 0 [1 ] Augmented FE formulation (Kikuchi, 1987) A MC (MC) T 0 x y = λ M 0 0 T 0 x y min{ λ i } = min λ i 0 {λ i} Computational aspects in Arbenz Geus (1999), Arbenz Geus Adam (2001) 16
17 [2 ] Modified FE formulation (Bespalov, 1988) Given a sym. nonsingular H R n c n c, solve (A + MCH 1 C T M)x = ηmx for suitable H, min i {η i } = min λi 0{λ i } 17
18 Augmented formulation A MC (MC) T 0 x y } {{ } A = λ M 0 x 0 T 0 y } {{ } M Az = λ Mz Spectral transformation: (A σm) 1 Mz = ηz η = 1 λ σ Apply inexact SI(σ) Lanczos 18
19 Augmented formulation At each iteration, solve Saddle-point linear system A σm MC x = Mv (MC) T 0 y 0 Natural inner preconditioners P = K (MC) T K1 1 (MC), K 1 = A 1 τm A 1 C = 0 Q = K 1 MC (MC) T 0 19
20 but Inexact SI(σ)-Lanczos applied to augmented formulation with inner preconditioner P or Q generates the same approximate eigenpairs as Inexact SI(σ)-Lanczos applied to Ax = λmx with inner preconditioner K 1 20
21 The modified formulation Original problem Ax = λmx Given a sym. nonsingular H R n c n c, solve (A + MCH 1 C T M)x = ηmx λ 0 µ s.t. µ = λ λ = 0 µ eigenvalue of (C T MC, H) Remark. No practical (numerical) advantages over solving Ax = λm x 21
22 A numerical example: Electromagnetic cavity resonator Variational formulation for the 2D computational model Find w h R s.t. 0 u h Σ h Σ: (rot(v 1, v 2 ) = (v 2 )x (v 1 )y ) (rot u h, rot v h ) = ωh(u 2 h, v h ) v h Σ h, Σ = {v [L 2 (Ω)] 2 : rot v L 2 (Ω), v t = 0 on Ω} t counterclockwise oriented tangent versor to the boundary Ω =]0, π[ 2 ω 2 = 1, 1, 2, 4, 4, 5, 5, 8, 9, 9, 10, 10,... ******************** FE method using edge elements Problem dimension = 3229 Number of zero eigenvalues = 1036 Boffi, Fernandes, Gastaldi, Perugia, SIAM J. Numer. Anal., v.36 (1999) 22
23 Norm of residual Ax (1) j λ (1) λ (1) j j Mx (1) j (A σm) 1 Mx = ηx (A σm) 1 Mz = ηz (A σm) 1 Mz = ηz m K 1 P Q
24 First wrap-up Original Augmented Formulation (n) Formulation (n + n c ) Same Approximation iterates 24
25 Computational considerations: Inexact Lanczos Starting vector v 1 M N (A) span(v j ) M N (A) for j = 1, 2,... w Mv j ˆv approx. solves (A σm)ˆv = w ˆv (I π)ˆv M-orthogonalize ˆv w.r. to {v 1,..., v j } v j+1 Compute approximate η (1) j,..., η (j) j to η 1, η 2,... Check convergence with Lanczos residuals Question: How accurate should be the solution of (A σm)ˆv = w? 25
26 Answer: You can decrease the accuracy as the Lanczos method converges imaginary part real part sherman5 MatrixMarket. Approx. min λ with inverted Arnoldi 26
27 Example inner residual norm magnitude ε exact res inexact residual number of outer iterations sherman5 MatrixMarket. Approx. min λ with inverted Arnoldi 27
28 At iteration j, solve (A σm)ˆv = w η (1) j 1 approx eig. after j 1 Lanczos its. r j 1 associated computed eigenvalue residual f j = (A σm)ˆv k w residual after k inner iterations (linear system) Stopping criterion for inner system solver Empirically (Bouras and Frayssé, t.r. 00): f j 10 α η (1) j 1 r j 1 ε, α = 0, 1, 2 28
29 At iteration j, solve (A σm)ˆv = w η (1) j 1 approx eig. after j 1 Lanczos its. r j 1 associated computed eigenvalue residual f j = (A σm)ˆv k w residual after k inner iterations (linear system) Stopping criterion for inner system solver New computable bound (Simoncini, t.r. 04): where δ (j 1) := f j min{ A σm, δ(j 1) } 2m η (1) j 1 r ε j 1 min η (k) η (k) j 1 Λ(H j 1)\{η (1) j 1 } j 1 η(1) j 1 29
30 The whole story If, for any k = 1,..., m, r k δ2 m,k 1 4 s m and f k δ m,k 1 2m η (1) j 1 r k 1 ε then after m iterations, (η (1) m, s) satisfies ((A σm) 1 MV m s η (1) m V m s) r m ε 30
31 [1 ] P. Arbenz and R. Geus, A comparison of solvers for large eigenvalue problems occuring in the design of resonant cavities, Numer. Linear Alg. with Appl., 6 (1999), pp [2 ] P. Arbenz, R. Geus, and S. Adam, Solving Maxwell eigenvalue problems for accelerating cavities, Physical Review special Topics - Accelerators and Beams, 4 (2001), pp [3 ] A. N. Bespalov, Finite element method for the eigenmode problem of a rf cavity resonator, Sov. J. Numer. Anal. Math. Mod., 3 (1988), pp [4 ] F. Kikuchi, Mixed and penalty formulations for finite element analysis of an eigenvalue problem in electromagnetism, Computer Methods in Appl. Mech. Eng., 64 (1987), pp [5 ] V. Simoncini, Algebraic formulations for the solution of the nullspace free eigenvalue problem using the inexact Shift and-invert Lanczos method, Numerical Linear Algebra w/appl, [6 ] V. Simoncini, Variable accuracy of matrix-vector products in projection methods for eigencomputation. March To appear in SIAM J. Numerical Analysis. valeria@dm.unibo.it 31
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