Draft Computation of frequency responses of PDEs in Chebfun - - PowerPoint PPT Presentation

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Draft Computation of frequency responses of PDEs in Chebfun - - PowerPoint PPT Presentation

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Draft Computation of frequency responses of PDEs in Chebfun Mihailo Jovanovi c www.umn.edu/ mihailo joint work with: Binh K. Lieu 2012 SIAM Annual Meeting Draft M. J OVANOVI C , UMN 1 Example: heat equation Distributed input

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Computation of frequency responses of PDEs in Chebfun Mihailo Jovanovi´ c www.umn.edu/∼mihailo joint work with: Binh K. Lieu 2012 SIAM Annual Meeting
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  • M. JOVANOVI ´
C, UMN 1 Example: heat equation
  • Distributed input and output fields
ϕt(y, t) = ϕyy(y, t) + d(y, t) ϕ(y, 0) = ϕ(±1, t) = ⋆ Harmonic forcing d(y, t) = d(y, ω) ejωt steady-state response − − − − − − − − − − − − − − − − − − − → ϕ(y, t) = ϕ(y, ω) ejωt ⋆ Frequency response operator ϕ(y, ω) = [ T (ω) d( · , ω) ] (y) =
  • (jωI − ∂yy)−1 d( · , ω)
  • (y)
= 1 −1 Tker(y, η; ω) d(η, ω) dη
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  • M. JOVANOVI ´
C, UMN 2 Two point boundary value realizations of T (ω)
  • Input-output differential equation
T (ω) : ϕ′′(y, ω) − jω ϕ(y, ω) = −d(y, ω) ϕ(±1, ω) =
  • Spatial state-space realization
T (ω) :                    x′ 1(y, ω) x′ 2(y, ω)
  • =
1 jω x1(y, ω) x2(y, ω)
  • +
  • −1
  • d(y, ω)
ϕ(y, ω) = 1 0 x1(y, ω) x2(y, ω)
  • 0 =
  • 1
x1(−1, ω) x2(−1, ω)
  • +
  • 1
x1(1, ω) x2(1, ω)
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  • M. JOVANOVI ´
C, UMN 3 Frequency response operator
  • Evolution equation
[E φt(·, t)] (y) = [F φ(·, t)] (y) + [G d(·, t)] (y), y ∈ [a, b] ϕ(y, t) = [H φ(·, t)] (y), t ∈ [0, ∞) ⋆ Spatial differential operators F = [Fij] = nij
  • k = 0
fij,k(y) dk dyk ⋆ Frequency response operator T = H (j ω E − F)−1 G
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  • M. JOVANOVI ´
C, UMN 4 Example: channel flow
  • Streamwise-constant fluctuations
  • ∆
I φ1t φ2t
  • =
  • (1/Re) ∆2
F21 (1/Re) ∆ φ1 φ2
  • Laplacian:
∆ = ∂yy − k2 z ”Square of Laplacian”: ∆2 = ∂yyyy − 2 k2 z ∂yy + k4 z Coupling: F21 = − jkz U ′(y)
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  • M. JOVANOVI ´
C, UMN 5 Singular value decomposition
  • Schmidt decomposition of a compact operator T (ω): Hin −
→ Hout ϕ(y, ω) = [T (ω) d(·, ω)] (y) = ∞
  • n = 1
σn(ω) un(y, ω) vn, d
  • Left and right singular functions
[T (ω) T ⋆(ω) un(·, ω)] (y) = σ2 n(ω) un(y, ω) [T ⋆(ω) T (ω) vn(·, ω)] (y) = σ2 n(ω) vn(y, ω) {un} orthonormal basis of Hout {vn} orthonormal basis of Hin
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  • M. JOVANOVI ´
C, UMN 6
  • Right singular functions
⋆ identify input directions with simple responses σ1(ω) ≥ σ2(ω) ≥ · · · > 0 ϕ(ω) = T (ω) d(ω) = ∞
  • n = 1
σn(ω) un(ω) vn(ω), d(ω)   d(ω) = vm(ω) ϕ(ω) = σm(ω) um(ω) σ1(ω): the largest amplification at any frequency
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  • M. JOVANOVI ´
C, UMN 7 Worst case amplification
  • H∞ norm: an induced L2 gain (of a system)
G = sup output energy input energy = sup ∞ ϕ(t), ϕ(t) dt ∞ d(t), d(t) dt = sup ω σ2 1 (T (ω)) σ2 1 (T (ω))
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  • M. JOVANOVI ´
C, UMN 8 Robustness interpretation ✲ d Nominal Dynamics s ✲ ϕ ✛ Γ modeling uncertainty (can be nonlinear or time-varying) small-gain theorem: stability for all Γ with max ω σ2 1 (Γ(ω)) ≤ γ2 ⇔ γ2 < 1 G

LARGE

worst case amplification

⇒

small stability margins closely related to pseudospectra of linear operators
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  • M. JOVANOVI ´
C, UMN 9 Pseudo-spectral methods
  • MATLAB Differentiation Matrix Suite
T (ω) =
  • jωI − D(2)−1
≈ N N
  • Advantages
⋆ superior accuracy compared to finite difference methods ⋆ ease-to-use MATLAB codes
  • Disadvantages
⋆ ill-conditioning of high-order differentiation matrices ⋆ implementation of boundary conditions may be non-trivial Weideman & Reddy, ACM. TOMS. ’00
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  • M. JOVANOVI ´
C, UMN 10 Alternative method
  • 1. Frequency response operator: two-point boundary value problem
  • 2. Integral form of differential equations
  • 3. State-of-the-art automatic spectral collocation techniques
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  • M. JOVANOVI ´
C, UMN 11 Advantages of Chebfun
  • Superior accuracy compared to currently available schemes
  • Avoids ill-conditioning of high-order differentiation matrices
  • Incorporates a wide range of boundary conditions
  • Easy-to-use MATLAB codes
Lieu & Jovanovi´ c “Computation of frequency responses of linear time-invariant PDEs on a compact interval”, submitted to J. Comput. Phys., 2011 Also arXiv:1112.0579v1
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  • M. JOVANOVI ´
C, UMN 12 2D inertialess flow of viscoelastic fluids 0 = − ∇p + (1 − β) ∇ · τ + β ∇2v + d 0 = ∇ · v τ t = ∇v + (∇v)T − τ + We (τ · ∇¯ v + ¯ τ · ∇v + (¯ τ · ∇v)T + (τ · ∇¯ v)T − v · ∇¯ τ − ¯ v · ∇τ) We = polymer relaxation time characteristic flow time
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  • M. JOVANOVI ´
C, UMN 13 Largest singular value of T σ1 (T ) We β = 0.5 kx = 1 ω = N = 50 (×) N = 100 (◦) N = 200 (+)
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  • M. JOVANOVI ´
C, UMN 14 σ1 (T ) We β = 0.5 kx = 1 ω = N = 50 (×) N = 100 (◦) N = 200 (+)
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  • M. JOVANOVI ´
C, UMN 15 Input-output differential equations
  • Frequency response operator
T (ω) :        [A0 φ] (y) = [B0 d] (y) ϕ(y) = [C0 φ] (y) 0 = N0 φ(y)
  • Adjoint of the frequency response operator
T ⋆(ω) :        [A⋆ 0 ψ] (y) = [C⋆ 0 f] (y) g(y) = [B⋆ 0 ψ] (y) 0 = N ⋆ 0 ψ(y)
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  • M. JOVANOVI ´
C, UMN 16 Composition of T with T ⋆
  • Cascade connection
T T ⋆ :      [A ξ] (y) = [B f] (y) ϕ(y) = [C ξ] (y) 0 = N ξ(y) ⋆ Do e-value decomposition of T T ⋆ and T ⋆ T in Chebfun [T (ω) T ⋆(ω) un(·, ω)] (y) = σ2 n(ω) un(y, ω) [T ⋆(ω) T (ω) vn(·, ω)] (y) = σ2 n(ω) vn(y, ω)
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  • M. JOVANOVI ´
C, UMN 17 Example: 1D heat equation φt(y, t) = φyy(y, t) + d(y, t), y ∈ [−1, 1] φ(±1, t) = 0
  • Frequency response and adjoint operators
T (ω) :      φ′′(y) − j ω φ(y) = −d(y) 1
  • E−1 +
1
  • E1
  • φ(y) =
  • T ⋆(ω) :
           ψ′′(y) + j ω ψ(y) = f(y) g(y) = −ψ(y) 1
  • E−1 +
1
  • E1
  • ψ(y) =
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  • M. JOVANOVI ´
C, UMN 18 Integral form of a differential equation Driscoll, J. Comput. Phys., 2010
  • 1D diffusion equation: differential form
  • D(2) − jωI
  • φ(y) = − d(y)
  • 1
  • E−1 +
  • 1
  • E1
  • φ(y) =
  • Auxiliary variable:
ν(y) =
  • D(2) φ
  • (y)
Integrate twice φ′(y) = y −1 ν(η1) dη1 + k1 =
  • J(1) ν
  • (y) + k1
φ(y) = y −1 η2 −1 ν(η1) dη1
  • dη2 +
1 (y + 1) k2 k1
  • =
  • J(2) ν
  • (y) + K(2) k
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  • M. JOVANOVI ´
C, UMN 19
  • 1D diffusion equation: integral form
  • I − jωJ(2)
ν(y) − jω K(2) k = − d(y)
  • 1
1 2 k2 k1
  • = −
  • 1
  • E−1 +
  • 1
  • E1
  • J(2)ν(y)
Eliminate k from the equations to obtain
  • I − jωJ(2) + 1
2 jω (y + 1) E1J(2)
  • ν(y) = −d(y)
☞ More suitable for numerical computations than differential form integral operators and point evaluation functionals are well-conditioned
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  • M. JOVANOVI ´
C, UMN 20 Online resources
  • Chebfun
http://www2.maths.ox.ac.uk/chebfun/ Google: “chebfun”
  • Computing frequency responses of PDEs
http://www.umn.edu/∼mihailo/software/chebfun-svd/ Google: “frequency responses pde”
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  • M. JOVANOVI ´
C, UMN 21 Summary
  • method for computing frequency responses of PDEs
  • easy-to-use mini-toolbox in MATLAB
⋆ enabling tool: Chebfun
  • two major advantages over currently available schemes
⋆ avoids ill-conditioning of high-order differentiation matrices ⋆ easy implementation of boundary conditions
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  • M. JOVANOVI ´
C, UMN 22 Acknowledgments TEAM: Binh Lieu SUPPORT: NSF CAREER Award CMMI-06-44793 SOFTWARE: http://www.umn.edu/∼mihailo/software/chebfun-svd/