Computable error bounds for nonlinear eigenvalue problems allowing for a minmax characterization

Transcription

1 Computable error bounds for nonlinear eigenvalue problems allowing for a minmax characterization Heinrich Voss voss@tuhh.de Joint work with Kemal Yildiztekin Hamburg University of Technology Institute of Mathematics TUHH Heinrich Voss Computable error bounds Valencia, June 20, / 30

2 Outline 1 Introduction 2 Minmax Characterization 3 A posteriori error bounds 4 Quadratic Problem 5 A rational eigenproblem TUHH Heinrich Voss Computable error bounds Valencia, June 20, / 30

3 Introduction Linear eigenvalue problems Theorem (Krylov, Bogoliubov 1929, Weinstein 1934) Let A = A H, x 0, and R(x) := x H Ax/x H x. Then here exists an eigenvalue η of A such that Ax R(x)x η R(x). x Theorem (Kato 1949, Temple 1929, 1952) Let A = A H with eigenvalues η 1 η 2 η n, and let α < β such that η j+1 α η j β η j 1. Let Ax R(x)x 2 (R(x) α)(β R(x)) x 2. Then it holds that R(x) Ax R(x)x 2 (β R(x)) x 2 η Ax R(x)x 2 j R(x) + (R(x) α) x 2. TUHH Heinrich Voss Computable error bounds Valencia, June 20, / 30

4 Minmax Characterization Nonlinear Eigenvalue Problem Let J R be an open interval (which may be unbounded), and T (λ), λ J be a family of Hermitian matrices. Nonlinear eigenvalue Problem: Find λ J and x 0 such that T (λ)x = 0. Then λ is called an eigenvalue of T ( ), and x a corresponding eigenvector. Problems of this type arise in damped vibrations of structures, conservative gyroscopic systems, lateral buckling problems, problems with retarded arguments, fluid-solid vibrations, and quantum dot heterostructures, e.g. TUHH Heinrich Voss Computable error bounds Valencia, June 20, / 30

5 Minmax Characterization Nonlinear minmax theory Assume that for fixed x C n, x 0 the real equation f (λ, x) := x H T (λ)x = 0 has at most one solution λ =: p(x) in J. Then equation f (λ, x) = 0 implicitly defines a functional p on some subset D of C n which we call the Rayleigh functional. Let (λ p(x))f (λ, x) > 0 for every λ p(x) and every x D. TUHH Heinrich Voss Computable error bounds Valencia, June 20, / 30

6 Minmax Characterization Overdamped problems If p is defined on D = C n \ {0} then the problem T (λ)x = 0 is called overdamped. Notation is motivated by the finite dimensional quadratic eigenvalue problem T (λ)x = λ 2 Mx + λcx + Kx = 0 where M, C and K are Hermitian and positive definite matrices. Theorem (Duffin 1955, Rogers 1964) Under the conditions above an overdamped problem has exactly n eigenvalues λ 1 λ 2 λ n which can be characterized by λ j = min dim V =j max p(x). x V \{0} TUHH Heinrich Voss Computable error bounds Valencia, June 20, / 30

7 Minmax Characterization Nonoverdamped problems For nonoverdamped eigenproblems the natural ordering to call the smallest eigenvalue the first one, the second smallest the second one, etc., is not appropriate. This is obvious if we make a linear eigenvalue T (λ)x := (λi A)x = 0 nonlinear by restricting it to an interval J which does not contain the smallest eigenvalue of A. Then all conditions are satisfied, p is the restriction of the Rayleigh quotient R A to D := {x 0 : R A (x) J}, and inf x D p(x) will in general not be an eigenvalue. TUHH Heinrich Voss Computable error bounds Valencia, June 20, / 30

8 Minmax Characterization Enumeration of eigenvalues If λ J is an eigenvalue of T ( ) then µ = 0 is an eigenvalue of the linear problem T (λ)y = µy, and therefore there exists l N such that 0 = max V H l min v V \{0} v H T (λ)v v 2 where H l denotes the set of all l dimensional subspaces of C n. In this case λ is called an l-th eigenvalue of T ( ). TUHH Heinrich Voss Computable error bounds Valencia, June 20, / 30

9 Minmax Characterization Minmax characterization (V., Werner 1982, V. 2009) Under the conditions given above it holds: (i) For every l N there is at most one l-th eigenvalue of T ( ) which can be characterized by λ l = min sup V H l, V D v V D p(v). ( ) The set of eigenvalues of T ( ) in J is at most countable. (ii) λ is an l-th eigenvalue if and only if µ = 0 is the l largest eigenvalue of the linear eigenproblem T ( λ)x = µx. (iii) The minimum in (*) is attained for the invariant subspace of T (λ l ) corresponding to its l largest eigenvalues. (iv) If T ( ) has an lth eigenvalue λ l J, then it holds for λ J < λ = λ x H T (λ)x l η l (λ) := sup min > dim V =l x V,x 0 x H x < = > 0. TUHH Heinrich Voss Computable error bounds Valencia, June 20, / 30

10 A posteriori error bounds Assume that T : J C n n allows for a minmax characterization of its eigenvalues in J, and let η j (λ) be the jth largest eigenvalue of the linear eigenproblem T (λ)y = η j (λ)y, j = 1,..., n, λ J. Lemma For λ, λ J, λ λ it holds that η j (λ) η j ( λ) λ λ y H (T (λ) T ( λ))y min y 0 (λ λ)y H y =: φ(λ, λ). Follows easily from the monotonicity result for eigenvalues of sums of Hermitian matrices A and B λ j+k 1 (A + B) λ j (A) + λ k (B), j, k = 1,..., n, j + k n + 1, λ j+k n (A + B) λ j (A) + λ k (B), j, k = 1,..., n, j + k n + 1, TUHH Heinrich Voss Computable error bounds Valencia, June 20, / 30

11 A posteriori error bounds Krylov,Bogoliubov, Weinstein Theorem Under the conditions of the minmax characterization let x D(p), and assume that for γ p(x) δ it holds that φ(p(x), γ)(p(x) γ) T (p(x))x x and φ(δ, p(x))(δ p(x)) T (p(x))x. x Then T (λ)x = 0 has an eigenvalue λ such that γ λ δ. TUHH Heinrich Voss Computable error bounds Valencia, June 20, / 30

12 A posteriori error bounds Krylov,Bogoliubov, Weinstein Proof The Rayleigh quotient for T (p(x))y = ηy at x is 0, and hence the (linear) Krylov, Bogoliubov, Weinstein Theorem yields the existence of some eigenvalue η j (p(x)) such that η j (p(x)) T (p(x))x. x If η j (p(x)) > 0, then it follows from the Lemma T (p(x))x x η j (p(x)) η j (γ) + φ(p(x))(p(x) γ) η j (γ) + i.e. η j (γ) 0, and there exists λ j [γ, p(x)] such that η j (λ j ) = 0. T (p(x))x, x Likewise for η j (p(x)) < 0 we get the existence of an eigenvalue λ j [p(x), δ]. TUHH Heinrich Voss Computable error bounds Valencia, June 20, / 30

13 Kato, Temple A posteriori error bounds Theorem Under the conditions of the minmax characterization let λ j, β J and x D(p). Assume that λ j p(x) β, let η j+1 (β) 0, and φ(λ, λ) 0 for λ, λ [λ j, β], λ λ. Then it holds that φ(p(x), λ j )(p(x) λ j ) T (p(x))x 2 φ(β, p(x))(β p(x)) x 2. Remarks 1. If λ j+1 J then η j+1 (β) 0 holds for β λ j+1 ; otherwise it is trivial for β J. 2. For T (λ) := λi K it holds φ(λ, λ) = 1, and with p(x) = x H Kx/x H x one gets (p(x) λ j )(β p(x)) T (p(x))x 2 / x 2, i.e. the Kato-Temple Theorem for linear problems. TUHH Heinrich Voss Computable error bounds Valencia, June 20, / 30

14 Kato, Temple A posteriori error bounds Proof From the Lemma we get η j+1 (p(x)) η j+1 (β) η j+1 (p(x)) φ(β, p(x))(β p(x)) 0. From λ j p(x) β and property (vi) of the minmax Theorem it follows that the conditions of the (linear) Kato- Temple Theorem are satisfied, and therefore η j (p(x)) T (p(x))x 2 ( η j+1 (p(x)) x 2 T (p(x))x 2 φ(β, p(x))(β p(x)) x 2 and applying the Lemma again yields T (p(x))x 2 φ(β, p(x))(β p(x)) x 2 η j (p(x)) = η j (p(x)) η j (λ j ) φ(p(x), λ j )(p(x) λ j ). TUHH Heinrich Voss Computable error bounds Valencia, June 20, / 30

15 A posteriori error bounds Kato, Temple for differentiable T Theorem Let T ( ) be differentiable and let the conditions of the minmax characterization be satisfied. Let x D(p), and assume that λ j, β J with λ j p(x) β, that η j+1 (β) 0, and y H T (λ)y ψ(λ) := min y 0 y H y 0 for every Λ [λ 1, λ 2 ]. Then it holds that p(x) λ j ψ(λ) dλ β p(x) ψ(λ) dλ T (p(x))x 2 x 2. For overdamped problems and the maximal eigenvalue of T ( ) this Kato-Temple bound was proved by Hadeler (1969). TUHH Heinrich Voss Computable error bounds Valencia, June 20, / 30

16 Proof A posteriori error bounds η j (λ) : J R, the j largest eigenvalue of T (λ)y(λ) = η(λ)y(λ) is a continuous and piecewise continuously differentiable function, and the corresponding eigenvector y(λ) can be chosen continuous and piecewise continuously differentiable. Multiplying by y(λ) H from the left yields T (λ)y(λ) + T (λ)y (λ) = η j (λ)y(λ) + η j (λ)y (λ) η j (λ) = y(λ)h T (λ)y(λ) y(λ) H. y(λ) TUHH Heinrich Voss Computable error bounds Valencia, June 20, / 30

17 Proof A posteriori error bounds Hence, β β η j+1 (p(x)) = η j+1 (β) η j (λ)dλ p(x) p(x) β ψ(λ)dλ =: γ 0. y(λ) H T (λ)y(λ) y(λ) H dλ y(λ) p(x) If γ = 0, then nothing is left to be shown. Otherwise, we have η j+1 (p(x)) γ < 0 η j (p(x)) where the last inequality follows from the minmax characterization, (iv). TUHH Heinrich Voss Computable error bounds Valencia, June 20, / 30

18 Proof A posteriori error bounds From the (linear) Kato-Temple Theorem for T (p(x)) (notice that the Rayleigh quotient of T (p(x)) at x is 0) we get T (p(x))x 2 η j (p(x)) γ x 2, and therefore 0 = η j (λ j ) = η j (p(x)) + λ j p(x) λ j η j T (p(x))x 2 (λ)dλ γ x 2 + p(x) ψ(λ)dλ, i.e. φ(p(x), λ j )(p(x) λ j ) T (p(x))x 2 φ(β, p(x))(β p(x)) x 2. TUHH Heinrich Voss Computable error bounds Valencia, June 20, / 30

19 Quadratic Problem Quadratic eigenproblem Q(λ)x := (λ 2 I + 2λB + C)x = 0, B = B T > 0, C = C T > 0. Assume that Q(σ) is indefinite for some σ < 0. Then there exist intervals J := (, σ ) and J + := (σ +, 0) such that all eigenvalue λ 1 λ 2 λ k in J are minmax values of p, and all eigenvalues λ + 1 λ+ 2 λ+ l are maxmin values of p + where p ± (x) = ( x T Bx ± (x T Bx) 2 x T Cx x T x)/ x 2. Let x D(p ) and λ j p(x) β λ j+1 such that β λ max(b). Then for T ( ) = Q( ) we have φ(λ, λ) = λ λ 2λ max (B) 0 for λ, λ λ max (B), and one gets (λ j +λ max (B)) 2 (p (x)+λ max (B)) 2 Q(p (x))x 2 + x 2 ( β p (x) 2λ max (B))(β p (x)) TUHH Heinrich Voss Computable error bounds Valencia, June 20, / 30

20 Quadratic Problem Numerical Example A = eye(20); B = randn(20); B = 0.5 B T B; C = randn(20); C = C T C; Then Q(λ)x = 0 has 26 real eigenvalues, 13 of either type, and the maximum of the eigenvalues of negative type is less than the minimum of the eigenvalue of positive type. So, all of them are minmax and maxmin values of p and p +, respectively. Only 4 real eigenvalues satisfy λ j bound applies. < λ max (B) such that the Kato-Temple For a random vector y and p = p (y) we projected Q( ) to the invariant subspace of Q(p) corresponding to the 4 largest eigenvalues, and chose as ansatz vectors the corresponding Ritz vectors. For the Kato-Temple bound of the kth eigenvalue we chose β = 0.5 (λ k + λ k+1 ). TUHH Heinrich Voss Computable error bounds Valencia, June 20, / 30

21 Quadratic Problem Numerical Example l.bound e.val u.bound e.val-l.b. u.b.-e.val e e e e e e e e-3 TUHH Heinrich Voss Computable error bounds Valencia, June 20, / 30

22 Quadratic Problem Quadratic eigenproblem l.bound e.val-l.b. u.bound u.b.-e.val e e e e e e e e e e e e e e e e-4 TUHH Heinrich Voss Computable error bounds Valencia, June 20, / 30 In a similar way one can prove a Kato-Temple Theorem yielding upper bounds for nonlinear eigenproblems allowing for a maxmin characterization of its eigenvalues. For Q(λ) = λ 2 + 2λB + C as before let x D(p + ), λ + j+1 β p +(x) λ + j, and β > λ min (B). Then it holds that (λ + j +λ min (B)) 2 (p + (x)+λ min (B)) 2 Q(p + (x))x 2 + x 2 (β + p + (x) + 2λ min (B))(p + (x) β). In our numerical example only one eigenvalue λ + 1 = 3.66e 3 satisfies λ + j > λ min (B) = 6.52e 3, and a very good lower bound p + (x) is needed to obtain a negative upper bound from the Kato-Temple Theorem.

23 Plateproblem A rational eigenproblem Consider vertical vibrations of a clamped plate with k identical elastically attached loads. Discretization with finite elements (with Bogner, Fox, Schmit elements) yields a rationale eigenvalue problem T (λ)x := λmx Kx + λ σ λ CCT x = 0 where K, M R n n are symmetric and positive definite, and C R n k, which is equivalent to A(λ)x := (λi E 1 KE T + λ σ λ (E 1 C)(E 1 C) T )x = 0, M = EE T. All eigenvalues allow for a minmax characterization, and for λ, λ < σ φ(λ, λ) = 1 + σ (σ λ)(σ λ) λ min(e 1 CC T E T ) = 1 > 0. TUHH Heinrich Voss Computable error bounds Valencia, June 20, / 30

24 Plateproblem A rational eigenproblem j [λ l, λ u] λ u λ l A(λ u)x / x 1 [0, ] 1.29e e [ , ] 8.28e e [ , ] 7.24e e 03 1 [ , ] e 05 2 [ , ] 4.07e e [ , ] 6.06e e 01 2 [ , ] 1.22e e 03 2 [ , ] 1.33e e 03 2 [ , ] 4.31e e 04 2 [ , ] 5.02e e 07 2 [ , ] 1.00e e 07 3 [ , ] 7.11e e [ , ] 2.33e e [ , ] 2.79e e 03 3 [ , ] e 06 TUHH Heinrich Voss Computable error bounds Valencia, June 20, / 30

25 Plateproblem A rational eigenproblem j [λ l, λ u] λ u λ l A(λ u)x / x 4 [ , ] 6.32e e [ , ] 2.53e e 01 4 [ , ] 1.09e e 04 4 [ , ] e 07 5 [ , ] 5.40e e [ , ] 9.27e e 01 5 [ , ] 5.52e e 04 5 [ , ] e 06 6 [ , ] 1.64e e [ , ] 1.33e e 01 6 [ , ] e 04 TUHH Heinrich Voss Computable error bounds Valencia, June 20, / 30