Numerical Analysis for Nonlinear Eigenvalue Problems

Transcription

1 Numerical Analysis for Nonlinear Eigenvalue Problems D. Bindel Department of Computer Science Cornell University 14 Sep 29

2 Outline Nonlinear eigenvalue problems Resonances via nonlinear eigenproblems Sensitivity and backward error analysis Resonances via Perfectly Matched Layers Perturbation bounds

3 Outline Nonlinear eigenvalue problems Resonances via nonlinear eigenproblems Sensitivity and backward error analysis Resonances via Perfectly Matched Layers Perturbation bounds

4 Linear eigenvalue problems Consider the linear system of differential equations y Ay = f, y() =. Taking the Laplace transform gives (s A)Y (s) = F(s). Singularities of s A correspond to solutions y(t) = e λt v to the unforced equation, where (A λi)v =.

5 Damping and delay Free vibrations correspond to nonlinear eigenvalues in more complicated settings. With simple damping: For systems with delay Mu + Bu + Ku = f. (s 2 M + sb + K )v =. u (t) Au(t) Bu(t τ) =. (s A e τs B)v =. General picture: study asymptotic free vibrations in terms of a nonlinear eigenvalue problem A(s)v = where A : C C n n is meromorphic.

6 Spectral Schur complements Nonlinear eigenvalue problems even arise from linear problems: [ ] A11 A A = 12. A 21 A 22 The spectral Schur complement is the inverse of a piece of the resolvent R(z) = (A zi) 1 : S(z) = (R 11 (z)) 1 = A 11 zi A 12 (A 22 zi) 1 A 21. Can use to reduce a large linear eigenvalue problem to a smaller nonlinear eigenvalue problem. Bounds on (A 22 zi) 1 lead to information about spectrum of A. Think of this as looking at a (block) transfer function.

7 Resonances and nonlinear eigenvalues For problems on unbounded domains: Can still have ordinary eigenvalues (bound states) Also have continuous spectrum (scattering states) Explore the continuous spectrum with a scattering matrix S(ω) relating incoming to outgoing waves S(ω) admits analytic continuation from R into C Places where (continued) S(ω) is singular are resonances Read off information about scattering from resonances

8 Example: Resonances and transmission ( d 2 ) dx 2 + V (x) ψ =, V (x) = 5 χ [,L] (x). T (E) imag(e) real(e)

9 Outline Nonlinear eigenvalue problems Resonances via nonlinear eigenproblems Sensitivity and backward error analysis Resonances via Perfectly Matched Layers Perturbation bounds

10 Simple 1D Problem Consider 1D Schrödinger (V nice, supp(v ) [a, b]): Hψ = ( d 2 ) dx 2 + V (x) ψ = Eψ. H self-adjoint with discrete spectrum on E <, continuous spectrum on E. Continuous spectrum is a branch cut for the resolvent. (Think χ(h E) 1 χ, χ a smooth cutoff, χ([a, b]) = 1. One moral analogue of a spectral Schur complement.) Second-sheet poles of the resolvent are resonances. Correspond to trapping, quasi-stable states.

11 Simple 1D Problem Consider 1D Schrödinger: ( d 2 ) dx 2 + V (x) ψ = Eψ. How do we: 1. Quickly compute resonances (nice enough V )? 2. Make sure the computations are correct?

12 Simple 1D Problem Consider 1D Schrödinger: If supp(v ) [a, b], write ( d 2 ) dx 2 + V (x) ψ = Eψ. ( d 2 ) dx 2 + V (x) k 2 ψ =, x (a, b) ( ) d dx ik ψ =, x = b ( ) d dx + ik ψ =, x = a E = k 2, Im k for eigenvalues, Im k < for resonances.

13 Pseudospectral Discretization Sample ψ at Chebyshev nodes and approximate dψ/dx by differentiating the interpolant: ( D 2 + V (x) k 2) ψ =, x (a, b) (D ik) ψ =, x = b (D + ik) ψ =, x = a Now linearize (introduce auxiliary variable φ = kψ) to get an ordinary generalized eigenvalue problem.

14 Is it that easy? 2 1 All eigenvalues Checked eigenvalues

15 Outline Nonlinear eigenvalue problems Resonances via nonlinear eigenproblems Sensitivity and backward error analysis Resonances via Perfectly Matched Layers Perturbation bounds

16 Backward Error Analysis 1. If ( ˆψ, Ê) is a numerical solution with above scheme, then there is some ˆV s.t. for x (a, b), (H ˆV Ê) ˆψ = ( d 2 ) dx 2 + ˆV (x) Ê ˆψ = together with corresponding radiation conditions. 2. Estimate ˆV explicitly by remapping residual to finer mesh 3. Original problem is a perturbation of computed problem. 4. Use first-order perturbation theory to correct Ê. Useful to take a variational approach.

17 More General Picture Consider Schrödinger with compactly supported V in R d. For E in resolvent set on appropriate Riemann surface, seek (H V E)ψ = f on Ω ψ B(E)ψ n = on Γ where B(E) is the Dirichlet-to-Neumann map on Ω. Solutions are stationary points for I(ψ) = 1 ( ) ( ψ) T ( ψ) + ψ(v E)ψ dω + 2 Ω 1 ψb(e)ψ dγ ψf dω. 2 Γ Ω

18 Variational Formulation Check variational formulation: I(ψ) = 1 ( ) ( ψ) T ( ψ) + ψ(v E)ψ dω 2 Ω 1 ψb(e)ψ dγ ψf dω. 2 Γ Use symmetry of form (note Γ φb(e)ψ = Γ ψb(e)φ) + integration by parts: δi(ψ) = δψ ( ψ + (V E)ψ f ) dω + Ω ( ) ψ δψ n B(E) ψ dγ. Γ Ω

19 Rayleigh Quotient Analogue Now define a residual for an approximate eigenpair: ( ) r(ψ, E) = ( ψ) T ( ψ) + ψ(v E)ψ ψb(e)ψ. Ω Take variations and use symmetry of B: δr(ψ, E) = 2 δψ [( + V E)ψ] + Ω [ ] ψ 2 δψ Γ n B(E)ψ + [ ] δe ψ 2 ψb (E)ψ Ω Γ For an eigenpair or resonance, r(ψ, E) = and δr(ψ, E) =. Γ

20 Rayleigh Quotient Analogue We now implicitly define a differentiable function Ẽ(φ) in the neighborhood of an eigenpair (ψ, E), with r(φ, E(φ)) = and Ẽ(ψ) = E. Such a function should exist if ψ 2 ψb (E)ψ Ω Stationary precisely when (ψ, E) an eigenpair. Γ

21 Sensitivity Now assume δv a compactly-supported perturbation, and look at effect of δv on Rayleigh quotient analogue. Gives that isolated eigenvalues change like Ω δv ψ2 δe = Ω ψ2 Γ ψb (E)ψ Can also write in terms of a residual for ψ as a solution for the potential V + δv : Ω ψ( + (V + δv ) E)ψ δe = Ω ψ2. Γ ψb (E)ψ

22 Backward Error Analysis Revisited 1. Compute approximate solution ( ˆψ, Ê). 2. Map ˆψ to high-resolution quadrature grid to evaluate Ω δe = ˆψ( + V Ê) ˆψ Ω ˆψ 2 ˆψB Γ (Ê) ˆψ. 3. If δe large, discard Ê as spurious; otherwise, accept E Ê + δe.

23 Some Computational Issues In general, using the domain equation + DtN map to find resonances is problematic because: 1. The DtN map is nonlocal, expensive to work with computationally. 2. The Green s function (and hence the DtN map) are hard to compute for some problems I care about (e.g. elastic half space problems). 3. Nonlinear eigenvalue problems are trickier than linear problems to solve.

24 Outline Nonlinear eigenvalue problems Resonances via nonlinear eigenproblems Sensitivity and backward error analysis Resonances via Perfectly Matched Layers Perturbation bounds

25 Perfectly Matched Layers For scattering computations / resonance computations, need an outgoing BC. We use perfectly matched layers: Complex coordinate transformation Generates a perfectly matched absorbing layer Rotates the continuous spectrum to reveal resonances Idea works with general linear wave equations Electromagnetics (Berengér, 1994) Quantum mechanics exterior complex scaling (Simon, 1979 originally used to define resonances) Elasticity in standard finite element framework (Basu and Chopra, 23)

26 Model Problem Domain: x [, ) Frequency-domain equation: d 2 û dx 2 + k 2 û = Solution: û = c out e ikx + c in e ikx

27 Model with Perfectly Matched Layer σ Regular domain x Transformed domain d x dx = λ(x) where λ(s) = 1 iσ(s) d 2 û d x 2 + k 2 û = û = c out e ik x ik x + c in e

28 Model with Perfectly Matched Layer σ Regular domain x Transformed domain d x dx = λ(x) where λ(s) = 1 iσ(s), 1 d λ dx ( 1 λ ) dû + k 2 û = dx û = c out e ikx kσ(x) + c in e ikx+kσ(x) Σ(x) = x σ(s) ds

29 Model with Perfectly Matched Layer σ Regular domain x Transformed domain If solution clamped at x = L then c in c out = O(e kγ ) where γ = Σ(L) = L σ(s) ds

30 Model Problem Illustrated 1 Outgoing exp( i x) 1 Incoming exp(i x) Transformed coordinate Re( x)

31 Model Problem Illustrated 1 Outgoing exp( i x) 3 Incoming exp(i x) Transformed coordinate Re( x)

32 Model Problem Illustrated 1 Outgoing exp( i x) 6 Incoming exp(i x) Transformed coordinate Re( x)

33 Model Problem Illustrated 1 Outgoing exp( i x) 15 Incoming exp(i x) Transformed coordinate Re( x)

34 Model Problem Illustrated 1 Outgoing exp( i x) 4 Incoming exp(i x) Transformed coordinate Re( x)

35 Model Problem Illustrated 1 Outgoing exp( i x) 1 Incoming exp(i x) Transformed coordinate Re( x)

36 Finite/Spectral Element Implementation ξ 2 x 2 x 2 x(ξ) x(x) Ω ξ 1 Ω e Ωe x 1 x 1 Combine PML and isoparametric mappings k e ij = ( N i ) T D( N j ) J dω Ω m e ij = ρn i N j J dω Ω Matrices are complex symmetric

37 DtN Approximation in PML Earlier (DtN) form - Neumann form = DtN term Γ ψbψ. Eliminate PML dofs - Neumann form DtN term Approximation is rational in k, good locally Would like to understand how good approximation to DtN + some form of stability leads to error bounds.

38 12 DtN Approximation in PML log 1 B(k) PML B(k) on circle of radius 3 in R 2. Order 3 spectral elements, PML goes [3, 4]

39 12 DtN Approximation in PML Axisymmetric resonances for ring barrier for r [1, 2]. Stars for small residual; circles for spurious resonance

40 Relation of PML to DtN Approach Numerically eliminate variables in PML = local (in k or E) rational approximation to DtN-like condition. Could also form explicit Padé approximation. Approximation is local in k where can we guarantee (for example) that we have approximated all resonances?

41 Outline Nonlinear eigenvalue problems Resonances via nonlinear eigenproblems Sensitivity and backward error analysis Resonances via Perfectly Matched Layers Perturbation bounds

42 General NEP Picture A : C C n n analytic in Ω Λ(A) := {z C : A(z) singular} Λ ɛ (A) := {z C : A(z) 1 ɛ 1 } Resonance calculation with DtN map or with eliminated PML as an example. Λ(A) and Λ ɛ (A) describe asymptotics, transients of some linear differential or difference equation. Lots of function theoretic proofs from analyzing ordinary eigenvalue problems carry over without change.

43 Counting Eigenvalues If A nonsingular on Γ, analytic inside, count eigs inside by W Γ (det(a)) = 1 d ln det(a(z)) dz 2πi Γ dz ( ) 1 = tr A(z) 1 A (z) dz 2πi Suppose E also analytic inside Γ. By continuity, W Γ (det(a)) = W Γ (det(a + se)) for s in neighborhood of such that A + se remains nonsingular on Γ. Γ

44 Function Theoretic Perturbation Recipe Winding number counts give continuity of eigenvalues = Should consider eigenvalues of A + se for s 1: Analyticity of A and E + Matrix nonsingularity test for A + se = Inclusion region for Λ(A + E) + Eigenvalue counts for connected components of region

45 Example: Matrix Rouché A 1 (z)e(z) < 1 on Γ = same eigenvalue count in Γ Proof: A 1 (z)e(z) < 1 = A(z) + se(z) invertible for s 1. (Gohberg and Sigal proved a more general version in 1971.)

46 Example: Nonlinear Gershgorin Define Then G i = z : a ii(z) < j i a ij (z) 1. Λ(A) i G i 2. Connected component m i=1 G i contains m eigs (if bounded and disjoint from Ω) Proof: Write A = D + F where D = diag(a). D + sf is diagonally dominant (so invertible) off i G i.

47 Example: Pseudospectral containment Define D = {z : E(z) < ɛ}. Then 1. Λ(A + E) Λ ɛ (A) D C 2. A bounded component of Λ ɛ (A) strictly inside D contains the same number of eigs of A and A + E.

48 Application: Lattice Schrödinger Consider the discrete analogue to Schrödinger s equation: Hψ = ( T + V )ψ = Eψ where (Hψ) k = ψ k 1 + 2ψ k ψ k+1 + V k ψ k. Assume V k = for k and k L. May be complex. Want to relate the spectrum for two variants: 1. Non-negative integers: ψ = and ψ l 2 2. Bounded: ψ k = for k = and k L + N.

49 Application: Lattice Schrödinger For V 1 =.1i and V 2 = 4, N = 2.

50 Application: Lattice Schrödinger For V 1 =.1i and V 2 = 4, N = 2.

51 Spectral Schur Complement Write H in either case as [ T11 + V H = 11 e L e1 T ] e L e1 T T 22 Then Λ(H) Λ( T 22 ) c = Λ(S), where ) S(z) = ( T 11 + V 11 ) zi (e 1 T ( T 22 zi) 1 e 1 e L el T Write S (N) (z) and S ( ) (z) for bounded and unbounded cases.

52 Spectral Schur Complement For z [, 4], choose ξ 2 (2 z)ξ + 1 =, ξ < 1. Then S ( ) (z) = ( T 11 + V 11 ) zi ξe L el T ( ) 1 ξ S (N) 2N (z) = ( T 11 + V 11 ) zi ξ 1 ξ 2(N+1) e L el T Convenient to write z = 2 ξ ξ 1, use ξ as primary variable.

53 Error Bounds Find S ( ) S (N) ɛ if ξ < ( ) log(3ɛ 1 ) = 1 O 2N + 1 ( log(ɛ 1 ) N ). Therefore, eigenvalues in bounded case (in ξ plane) either 1. Are within O(log(ɛ 1 )/N) of circle (continuous spectrum) 2. Are in Λ ɛ (S ( ) ). Get exponential convergence to discrete spectrum, linear convergence to continuous spectrum.

54 Error Estimate If S ( ) has an isolated eigenvalue at γ, then S (N) asymptotically has eigenvalues γ (N) γ with γ (N) γ = γ 2N w e L el T v L (1 γ 2 )w v w e L el T v + O(γ2N+1 ) where S ( ) (γ)v = and w S ( ) (γ) =.

55 Similar Applications Linear stability analysis for traveling waves. Bounds on distance to instability via subspace projections. Estimates of damping in MEMS resonators.

56 Conclusions Nonlinear eigenproblems tell us interesting information about dynamics. Analytic structure of the eigenproblem is key to error analysis. Variational characterization gives easy first-order perturbation theory Also get analogues to standard perturbation bounds (Rouché, Gerschgorin, pseudospectral) Get interesting estimates via approximation of spectral Schur complements