nonlinear eigenvalue problems: theory and applicationsbindel/present/2017-01-berkeley.pdftheory and...
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Nonlinear Eigenvalue Problems:Theory and Applications
David Bindel1
Department of Computer ScienceCornell University
12 January 2017
1Joint work with Amanda HoodBerkeley 1 / 37
The Computational Science & Engineering Picture
Application
ComputationAnalysis
MEMS
Smart grids
Networks
Systems
Linear algebra
Approximation theory
Symmetry + structure
Optimization
HPC / cloud
Simulators
Solvers
Frameworks
Berkeley 2 / 37
Why eigenvalues?
The polynomial connection
The optimization connection
The approximation connection
The Fourier/quadrature/special function connection
The dynamics connection
Berkeley 3 / 37
Why nonlinear eigenvalues?
y ′ − Ay = 0y=eλtv−−−−−−−−→ (λI − A)v = 0
y ′′ + By ′ + Ky = 0y=eλtv−−−−−−−−→ (λ2I + λB + K )v = 0
y ′ − Ay − By(t − 1) = 0y=eλtv−−−−−−−−→ (λI − A− Be−λ)v = 0
T (d/dt)y = 0y=eλtv−−−−−−−−→ T (λ)v = 0
Higher-order ODEs
Dynamic element formulations
Delay differential equations
Boundary integral equation eigenproblems
Radiation boundary conditions
Berkeley 4 / 37
Big and little
y ′′ + By ′ + Ky = 0v=y ′
−−−−−→[vy
]′+
[B K−I 0
] [vy
]= 0
(λ2I + λB + K )u = 0v=λu−−−−−→
(λI +
[B K−I 0
])[vu
]= 0
Tradeoff: more variables = more linear.
Berkeley 5 / 37
The big picture
T (λ)v = 0, v 6= 0.
where
T : Ω→ Cn×n analytic, Ω ⊂ C simply connected
Regularity: det(T ) 6≡ 0
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
Standard solver strategies
T (λ)v = 0, v 6= 0.
A common approach:
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
Analyticity to the rescue
T (λ)v = 0, v 6= 0.
where
T : Ω→ Cn×n analytic, Ω ⊂ C simply connected
Regularity: det(T ) 6≡ 0
Nonlinear spectrum: Λ(T ) = z ∈ Ω : T (z) singular.
Goal: Use analyticity to compare and to count
Berkeley 8 / 37
Winding and Cauchy’s argument principle
Ω
Γ
WΓ(f ) =1
2πi
∫Γ
f ′(z)
f (z)dz
= # zeros−# poles
Berkeley 9 / 37
Winding, Rouche, and Gohberg-Sigal
ΩΓ
sΓ
|f ||f + g |
Analytic f , g : Ω→ C T ,E : Ω→ Cn×n
Winding # 12πi
∫Γ
f ′(z)f (z) dz tr
(1
2πi
∫Γ T (z)−1T ′(z) dz
)Theorem Rouche (1862): Gohberg-Sigal (1971):
|g | < |f | on Γ =⇒ ‖T−1E‖ < 1 on Γ =⇒same # zeros of f , f + g same # eigs of T ,T + E
Berkeley 10 / 37
Comparing NEPs
Suppose
T ,E : Ω→ Cn×n analytic
Γ ⊂ Ω a simple closed contour
T (z) + sE (z) nonsingular ∀s ∈ [0, 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
Nonlinear pseudospectra
Λε(T ) ≡ z ∈ Ω : ‖T (z)−1‖ > ε−1
−20 −15 −10 −5 0 5 10 15 20−20
−15
−10
−5
0
5
10
15
20
Berkeley 12 / 37
Pseudospectral comparison
Ω
Ωε
Uε
E analytic, ‖E (z)‖ < ε on Ωε. Then
Λ(T + E ) ∩ Ωε ⊂ Λε(T ) ∩ Ωε
Also, if Uε a component of Λε and Uε ⊂ Ωε, then
|Λ(T + E ) ∩ Uε| = |Λ(T ) ∩ Uε|Berkeley 13 / 37
Pseudospectral comparison
Ω
Ωε
Uε
Most useful when T is linear
Even then, can be expensive to compute!
What about related tools?
Berkeley 14 / 37
The Gershgorin picture (linear case)
A = D + F , D = diag(di ), ρi =∑j
|fij |
d1
ρ1
d2
ρ2
d3
ρ3
Berkeley 15 / 37
Gershgorin (+ε)
Write A = D + F , D = diag(d1, . . . , dn). Gershgorin disks are:
Gi =
z ∈ C : |z − di | ≤∑j
|fij |
.
Useful facts:
Spectrum of A lies in⋃m
i=1 Gi⋃i∈I Gi disjoint from other disks =⇒ contains |I| eigenvalues.
Pf:A− zI strictly diagonally dominant outside
⋃mi=1 Gi .
Eigenvalues of D − sF , 0 ≤ s ≤ 1, are continuous.
Berkeley 16 / 37
Nonlinear Gershgorin
Write T (z) = D(z) + F (z). Gershgorin regions are
Gi =
z ∈ C : |di (z)| ≤∑j
|fij(z)|
.
Useful facts:
Spectrum of T lies in⋃m
i=1 Gi
Bdd connected component of⋃m
i=1 Gi strictly in Ω=⇒ same number of eigs of D and T in component=⇒ at least one eig per component of Gi involved
Pf: Strict diag dominance test + continuity of eigs
Berkeley 17 / 37
Mini-example: Counting contributions
G12G1
2 G11
T (z) =
z 1 00 z2 − 1 0.50 0 1
.
Berkeley 18 / 37
Mini-example: Domain boundaries
G11 G1
2
T (z) =
[z − 0.2
√z + 1 −1
0.4√z 1
]Ω = C− (−∞, 0]
det(D(z)) =(√z − 0.1− i
√0.99)
(√z − 0.1 + i
√0.99)
det(T (z)) =(√z + 0.1− i
√0.99)
(√z + 0.1 + i
√0.99)
D has two eigenvalues in Ω;T hides both eigenvalues behind a branch cut.
Berkeley 19 / 37
Example I: Hadeler
T (z) = (ez − 1)B + z2A− αI , A,B ∈ R8×8
−20 −15 −10 −5 0 5 10 15 20−20
−15
−10
−5
0
5
10
15
20
Berkeley 20 / 37
Comparison to simplified problem
Bauer-Fike idea: apply a similarity!
T (z) = (ez − 1)B + z2A− αI
T (z) = UTT (z)U
= (ez − 1)DB + z2I − αE= D(z)− αE
Gi = z : |βi (ez − 1) + z2| < ρi.
Berkeley 21 / 37
Gershgorin regions
−20 −15 −10 −5 0 5 10 15 20−20
−15
−10
−5
0
5
10
15
20
Berkeley 22 / 37
A different comparison
Approximate ez − 1 by a Chebyshev interpolant:
T (z) = (ez − 1)B + z2A− αIT (z) = q(z)B + z2A− α
T (z) = T (z) + r(z)B
Linearize T and transform both:
T (z) 7→ DC − zI
T (z) 7→ DC − zI + r(z)E
Restrict to Ωε = z : |r(z)| < ε:
Gi ⊂ Gi = z : |z − µi | < ρiε, ρi =∑j
|eij |
Berkeley 23 / 37
Spectrum of T
−10 −8 −6 −4 −2 0 2 4 6 8 10−10
−8
−6
−4
−2
0
2
4
6
8
10
Berkeley 24 / 37
Gi for ε < 10−10
−10 −8 −6 −4 −2 0 2 4 6 8 10−10
−8
−6
−4
−2
0
2
4
6
8
10
Berkeley 25 / 37
Gi for ε = 0.1
−10 −8 −6 −4 −2 0 2 4 6 8 10−10
−8
−6
−4
−2
0
2
4
6
8
10
Berkeley 25 / 37
Gi for ε = 1.6
−10 −8 −6 −4 −2 0 2 4 6 8 10−10
−8
−6
−4
−2
0
2
4
6
8
10
Berkeley 25 / 37
Example II: Resonance problem
V0
a b
ψ(0) = 0(− d2
dx2+ V − λ
)ψ = 0 on (0, b),
ψ′(b) = i√λψ(b),
Berkeley 26 / 37
Reduction via shooting
V0
a b
R0a(λ) Rab(λ)
ψ(0) = 0,
R0a(λ)
[ψ(0)ψ′(0)
]=
[ψ(a)ψ′(a)
],
Rab(λ)
[ψ(b)ψ′(b)
]=
[ψ(b)ψ′(b)
],
ψ′(b) = i√λψ(b)
Berkeley 27 / 37
Reduction via shooting
First-order form:
du
dx=
[0 1
V − λ 0
]u, where u(x) ≡
[ψ(x)ψ′(x)
].
On region (c , d) where V is constant:
u(d) = Rcd(λ)u(c), Rcd(λ) = exp
((d − c)
[0 1
V − λ 0
])Reduce resonance problem to 6D NEP:
T (λ)uall ≡
R0a(λ) −I 0
0 Rab(λ) −I[1 00 0
]0
[0 0
−i√λ 1
]u(0)u(a)u(b)
= 0.
Berkeley 28 / 37
Expansion via rational approximation
Consider the equation ([A BC D
]− λI
)[uv
]= 0
Partial Gaussian elimination gives the spectral Schur complement(A− λI − B(D − λI )−1C
)u = 0
Idea: GivenT (λ) = A− λI − F (λ),
find a rational approximation F (λ) ≈ B(D − λI )−1C .
Berkeley 29 / 37
Expansion via rational approximation
u(0)u(a)u(b)
...
× = 0
Berkeley 30 / 37
Analyzing the expanded system
T (z) is a Schur complement in K − zM
So Λ(T ) is easy to compute.
Or: think T (z) is a Schur complement in K − zM + E (z)
Compare T (z) to T (z) or compare K − zM + E (z) to K − zM
Berkeley 31 / 37
Analyzing the expanded system
Q: Can we find all eigs in a region not missing anything?
Concrete plan (ε = 10−8)
T = shooting system
T = rational approximation
Find region D with boundary Γ s.t.
D ⊂ Ωε (i.e. ‖T − T‖ < ε on D)Γ does not intersect Λε(T )
=⇒ Same eigenvalue counts for T , T
=⇒ Eigs of T in components of Λε(T )
Converse holds if D ⊂ Ωε/2
Can refine eigs of T in D via Newton.
Berkeley 32 / 37
Resonance approximation
−50 0 50 100 150 200 250 300 350 400−50
−40
−30
−20
−10
0
10
20
30
40
50
−10
−10
−10
−10
−10
−8
−8
−8
−8
−6
−6
−6
−6
−4
−4
−2
−2
0
0
0
Berkeley 33 / 37
Resonance approximation
−10 −8 −6 −4 −2 0 2 4 6 8 10−5
−4
−3
−2
−1
0
1
2
−10
−10
−8
−8
−6
−6
−4
−4
−2
−2
−2
0
0
0
0
Berkeley 33 / 37
Example III: Bounding dynamics
DDE model (of a laser with feedback)
u = Au(t) + Bu(t − 1).
Solution with u(0+) = u0 and u(t) = φ(t) on [−1, 0):
u(t) = Ψ(t)u0 +
∫ 1
0Ψ(t − s)Bφ(s − t) ds
where Ψ(t) is a fundamental matrix for “shock” solutions.
Berkeley 34 / 37
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
2πi
∫ΓT (z)−1ezt dz .
for contour Γ right of the spectrum.
Berkeley 35 / 37
Bounding pseudospectra =⇒ bounding dynamics
−10 −8 −6 −4 −2 0 2 4 6 8 10
−60
−40
−20
0
20
40
60
0 1 2 3 4 5 6 7 8 9 100
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Bound fundamental solution Ψ via
‖Ψ(t)‖ =
∥∥∥∥ 1
2πi
∫ΓT (z)−1ezt dz
∥∥∥∥ ≤ 1
2π
∫Γ‖T (z)−1‖|ezt ||dz |.
Berkeley 36 / 37
For more
Localization theorems for nonlinear eigenvalues.David Bindel and Amanda Hood, SIAM Review 57(4), Dec 2015
Pseudospectral bounds on transient growth for higher order and constantdelay differential equations.Amanda Hood and David Bindel, submitted to SIMAXarXiv:1611.05130, Nov 2016
Berkeley 37 / 37
Trailers!
Application
Analysis Computation
Berkeley 37 / 37
Spectral Topic Modeling
C ≈ B
× A × BT
Berkeley 37 / 37
Music of the Microspheres
Berkeley 37 / 37
Fast Fingerprints for Power Systems
2426
27
28
Berkeley 37 / 37
Response Surfaces for Global Optimization
Berkeley 37 / 37
Graph Densities of States
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 10
20
40
60
80
100
120
140
160
180
200
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 10
50
100
150
200
250
300
350
400
450
500
−0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1−200
0
200
400
600
800
1000
1200
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 10
50
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500
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 10
50
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500
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 10
10
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50
60
70
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90
100
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 10
2000
4000
6000
8000
10000
12000
14000
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 10
500
1000
1500
2000
2500
3000
3500
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 10
100
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300
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500
600
700
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 10
1
2
3
4
5
6x 10
4
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 10
0.5
1
1.5
2
2.5
3
3.5
4x 10
5
Berkeley 37 / 37
