Nonlinear Eigenvalue Problems: Theory and Applications

Transcription

1 Nonlinear Eigenvalue Problems: Theory and Applications David Bindel 1 Department of Computer Science Cornell University 12 January Joint work with Amanda Hood Berkeley 1 / 37

2 The Computational Science & Engineering Picture Application Analysis Computation MEMS Smart grids Networks Systems Linear algebra Approximation theory Symmetry + structure Optimization HPC / cloud Simulators Solvers Frameworks Berkeley 2 / 37

3 Why eigenvalues? The polynomial connection The optimization connection The approximation connection The Fourier/quadrature/special function connection The dynamics connection Berkeley 3 / 37

4 Why nonlinear eigenvalues? y Ay = y + By + Ky = y Ay By(t 1) = T (d/dt)y = y=e λt v (λi A)v = y=e λt v (λ 2 I + λb + K)v = y=e λt v (λi A Be λ )v = y=e λt v T (λ)v = Higher-order ODEs Dynamic element formulations Delay differential equations Boundary integral equation eigenproblems Radiation boundary conditions Berkeley 4 / 37

5 Big and little y + By + Ky = (λ 2 I + λb + K)u = v=y [ ] v + y ( v=λu λi + [ B K I [ B K I ] [ ] v = y ]) [ ] v = u Tradeoff: more variables = more linear. Berkeley 5 / 37

6 The big picture T (λ)v =, v. where T : Ω C n n analytic, Ω C simply connected Regularity: det(t ) Nonlinear spectrum: Λ(T ) = {z Ω : T (z) singular}. What do we want? Qualitative information (e.g. no eigenvalues in right half plane) Error bounds on computed/estimated eigenvalues Assurances that we know all the eigenvalues in some region Berkeley 6 / 37

7 Standard solver strategies A common approach: T (λ)v =, v. 1 Approximate T locally (linear, rational, etc) 2 Solve approximate problem. 3 Repeat as needed. How should we choose solver parameters? What about global behavior? What can we trust? What might we miss? Berkeley 7 / 37

8 Analyticity to the rescue T (λ)v =, v. where T : Ω C n n analytic, Ω C simply connected Regularity: det(t ) Nonlinear spectrum: Λ(T ) = {z Ω : T (z) singular}. Goal: Use analyticity to compare and to count Berkeley 8 / 37

9 Winding and Cauchy s argument principle Γ Ω W Γ (f ) = 1 f (z) 2πi Γ f (z) dz = # zeros # poles Berkeley 9 / 37

10 Winding, Rouché, and Gohberg-Sigal f f + g Ω Γ s Γ Analytic f, g : Ω C T, E : Ω C n n Winding # 1 f (z) 2πi Γ f (z) dz tr ( 1 2πi Γ T (z) 1 T (z) dz ) Theorem Rouché (1862): Gohberg-Sigal (1971): g < f on Γ = T 1 E < 1 on Γ = same # zeros of f, f + g same # eigs of T, T + E Berkeley 1 / 37

11 Comparing NEPs Suppose T, E : Ω C n n analytic Γ Ω a simple closed contour T (z) + se(z) nonsingular s [, 1], z Γ Then T and T + E have the same number of eigenvalues inside Γ. Pf: Constant winding number around Γ. Berkeley 11 / 37

12 Nonlinear pseudospectra Λ ɛ (T ) {z Ω : T (z) 1 > ɛ 1 } Berkeley 12 / 37

13 Pseudospectral comparison U ɛ Ω Ω ɛ E analytic, E(z) < ɛ on Ω ɛ. Then Λ(T + E) Ω ɛ Λ ɛ (T ) Ω ɛ Also, if U ɛ a component of Λ ɛ and Ū ɛ Ω ɛ, then Λ(T + E) U ɛ = Λ(T ) U ɛ Berkeley 13 / 37

14 Pseudospectral comparison U ɛ Ω Ω ɛ Most useful when T is linear Even then, can be expensive to compute! What about related tools? Berkeley 14 / 37

15 The Gershgorin picture (linear case) A = D + F, D = diag(d i ), ρ i = j f ij ρ 1 ρ 2 ρ 3 d 1 d 2 d 3 Berkeley 15 / 37

16 Gershgorin (+ɛ) Write A = D + F, D = diag(d 1,..., d n ). Gershgorin disks are: G i = z C : z d i f ij. j Useful facts: Pf: Spectrum of A lies in m i=1 G i i I G i disjoint from other disks = contains I eigenvalues. A zi strictly diagonally dominant outside m i=1 G i. Eigenvalues of D sf, s 1, are continuous. Berkeley 16 / 37

17 Nonlinear Gershgorin Write T (z) = D(z) + F (z). Gershgorin regions are G i = z C : d i(z) f ij (z). j Useful facts: Spectrum of T lies in m i=1 G i Bdd connected component of m i=1 G i strictly in Ω = same number of eigs of D and T in component = at least one eig per component of G i involved Pf: Strict diag dominance test + continuity of eigs Berkeley 17 / 37

18 Mini-example: Counting contributions G 1 2 G 1 1 G 1 2 z 1 T (z) = z Berkeley 18 / 37

19 Mini-example: Domain boundaries T (z) = [ ] z.2 z z 1 Ω = C (, ] G 1 1 G 1 2 det(d(z)) =( z.1 i.99) ( z.1 + i.99) det(t (z)) =( z +.1 i.99) D has two eigenvalues in Ω; T hides both eigenvalues behind a branch cut. ( z i.99) Berkeley 19 / 37

20 Example I: Hadeler T (z) = (e z 1)B + z 2 A αi, A, B R Berkeley 2 / 37

21 Comparison to simplified problem Bauer-Fike idea: apply a similarity! T (z) = (e z 1)B + z 2 A αi T (z) = U T T (z)u = (e z 1)D B + z 2 I αe = D(z) αe G i = {z : β i (e z 1) + z 2 < ρ i }. Berkeley 21 / 37

22 Gershgorin regions Berkeley 22 / 37

23 A different comparison Approximate e z 1 by a Chebyshev interpolant: T (z) = (e z 1)B + z 2 A αi T (z) = q(z)b + z 2 A α T (z) = T (z) + r(z)b Linearize T and transform both: T (z) D C zi T (z) D C zi + r(z)e Restrict to Ω ɛ = {z : r(z) < ɛ}: G i Ĝ i = {z : z µ i < ρ i ɛ}, ρ i = j e ij Berkeley 23 / 37

24 Spectrum of T Berkeley 24 / 37

25 Ĝ i for ɛ < Berkeley 25 / 37

26 Ĝ i for ɛ = Berkeley 25 / 37

27 Ĝ i for ɛ = Berkeley 25 / 37

28 Example II: Resonance problem V a b ψ() = ( d 2 ) dx 2 + V λ ψ = on (, b), ψ (b) = i λψ(b), Berkeley 26 / 37

29 Reduction via shooting V R a (λ) a b R ab (λ) R a (λ) R ab (λ) ψ() =, [ ] ψ() ψ = () [ ] ψ(b) ψ = (b) [ ] ψ(a) ψ, (a) [ ] ψ(b) ψ, (b) ψ (b) = i λψ(b) Berkeley 27 / 37

30 Reduction via shooting First-order form: du dx = [ ] 1 u, where u(x) V λ [ ] ψ(x) ψ. (x) On region (c, d) where V is constant: ( [ ]) 1 u(d) = R cd (λ)u(c), R cd (λ) = exp (d c) V λ Reduce resonance problem to 6D NEP: R a (λ) I T (λ)u all [ R ab (λ) I u() ] [ ] 1 i u(a) =. u(b) λ 1 Berkeley 28 / 37

31 Expansion via rational approximation Consider the equation ([ A B C D ] λi ) [ ] u = v Partial Gaussian elimination gives the spectral Schur complement ( A λi B(D λi ) 1 C ) u = Idea: Given T (λ) = A λi F (λ), find a rational approximation F (λ) B(D λi ) 1 C. Berkeley 29 / 37

32 Expansion via rational approximation u() u(a) u(b) =. Berkeley 3 / 37

33 Analyzing the expanded system ˆT (z) is a Schur complement in K zm So Λ( ˆT ) is easy to compute. Or: think T (z) is a Schur complement in K zm + E(z) Compare ˆT (z) to T (z) or compare K zm + E(z) to K zm Berkeley 31 / 37

34 Analyzing the expanded system Q: Can we find all eigs in a region not missing anything? Concrete plan (ɛ = 1 8 ) T = shooting system ˆT = rational approximation Find region D with boundary Γ s.t. D Ω ɛ (i.e. T ˆT < ɛ on D) Γ does not intersect Λ ɛ (T ) = Same eigenvalue counts for T, ˆT = Eigs of ˆT in components of Λ ɛ (T ) Converse holds if D Ω ɛ/2 Can refine eigs of ˆT in D via Newton. Berkeley 32 / 37

35 6 8 Resonance approximation Berkeley 33 / 37

36 8 1 Resonance approximation Berkeley 33 / 37

37 Example III: Bounding dynamics DDE model (of a laser with feedback) u = Au(t) + Bu(t 1). Solution with u(+) = u and u(t) = φ(t) on [ 1, ): u(t) = Ψ(t)u + 1 Ψ(t s)bφ(s t) ds where Ψ(t) is a fundamental matrix for shock solutions. Berkeley 34 / 37

38 Representing Ψ(t) Associated nonlinear eigenvalue problem involves T (z) = zi A Be z Assume all eigs in left half plane; via inverse Laplace transform, Ψ(t) = 1 T (z) 1 e zt dz. 2πi for contour Γ right of the spectrum. Γ Berkeley 35 / 37

39 Bounding pseudospectra = bounding dynamics Bound fundamental solution Ψ via Ψ(t) = 1 T (z) 1 e zt dz 2πi 1 2π Γ Γ T (z) 1 e zt dz. Berkeley 36 / 37

40 For more Localization theorems for nonlinear eigenvalues. David Bindel and Amanda Hood, SIAM Review 57(4), Dec 215 Pseudospectral bounds on transient growth for higher order and constant delay differential equations. Amanda Hood and David Bindel, submitted to SIMAX arxiv: , Nov 216 Berkeley 37 / 37

41 Trailers! Application Analysis Computation Berkeley 37 / 37

42 Spectral Topic Modeling A B T C B Berkeley 37 / 37

43 Music of the Microspheres Berkeley 37 / 37

44 Fast Fingerprints for Power Systems Berkeley 37 / 37

45 Response Surfaces for Global Optimization Berkeley 37 / 37

46 x x Graph Densities of States Berkeley 37 / 37

47 Fin Berkeley 37 / 37