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A projection method for the eigenvalues of Elias Jarlebring A projection method for the eigenvalues of Hot shower problem a delay-differential equation Lambert W Solution operator Infinitesimal generator 8th GAMM Workshop - Applied and

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A projection method for the eigenvalues

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Elias Jarlebring Hot shower problem Lambert W Solution operator Infinitesimal generator A Projection method Numerical examples

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A projection method for the eigenvalues of a delay-differential equation

8th GAMM Workshop - Applied and numerical linear algebra Hamburg, 11-12th September, 2008 Elias Jarlebring Institut Computational Mathematics Technische Universtität Braunschweig

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A projection method for the eigenvalues

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Elias Jarlebring Hot shower problem Lambert W Solution operator Infinitesimal generator A Projection method Numerical examples

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Delay-differential equations (DDEs) Retarded DDE with a single delay Σ =

  • ˙

x(t) = A0x(t) + A1x(t − τ), t ≥ 0 x(t) = ϕ(t), t ∈ [−τ, 0] (1) where A0, A1 ∈ Rn×n and ϕ : [−τ, 0] → Rn.

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A projection method for the eigenvalues

  • f

Elias Jarlebring Hot shower problem Lambert W Solution operator Infinitesimal generator A Projection method Numerical examples

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Delay-differential equations (DDEs) Retarded DDE with a single delay Σ =

  • ˙

x(t) = A0x(t) + A1x(t − τ), t ≥ 0 x(t) = ϕ(t), t ∈ [−τ, 0] (1) where A0, A1 ∈ Rn×n and ϕ : [−τ, 0] → Rn. Ansatz: x(t) = vest ⇒

  • −sI + A0 + A1e−sτ

v = 0

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A projection method for the eigenvalues

  • f

Elias Jarlebring Hot shower problem Lambert W Solution operator Infinitesimal generator A Projection method Numerical examples

2/20

Delay-differential equations (DDEs) Retarded DDE with a single delay Σ =

  • ˙

x(t) = A0x(t) + A1x(t − τ), t ≥ 0 x(t) = ϕ(t), t ∈ [−τ, 0] (1) where A0, A1 ∈ Rn×n and ϕ : [−τ, 0] → Rn. Ansatz: x(t) = vest ⇒

  • −sI + A0 + A1e−sτ

v = 0

Definition

Spectrum: σ(Σ) := {s ∈ C : det (−sI + A0 + A1e−sτ) = 0}

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A projection method for the eigenvalues

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Elias Jarlebring Hot shower problem Lambert W Solution operator Infinitesimal generator A Projection method Numerical examples

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The hot-shower example

  • x(t) = Temp diff from optimal Temp
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A projection method for the eigenvalues

  • f

Elias Jarlebring Hot shower problem Lambert W Solution operator Infinitesimal generator A Projection method Numerical examples

3/20

The hot-shower example

  • x(t) = Temp diff from optimal Temp
  • α = Human sensitivity
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A projection method for the eigenvalues

  • f

Elias Jarlebring Hot shower problem Lambert W Solution operator Infinitesimal generator A Projection method Numerical examples

3/20

The hot-shower example

  • x(t) = Temp diff from optimal Temp
  • α = Human sensitivity
  • τ = Length of pipe

Speed of water

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A projection method for the eigenvalues

  • f

Elias Jarlebring Hot shower problem Lambert W Solution operator Infinitesimal generator A Projection method Numerical examples

3/20

The hot-shower example

  • x(t) = Temp diff from optimal Temp
  • α = Human sensitivity
  • τ = Length of pipe

Speed of water

Delayed negative feedback: ˙ x(t) = −αx(t − τ)

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SLIDE 9

A projection method for the eigenvalues

  • f

Elias Jarlebring Hot shower problem Lambert W Solution operator Infinitesimal generator A Projection method Numerical examples

3/20

The hot-shower example

  • x(t) = Temp diff from optimal Temp
  • α = Human sensitivity
  • τ = Length of pipe

Speed of water

Delayed negative feedback: ˙ x(t) = −αx(t − τ) , τ = 1 5 10 15 −4 −2 2 4 t x α=1.4~ α=2 Solution x(t)

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A projection method for the eigenvalues

  • f

Elias Jarlebring Hot shower problem Lambert W Solution operator Infinitesimal generator A Projection method Numerical examples

3/20

The hot-shower example

  • x(t) = Temp diff from optimal Temp
  • α = Human sensitivity
  • τ = Length of pipe

Speed of water

Delayed negative feedback: ˙ x(t) = −αx(t − τ) , τ = 1 5 10 15 −4 −2 2 4 t x α=1.4~ α=2 Solution x(t) −4 −2 2 −50 50 Real Imag −4 −2 2 −50 50 Real Imag Spectrum σ(Σ) = {s ∈ C : −s − αe−τs = 0}

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A projection method for the eigenvalues

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Elias Jarlebring Hot shower problem Lambert W Solution operator Infinitesimal generator A Projection method Numerical examples

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Computing the spectrum

Example ([Ebenbauer, Allgöwer ’06])

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A projection method for the eigenvalues

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Elias Jarlebring Hot shower problem Lambert W Solution operator Infinitesimal generator A Projection method Numerical examples

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Computing the spectrum

Example ([Ebenbauer, Allgöwer ’06])

Spectrum σ(Σ): s ∈ C such that

det @−sI + @ −1 13.5 −1 −3 −1 −2 −2 −1 −4 1 A + @ −5.9 7.1 −70.3 2 −1 5 2 6 1 A e−τs 1 A = 0

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A projection method for the eigenvalues

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Elias Jarlebring Hot shower problem Lambert W Solution operator Infinitesimal generator A Projection method Numerical examples

4/20

Computing the spectrum

Example ([Ebenbauer, Allgöwer ’06])

Spectrum σ(Σ): s ∈ C such that

det @−sI + @ −1 13.5 −1 −3 −1 −2 −2 −1 −4 1 A + @ −5.9 7.1 −70.3 2 −1 5 2 6 1 A e−τs 1 A = 0

−2 −1 1 −50 50 Real σ(Σ) Imag σ(Σ) τ=1

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A projection method for the eigenvalues

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Elias Jarlebring Hot shower problem Lambert W Solution operator Infinitesimal generator A Projection method Numerical examples

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Methods for σ(Σ) := {s ∈ C : det (−sI + A0 + A1e−sτ) = 0}

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A projection method for the eigenvalues

  • f

Elias Jarlebring Hot shower problem Lambert W Solution operator Infinitesimal generator A Projection method Numerical examples

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Methods for σ(Σ) := {s ∈ C : det (−sI + A0 + A1e−sτ) = 0}

Method 1: Lambert W

Analytic method for scalar and simultaneously triangularizable systems.

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A projection method for the eigenvalues

  • f

Elias Jarlebring Hot shower problem Lambert W Solution operator Infinitesimal generator A Projection method Numerical examples

5/20

Methods for σ(Σ) := {s ∈ C : det (−sI + A0 + A1e−sτ) = 0}

Method 1: Lambert W

Analytic method for scalar and simultaneously triangularizable systems.

Method 2: Solution operator

Numerical method based on the discretization of the solution

  • perator.
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SLIDE 17

A projection method for the eigenvalues

  • f

Elias Jarlebring Hot shower problem Lambert W Solution operator Infinitesimal generator A Projection method Numerical examples

5/20

Methods for σ(Σ) := {s ∈ C : det (−sI + A0 + A1e−sτ) = 0}

Method 1: Lambert W

Analytic method for scalar and simultaneously triangularizable systems.

Method 2: Solution operator

Numerical method based on the discretization of the solution

  • perator.

Method 3: Infinitesimal generator

Numerical method based on the discretization of the infinitesimal generator.

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SLIDE 18

A projection method for the eigenvalues

  • f

Elias Jarlebring Hot shower problem Lambert W Solution operator Infinitesimal generator A Projection method Numerical examples

5/20

Methods for σ(Σ) := {s ∈ C : det (−sI + A0 + A1e−sτ) = 0}

Method 1: Lambert W

Analytic method for scalar and simultaneously triangularizable systems.

Method 2: Solution operator

Numerical method based on the discretization of the solution

  • perator.

Method 3: Infinitesimal generator

Numerical method based on the discretization of the infinitesimal generator.

Method 4: Subspace accelerated residual inverse iteration

A projection method based on residual inverse iteration.

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SLIDE 19

A projection method for the eigenvalues

  • f

Elias Jarlebring Hot shower problem Lambert W Solution operator Infinitesimal generator A Projection method Numerical examples

5/20

Methods for σ(Σ) := {s ∈ C : det (−sI + A0 + A1e−sτ) = 0}

Method 1: Lambert W

Analytic method for scalar and simultaneously triangularizable systems.

Method 2: Solution operator

Numerical method based on the discretization of the solution

  • perator.

Method 3: Infinitesimal generator

Numerical method based on the discretization of the infinitesimal generator.

Method 4: Subspace accelerated residual inverse iteration

A projection method based on residual inverse iteration.

  • Two examples indicate that Method 4 should be used for

large problems.

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SLIDE 20

A projection method for the eigenvalues

  • f

Elias Jarlebring Hot shower problem Lambert W Solution operator Infinitesimal generator A Projection method Numerical examples

5/20

Methods for σ(Σ) := {s ∈ C : det (−sI + A0 + A1e−sτ) = 0}

Method 1: Lambert W

Analytic method for scalar and simultaneously triangularizable systems.

Method 2: Solution operator

Numerical method based on the discretization of the solution

  • perator.

Method 3: Infinitesimal generator

Numerical method based on the discretization of the infinitesimal generator.

Method 4: Subspace accelerated residual inverse iteration

A projection method based on residual inverse iteration.

  • Two examples indicate that Method 4 should be used for

large problems.

  • Previously in the literature n = 131, Method 4: n = 106.
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A projection method for the eigenvalues

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Elias Jarlebring Hot shower problem Lambert W Solution operator Infinitesimal generator A Projection method Numerical examples

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Method 1: The Lambert W function Scalar single delay: −s + a0 + a1e−sτ = 0

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A projection method for the eigenvalues

  • f

Elias Jarlebring Hot shower problem Lambert W Solution operator Infinitesimal generator A Projection method Numerical examples

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Method 1: The Lambert W function Scalar single delay: −s + a0 + a1e−sτ = 0 Multiply with τeτs−τa0 and rearrange terms ⇒ τa1e−τa0 = τ(s − a0)eτ(s−a0)

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A projection method for the eigenvalues

  • f

Elias Jarlebring Hot shower problem Lambert W Solution operator Infinitesimal generator A Projection method Numerical examples

6/20

Method 1: The Lambert W function Scalar single delay: −s + a0 + a1e−sτ = 0 Multiply with τeτs−τa0 and rearrange terms ⇒ τa1e−τa0 = τ(s − a0)eτ(s−a0) Lambert W: Multivalued inverse of wew, i.e., Wk(x)eWk(x) = x

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A projection method for the eigenvalues

  • f

Elias Jarlebring Hot shower problem Lambert W Solution operator Infinitesimal generator A Projection method Numerical examples

6/20

Method 1: The Lambert W function Scalar single delay: −s + a0 + a1e−sτ = 0 Multiply with τeτs−τa0 and rearrange terms ⇒ τa1e−τa0 = τ(s − a0)eτ(s−a0) Lambert W: Multivalued inverse of wew, i.e., Wk(x)eWk(x) = x Matlab & Maple:lambertw [Corless et al ’96][Jeffrey et al 96]

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A projection method for the eigenvalues

  • f

Elias Jarlebring Hot shower problem Lambert W Solution operator Infinitesimal generator A Projection method Numerical examples

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Method 1: The Lambert W function Scalar single delay: −s + a0 + a1e−sτ = 0 Multiply with τeτs−τa0 and rearrange terms ⇒ τa1e−τa0 = τ(s − a0)eτ(s−a0) Lambert W: Multivalued inverse of wew, i.e., Wk(x)eWk(x) = x Matlab & Maple:lambertw [Corless et al ’96][Jeffrey et al 96] Wk(τa1e−τa0) = Wk

  • τ(s − a0)eτ(s−a0)
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A projection method for the eigenvalues

  • f

Elias Jarlebring Hot shower problem Lambert W Solution operator Infinitesimal generator A Projection method Numerical examples

6/20

Method 1: The Lambert W function Scalar single delay: −s + a0 + a1e−sτ = 0 Multiply with τeτs−τa0 and rearrange terms ⇒ τa1e−τa0 = τ(s − a0)eτ(s−a0) Lambert W: Multivalued inverse of wew, i.e., Wk(x)eWk(x) = x Matlab & Maple:lambertw [Corless et al ’96][Jeffrey et al 96] Wk(τa1e−τa0) = Wk

  • τ(s − a0)eτ(s−a0)

= τ(s − a0)

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SLIDE 27

A projection method for the eigenvalues

  • f

Elias Jarlebring Hot shower problem Lambert W Solution operator Infinitesimal generator A Projection method Numerical examples

6/20

Method 1: The Lambert W function Scalar single delay: −s + a0 + a1e−sτ = 0 Multiply with τeτs−τa0 and rearrange terms ⇒ τa1e−τa0 = τ(s − a0)eτ(s−a0) Lambert W: Multivalued inverse of wew, i.e., Wk(x)eWk(x) = x Matlab & Maple:lambertw [Corless et al ’96][Jeffrey et al 96] Wk(τa1e−τa0) = Wk

  • τ(s − a0)eτ(s−a0)

= τ(s − a0) ⇒ Well known closed form: s = 1

τ Wk(τa1e−τa0) + a0

[Corless et al ’96], [Shinozaki, Mori ’06]

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A projection method for the eigenvalues

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Elias Jarlebring Hot shower problem Lambert W Solution operator Infinitesimal generator A Projection method Numerical examples

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Problem

Does the formula σ(Σ) =

  • k∈Z

1 τ Wk(τa1e−τa0) + a0 hold for systems with n > 1? We have shown in [J.,Damm ’07]:

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A projection method for the eigenvalues

  • f

Elias Jarlebring Hot shower problem Lambert W Solution operator Infinitesimal generator A Projection method Numerical examples

7/20

Problem

Does the formula σ(Σ) =

  • k∈Z

1 τ Wk(τa1e−τa0) + a0 hold for systems with n > 1? We have shown in [J.,Damm ’07]:

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A projection method for the eigenvalues

  • f

Elias Jarlebring Hot shower problem Lambert W Solution operator Infinitesimal generator A Projection method Numerical examples

7/20

Problem

Does the formula σ(Σ) =

  • k∈Z

1 τ Wk(τa1e−τa0) + a0 hold for systems with n > 1? We have shown in [J.,Damm ’07]: i) Answer: Yes, if A0, A1 are simultaneously triangularizable

  • r in particular if A0A1 = A1A0.
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SLIDE 31

A projection method for the eigenvalues

  • f

Elias Jarlebring Hot shower problem Lambert W Solution operator Infinitesimal generator A Projection method Numerical examples

7/20

Problem

Does the formula σ(Σ) =

  • k∈Z

1 τ Wk(τa1e−τa0) + a0 hold for systems with n > 1? We have shown in [J.,Damm ’07]: i) Answer: Yes, if A0, A1 are simultaneously triangularizable

  • r in particular if A0A1 = A1A0.

ii) Answer: No, for the general case.

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A projection method for the eigenvalues

  • f

Elias Jarlebring Hot shower problem Lambert W Solution operator Infinitesimal generator A Projection method Numerical examples

7/20

Problem

Does the formula σ(Σ) =

  • k∈Z

1 τ Wk(τa1e−τa0) + a0 hold for systems with n > 1? We have shown in [J.,Damm ’07]: i) Answer: Yes, if A0, A1 are simultaneously triangularizable

  • r in particular if A0A1 = A1A0.

ii) Answer: No, for the general case. Other literature [Asl, Ulsoy ’03], with citation count 12, is not valid in the stated generality.

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A projection method for the eigenvalues

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Elias Jarlebring Hot shower problem Lambert W Solution operator Infinitesimal generator A Projection method Numerical examples

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The Lambert W formula

Theorem ([J., Damm ’07])

If A0 and A1 are simultaneously triangularizable, then σ(Σ) =

  • k∈Z

σ 1 τ Wk(A1τe−A0τ) + A0

  • .
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A projection method for the eigenvalues

  • f

Elias Jarlebring Hot shower problem Lambert W Solution operator Infinitesimal generator A Projection method Numerical examples

8/20

The Lambert W formula

Theorem ([J., Damm ’07])

If A0 and A1 are simultaneously triangularizable, then σ(Σ) =

  • k∈Z

σ 1 τ Wk(A1τe−A0τ) + A0

  • .

Corollary

If A0A1 = A1A0, then σ(Σ) =

  • k∈Z

σ 1 τ Wk(A1τe−A0τ) + A0

  • .
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A projection method for the eigenvalues

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Elias Jarlebring Hot shower problem Lambert W Solution operator Infinitesimal generator A Projection method Numerical examples

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Method 2: Solution operator discretization (SOD) ˙ x(t) = −αx(t − 1), ϕ(t) = 0.5

−1 1 2 3 4 −1 −0.5 0.5 1 t x α=1 −1 −0.5 −0.5 0.5

y(0) = ϕ

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A projection method for the eigenvalues

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Elias Jarlebring Hot shower problem Lambert W Solution operator Infinitesimal generator A Projection method Numerical examples

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Method 2: Solution operator discretization (SOD) ˙ x(t) = −αx(t − 1), ϕ(t) = 0.5

−1 1 2 3 4 −1 −0.5 0.5 1 t x α=1 −1 −0.5 −0.5 0.5

y(0) = ϕ

−1 −0.5 −0.5 0.5

y(1) = T(1)ϕ

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A projection method for the eigenvalues

  • f

Elias Jarlebring Hot shower problem Lambert W Solution operator Infinitesimal generator A Projection method Numerical examples

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Method 2: Solution operator discretization (SOD) ˙ x(t) = −αx(t − 1), ϕ(t) = 0.5

−1 1 2 3 4 −1 −0.5 0.5 1 t x α=1 −1 −0.5 −0.5 0.5

y(0) = ϕ

−1 −0.5 −0.5 0.5

y(1) = T(1)ϕ

−1 −0.5 −0.5 0.5

y(2) = T(2)ϕ

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A projection method for the eigenvalues

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Elias Jarlebring Hot shower problem Lambert W Solution operator Infinitesimal generator A Projection method Numerical examples

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Method 2: Solution operator discretization (SOD) ˙ x(t) = −αx(t − 1), ϕ(t) = 0.5

−1 1 2 3 4 −1 −0.5 0.5 1 t x α=1 −1 −0.5 −0.5 0.5

y(0) = ϕ

−1 −0.5 −0.5 0.5

y(1) = T(1)ϕ

−1 −0.5 −0.5 0.5

y(2) = T(2)ϕ

−1 −0.5 −0.5 0.5

y(3) = T(3)ϕ

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A projection method for the eigenvalues

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Elias Jarlebring Hot shower problem Lambert W Solution operator Infinitesimal generator A Projection method Numerical examples

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Hot shower problem: T(1)y = −α t

−1

y(t) dt + y(0)

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A projection method for the eigenvalues

  • f

Elias Jarlebring Hot shower problem Lambert W Solution operator Infinitesimal generator A Projection method Numerical examples

10/20

Hot shower problem: T(1)y = −α t

−1

y(t) dt + y(0)

More generally if h ≤ τ, (T(h)ϕ)(θ) = ( ψ(θ) = ϕ(θ − h) θ ≤ −h Solution of ˙ ψ(θ) = A0ψ(θ) + A1ϕ(θ + h − τ) θ > −h, Discretize with Runge-Kutta or Linear Multistep, stepsize h ⇒ SN ∈ RnN×nN.

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A projection method for the eigenvalues

  • f

Elias Jarlebring Hot shower problem Lambert W Solution operator Infinitesimal generator A Projection method Numerical examples

10/20

Hot shower problem: T(1)y = −α t

−1

y(t) dt + y(0)

More generally if h ≤ τ, (T(h)ϕ)(θ) = ( ψ(θ) = ϕ(θ − h) θ ≤ −h Solution of ˙ ψ(θ) = A0ψ(θ) + A1ϕ(θ + h − τ) θ > −h, Discretize with Runge-Kutta or Linear Multistep, stepsize h ⇒ SN ∈ RnN×nN.

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A projection method for the eigenvalues

  • f

Elias Jarlebring Hot shower problem Lambert W Solution operator Infinitesimal generator A Projection method Numerical examples

10/20

Hot shower problem: T(1)y = −α t

−1

y(t) dt + y(0)

More generally if h ≤ τ, (T(h)ϕ)(θ) = ( ψ(θ) = ϕ(θ − h) θ ≤ −h Solution of ˙ ψ(θ) = A0ψ(θ) + A1ϕ(θ + h − τ) θ > −h, Discretize with Runge-Kutta or Linear Multistep, stepsize h ⇒ SN ∈ RnN×nN.

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SLIDE 43

A projection method for the eigenvalues

  • f

Elias Jarlebring Hot shower problem Lambert W Solution operator Infinitesimal generator A Projection method Numerical examples

10/20

Hot shower problem: T(1)y = −α t

−1

y(t) dt + y(0)

More generally if h ≤ τ, (T(h)ϕ)(θ) = ( ψ(θ) = ϕ(θ − h) θ ≤ −h Solution of ˙ ψ(θ) = A0ψ(θ) + A1ϕ(θ + h − τ) θ > −h, Discretize with Runge-Kutta or Linear Multistep, stepsize h ⇒ SN ∈ RnN×nN. ⇒ If s ∈ 1 h ln(σ(SN)) and discretization sufficiently fine then s ≈ z for some z ∈ σ(Σ)

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A projection method for the eigenvalues

  • f

Elias Jarlebring Hot shower problem Lambert W Solution operator Infinitesimal generator A Projection method Numerical examples

10/20

Hot shower problem: T(1)y = −α t

−1

y(t) dt + y(0)

More generally if h ≤ τ, (T(h)ϕ)(θ) = ( ψ(θ) = ϕ(θ − h) θ ≤ −h Solution of ˙ ψ(θ) = A0ψ(θ) + A1ϕ(θ + h − τ) θ > −h, Discretize with Runge-Kutta or Linear Multistep, stepsize h ⇒ SN ∈ RnN×nN. ⇒ If s ∈ 1 h ln(σ(SN)) and discretization sufficiently fine then s ≈ z for some z ∈ σ(Σ) LMS, DDE-BIFTOOL [Engelborghs, Roose ’99, ’02], RK [Breda ’05]

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A projection method for the eigenvalues

  • f

Elias Jarlebring Hot shower problem Lambert W Solution operator Infinitesimal generator A Projection method Numerical examples

11/20

SOD(LMS) is a Padé approximation of the logarithm By change of variables: det(−sI + A0 + A1e−τs) = 0

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SLIDE 46

A projection method for the eigenvalues

  • f

Elias Jarlebring Hot shower problem Lambert W Solution operator Infinitesimal generator A Projection method Numerical examples

11/20

SOD(LMS) is a Padé approximation of the logarithm By change of variables: det(−sI + A0 + A1e−τs) = 0 ⇔ det

  • −1

hµN ln µ + A0µN + A1

  • = 0

(2) where µ = ehs and h = τ/N.

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SLIDE 47

A projection method for the eigenvalues

  • f

Elias Jarlebring Hot shower problem Lambert W Solution operator Infinitesimal generator A Projection method Numerical examples

11/20

SOD(LMS) is a Padé approximation of the logarithm By change of variables: det(−sI + A0 + A1e−τs) = 0 ⇔ det

  • −1

hµN ln µ + A0µN + A1

  • = 0

(2) where µ = ehs and h = τ/N. Approximate ln(µ) with (2,2)-Padé approximation in µ = 1 α(µ) β(µ) = µ2 − 1

1 3µ2 + 4 3µ + 1 3

⇒ (2) turns into polynomial eigenvalue problem

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SLIDE 48

A projection method for the eigenvalues

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Elias Jarlebring Hot shower problem Lambert W Solution operator Infinitesimal generator A Projection method Numerical examples

11/20

SOD(LMS) is a Padé approximation of the logarithm By change of variables: det(−sI + A0 + A1e−τs) = 0 ⇔ det

  • −1

hµN ln µ + A0µN + A1

  • = 0

(2) where µ = ehs and h = τ/N. Approximate ln(µ) with (2,2)-Padé approximation in µ = 1 α(µ) β(µ) = µ2 − 1

1 3µ2 + 4 3µ + 1 3

⇒ (2) turns into polynomial eigenvalue problem Companion linearization ⇒ LMS(Milne-Simpson) used in DDE-BIFTOOL.

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A projection method for the eigenvalues

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Elias Jarlebring Hot shower problem Lambert W Solution operator Infinitesimal generator A Projection method Numerical examples

12/20

Method 3: Infinitesimal generator discretization (IGD) PDE-representation: e.g. [Diekman, van Gils ’95]

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A projection method for the eigenvalues

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Elias Jarlebring Hot shower problem Lambert W Solution operator Infinitesimal generator A Projection method Numerical examples

12/20

Method 3: Infinitesimal generator discretization (IGD) PDE-representation: e.g. [Diekman, van Gils ’95]

∂u ∂t = ∂u ∂θ

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SLIDE 51

A projection method for the eigenvalues

  • f

Elias Jarlebring Hot shower problem Lambert W Solution operator Infinitesimal generator A Projection method Numerical examples

12/20

Method 3: Infinitesimal generator discretization (IGD) PDE-representation: e.g. [Diekman, van Gils ’95]

∂u ∂t = ∂u ∂θ

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A projection method for the eigenvalues

  • f

Elias Jarlebring Hot shower problem Lambert W Solution operator Infinitesimal generator A Projection method Numerical examples

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Method 3: Infinitesimal generator discretization (IGD) PDE-representation: e.g. [Diekman, van Gils ’95]

∂u ∂t = ∂u ∂θ IV:u(0, θ) = ϕ(θ) BC:∂u ∂t (t, 0) = A0u(t, 0) + A1u(t, −τ)

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PDE:            ∂u ∂t = ∂u ∂θ IV:u(0, θ) = ϕ(θ) BC:∂u ∂t (t, 0) = A0u(t, 0) + A1u(t, −τ)

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Elias Jarlebring Hot shower problem Lambert W Solution operator Infinitesimal generator A Projection method Numerical examples

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PDE:            ∂u ∂t = ∂u ∂θ IV:u(0, θ) = ϕ(θ) BC:∂u ∂t (t, 0) = A0u(t, 0) + A1u(t, −τ) Semi-discretization with Euler [Bellen, Maset ’00],

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Elias Jarlebring Hot shower problem Lambert W Solution operator Infinitesimal generator A Projection method Numerical examples

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PDE:            ∂u ∂t = ∂u ∂θ IV:u(0, θ) = ϕ(θ) BC:∂u ∂t (t, 0) = A0u(t, 0) + A1u(t, −τ) Semi-discretization with Euler [Bellen, Maset ’00], or Runge-Kutta [Breda, Maset, Vermiglio ’03]

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Elias Jarlebring Hot shower problem Lambert W Solution operator Infinitesimal generator A Projection method Numerical examples

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PDE:            ∂u ∂t = ∂u ∂θ IV:u(0, θ) = ϕ(θ) BC:∂u ∂t (t, 0) = A0u(t, 0) + A1u(t, −τ) Semi-discretization with Euler [Bellen, Maset ’00], or Runge-Kutta [Breda, Maset, Vermiglio ’03] Alternative derivation for [Bellen, Maset ’00]: Replace e−τs in σ(Σ) with (1 + τs/N)−N and companion linearization.

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Method 4:Subspace accelerated residual inverse iteration In the context of nonlinear eigenvalue problems (NLEVPs) 0 = (−sI + A0 + A1e−τs)v = T(s)v.

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Method 4:Subspace accelerated residual inverse iteration In the context of nonlinear eigenvalue problems (NLEVPs) 0 = (−sI + A0 + A1e−τs)v = T(s)v.

Residual inverse iteration [Neumaier ’85]

Input: v0 ∈ Cn shift µ ∈ C

1: for k = 1 . . . convergence do 2:

Solve v∗

k T(s)vk = 0 numerically

3:

vk+1 = vk − T(µ)−1T(s)vk and normalize

4: end for

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Elias Jarlebring Hot shower problem Lambert W Solution operator Infinitesimal generator A Projection method Numerical examples

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Method 4:Subspace accelerated residual inverse iteration In the context of nonlinear eigenvalue problems (NLEVPs) 0 = (−sI + A0 + A1e−τs)v = T(s)v.

Residual inverse iteration [Neumaier ’85]

Input: v0 ∈ Cn shift µ ∈ C

1: for k = 1 . . . convergence do 2:

Solve v∗

k T(s)vk = 0 numerically

3:

vk+1 = vk − T(µ)−1T(s)vk and normalize

4: end for

Idea: Expand a subspace [Voss ’04] [Meerbergen ’01]

Subspace accelerated residual inverse iteration (SRI)

1: for all k = 1 . . . convergence do 2:

Solve V ∗T(s)Vy = 0 numerically

3:

v = T(µ)−1T(s)Vy and orthogonalize

4:

V = [V, v/v]

5: end for

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Full featured SRI

1: Factorize LU = PT(σ)Q 2: Compute initial vectors V 3: for m = 1 . . . mmax do 4:

while relres > RESTOL do

5:

Update Aproj such that Aproj(k) = V ∗AkV for all k.

6:

Solve projected nonlinear eigenvalue problem [µ, y, µnext, ynext] = projectedsolve(V, Aproj, µ, σ, σt, Vl, µl) where (µ, y) is the “best solution candidate”.

7:

Compute v = T(σ)−1T(µ)Vy with factorization of T(σ).

8:

Orthogonalize v against V

9:

Expand search space V = [V, v/v]

10:

end while

11:

Store locked values Vl = [Vl, v], µl = [µl, µ].

12:

If dimension V > RESTARTDIMENSION restart: [V, µ] = restart(V, Vl, µ, y, µnext, ynext)

13: end for

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Example 1 A PDE with delay,    ∂u ∂t = ∂2u ∂x2 + a0(x)u + a1(x)u(x, t − τ), u(0, t) = u(π, t) = 0, t ≥ 0 (3) where a0(x) = a0 + α0 sin(x), a1(x) = a1 + α1x(1 − ex−π) + α2x(π − x).

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Example 1 A PDE with delay,    ∂u ∂t = ∂2u ∂x2 + a0(x)u + a1(x)u(x, t − τ), u(0, t) = u(π, t) = 0, t ≥ 0 (3) where a0(x) = a0 + α0 sin(x), a1(x) = a1 + α1x(1 − ex−π) + α2x(π − x). Central difference and uniform stepsize ⇒ ˙ v(t) = (n + 1)2 π2    −2 1 1 ... 1 1 −2    v(t)+    a0(x1) ... a0(xn)    v(t)+    a1(x1) ... a1(xn)    v(t−τ)

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Example 1: CPU consumption

Method n = 100 n = 103 n = 104 n = 106

  • nof. eigs

CPU

  • nof. eigs

CPU

  • nof. eigs

CPU

  • nof. eigs

CPU DDEBIFTOOL 2.03 minrealpart=-20 5 18.0s MEMERR SOD(MS,N = 4) 6 0.2s 6 3.3s MEMERR SOD(MS,N = 8) 10 0.4s 9 6.1s MEMERR IGD(PS,N = 5) 5 0.3s 5 2.6s MEMERR IGD(PS,N = 10) 12 0.4s 12 27.4s MEMERR IGD(Euler,N = 10) 4 0.1s 4 1.2s MEMERR IGD(Euler,N = 100) 5 0.9s 5 12.9s MEMERR SRI(RI) restart=0 12 1.3s 12 0.5s 12 2.8s 12 361.7s SRI(RI) restart=20 12 0.3s 12 0.3s 12 1.9s 12 306.5s SRI(RI) restart=∞ 12 0.2s 12 0.3s 12 1.9s 12 292.5s

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Example 1: CPU consumption

Method n = 100 n = 103 n = 104 n = 106

  • nof. eigs

CPU

  • nof. eigs

CPU

  • nof. eigs

CPU

  • nof. eigs

CPU DDEBIFTOOL 2.03 minrealpart=-20 5 18.0s MEMERR SOD(MS,N = 4) 6 0.2s 6 3.3s MEMERR SOD(MS,N = 8) 10 0.4s 9 6.1s MEMERR IGD(PS,N = 5) 5 0.3s 5 2.6s MEMERR IGD(PS,N = 10) 12 0.4s 12 27.4s MEMERR IGD(Euler,N = 10) 4 0.1s 4 1.2s MEMERR IGD(Euler,N = 100) 5 0.9s 5 12.9s MEMERR SRI(RI) restart=0 12 1.3s 12 0.5s 12 2.8s 12 361.7s SRI(RI) restart=20 12 0.3s 12 0.3s 12 1.9s 12 306.5s SRI(RI) restart=∞ 12 0.2s 12 0.3s 12 1.9s 12 292.5s

In the literature: Large scale DDE: n = 131 [Verheyden, Green, Roose, ’04].

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Example 2 ˙ x(t) = A0x(t) + A1x(t − 1), t > 0 x(t) = ϕ(t), t ∈ [−1, 0] Random matrices: rand(’seed’,0); A0=alpha*sprandn(n,n,beta); A1=alpha*sprandn(n,n,beta);

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Example 2 ˙ x(t) = A0x(t) + A1x(t − 1), t > 0 x(t) = ϕ(t), t ∈ [−1, 0] Random matrices: rand(’seed’,0); A0=alpha*sprandn(n,n,beta); A1=alpha*sprandn(n,n,beta);

Method n = 100 n = 1000 n = 2000 n = 4000 n = 10000 α = 0.7 α = 6 α = 9 α = 10 α = 15 β = 0.1 β = 0.1 β = 0.1 β = 0.05 β = 0.001 Accuracy CPU Accuracy CPU Accuracy CPU Accuracy CPU Accuracy CPU SOD(MS,N = 2) 4.8e-05 0.3s 8.4e-06 19.3s 8.5e-06 109.7s MEMERR SOD(MS,N = 4) 6.1e-07 0.3s 1.0e-07 38.5s 1.0e-07 193.2s MEMERR IGD(PS,N = 3) 6.3e-03 0.3s 5.5e-03 9.8s 7.7e-03 47.8s 4.9e-03 321.5s MEMERR IGD(PS,N = 5) 2.8e-06 0.1s 2.3e-06 11.7s 3.7e-06 105.4s MEMERR IGD(Euler,N = 10) 1.0e-02 0.6s 1.1e-02 131.1s 1.2e-02 399.0s MEMERR IGD(Euler,N = 100) 1.0e-03 5.4s 1.1e-03 680.8s MEMERR SRI(IGD) restart=0 3.8e-04 14.0s 2.6e-05 55.3s 1.0e-04 69.3s 1.1e-04 225.0s 1.1e-03 523.0s SRI(IGD) restart=50 3.8e-04 29.9s 4.5e-05 91.2s 1.1e-04 75.3s 4.6e-05 245.5s 2.0e-04 521.4s SRI(IGD) restart=∞ 3.8e-04 38.1s 4.5e-05 113.5s 1.1e-04 91.0s 2.3e-01 286.3s 2.0e-04 518.7s

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Thanks for your attention!