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Meshfree Approximation with MATLABLecture VI: Nonlinear Problems: Nash Iteration and Implicit
Smoothing
Greg Fasshauer
Department of Applied MathematicsIllinois Institute of Technology
Dolomites Research Week on ApproximationSeptember 8–11, 2008
[email protected] Lecture VI Dolomites 2008
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Outline
[email protected] Lecture VI Dolomites 2008
1 Nonlinear Elliptic PDE
2 Examples of RBFs and MATLAB code
3 Operator Newton Method
4 Smoothing
5 RBF-Collocation
6 Numerical Illustration
7 Conclusions and Future Work
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Nonlinear Elliptic PDE
Generic nonlinear elliptic PDE
Lu = f on Ω ⊂ Rs
Approximate Newton Iteration
uk = uk−1 − Thk (uk−1)F (uk−1), k ≥ 1
F (u) = Lu − f (residual),Thk numerical inversion operator, approximates (F ′)−1
−→ RBF collocation
Nash-Moser Iteration [Nash (1956), Moser (1966),Hörmander (1976), Jerome (1985), F. & Jerome (1999)]
uk = uk−1 − Sθk Thk (uk−1)F (uk−1), k ≥ 1
Sθk additional smoothing for accelerated convergence(separated from numerical inversion)−→ implicit RBF smoothing
[email protected] Lecture VI Dolomites 2008
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Nonlinear Elliptic PDE
Generic nonlinear elliptic PDE
Lu = f on Ω ⊂ Rs
Approximate Newton Iteration
uk = uk−1 − Thk (uk−1)F (uk−1), k ≥ 1
F (u) = Lu − f (residual),
Thk numerical inversion operator, approximates (F ′)−1
−→ RBF collocation
Nash-Moser Iteration [Nash (1956), Moser (1966),Hörmander (1976), Jerome (1985), F. & Jerome (1999)]
uk = uk−1 − Sθk Thk (uk−1)F (uk−1), k ≥ 1
Sθk additional smoothing for accelerated convergence(separated from numerical inversion)−→ implicit RBF smoothing
[email protected] Lecture VI Dolomites 2008
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Nonlinear Elliptic PDE
Generic nonlinear elliptic PDE
Lu = f on Ω ⊂ Rs
Approximate Newton Iteration
uk = uk−1 − Thk (uk−1)F (uk−1), k ≥ 1
F (u) = Lu − f (residual),Thk numerical inversion operator, approximates (F ′)−1
−→ RBF collocation
Nash-Moser Iteration [Nash (1956), Moser (1966),Hörmander (1976), Jerome (1985), F. & Jerome (1999)]
uk = uk−1 − Sθk Thk (uk−1)F (uk−1), k ≥ 1
Sθk additional smoothing for accelerated convergence(separated from numerical inversion)−→ implicit RBF smoothing
[email protected] Lecture VI Dolomites 2008
Page 6
Nonlinear Elliptic PDE
Generic nonlinear elliptic PDE
Lu = f on Ω ⊂ Rs
Approximate Newton Iteration
uk = uk−1 − Thk (uk−1)F (uk−1), k ≥ 1
F (u) = Lu − f (residual),Thk numerical inversion operator, approximates (F ′)−1
−→ RBF collocation
Nash-Moser Iteration [Nash (1956), Moser (1966),Hörmander (1976), Jerome (1985), F. & Jerome (1999)]
uk = uk−1 − Sθk Thk (uk−1)F (uk−1), k ≥ 1
Sθk additional smoothing for accelerated convergence(separated from numerical inversion)−→ implicit RBF smoothing
[email protected] Lecture VI Dolomites 2008
Page 7
Nonlinear Elliptic PDE
Generic nonlinear elliptic PDE
Lu = f on Ω ⊂ Rs
Approximate Newton Iteration
uk = uk−1 − Thk (uk−1)F (uk−1), k ≥ 1
F (u) = Lu − f (residual),Thk numerical inversion operator, approximates (F ′)−1
−→ RBF collocation
Nash-Moser Iteration [Nash (1956), Moser (1966),Hörmander (1976), Jerome (1985), F. & Jerome (1999)]
uk = uk−1 − Sθk Thk (uk−1)F (uk−1), k ≥ 1
Sθk additional smoothing for accelerated convergence(separated from numerical inversion)
−→ implicit RBF smoothing
[email protected] Lecture VI Dolomites 2008
Page 8
Nonlinear Elliptic PDE
Generic nonlinear elliptic PDE
Lu = f on Ω ⊂ Rs
Approximate Newton Iteration
uk = uk−1 − Thk (uk−1)F (uk−1), k ≥ 1
F (u) = Lu − f (residual),Thk numerical inversion operator, approximates (F ′)−1
−→ RBF collocation
Nash-Moser Iteration [Nash (1956), Moser (1966),Hörmander (1976), Jerome (1985), F. & Jerome (1999)]
uk = uk−1 − Sθk Thk (uk−1)F (uk−1), k ≥ 1
Sθk additional smoothing for accelerated convergence(separated from numerical inversion)−→ implicit RBF smoothing
[email protected] Lecture VI Dolomites 2008
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Examples of RBFs and MATLAB code Matérn RBFs
Matérn Radial Basic FunctionsDefinition
Φs,β(x) =Kβ− s
2(‖x‖)‖x‖β−
s2
2β−1Γ(β), β >
s2
Kν : modified Bessel function of the second kind of order ν.
Properties:Φs,β strictly positive definite on Rs for all s < 2β since
Φs,β(ω) =(
1 + ‖ω‖2)−β
> 0
κ(x ,y) = Φs,β(x − y) are reproducing kernels of Sobolev spacesW β
2 (Ω)
Kν > 0 =⇒ Φs,β > 0
[email protected] Lecture VI Dolomites 2008
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Examples of RBFs and MATLAB code Matérn RBFs
Matérn Radial Basic FunctionsDefinition
Φs,β(x) =Kβ− s
2(‖x‖)‖x‖β−
s2
2β−1Γ(β), β >
s2
Kν : modified Bessel function of the second kind of order ν.
Properties:Φs,β strictly positive definite on Rs for all s < 2β since
Φs,β(ω) =(
1 + ‖ω‖2)−β
> 0
κ(x ,y) = Φs,β(x − y) are reproducing kernels of Sobolev spacesW β
2 (Ω)
Kν > 0 =⇒ Φs,β > 0
[email protected] Lecture VI Dolomites 2008
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Examples of RBFs and MATLAB code Matérn RBFs
Matérn Radial Basic FunctionsDefinition
Φs,β(x) =Kβ− s
2(‖x‖)‖x‖β−
s2
2β−1Γ(β), β >
s2
Kν : modified Bessel function of the second kind of order ν.
Properties:Φs,β strictly positive definite on Rs for all s < 2β since
Φs,β(ω) =(
1 + ‖ω‖2)−β
> 0
κ(x ,y) = Φs,β(x − y) are reproducing kernels of Sobolev spacesW β
2 (Ω)
‖f − Pf‖W k2 (Ω) ≤ Chβ−k‖f‖Wβ
2 (Ω), k ≤ β
[Wu & Schaback (1993)]
Kν > 0 =⇒ Φs,β > 0
[email protected] Lecture VI Dolomites 2008
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Examples of RBFs and MATLAB code Matérn RBFs
Matérn Radial Basic FunctionsDefinition
Φs,β(x) =Kβ− s
2(‖x‖)‖x‖β−
s2
2β−1Γ(β), β >
s2
Kν : modified Bessel function of the second kind of order ν.
Properties:Φs,β strictly positive definite on Rs for all s < 2β since
Φs,β(ω) =(
1 + ‖ω‖2)−β
> 0
κ(x ,y) = Φs,β(x − y) are reproducing kernels of Sobolev spacesW β
2 (Ω)
‖f − Pf‖W kq (Ω) ≤ Chβ−k−s(1/2−1/q)+‖f‖Cβ(Ω), k ≤ β
[Narcowich, Ward & Wendland (2005)]
Kν > 0 =⇒ Φs,β > 0
[email protected] Lecture VI Dolomites 2008
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Examples of RBFs and MATLAB code Matérn RBFs
Matérn Radial Basic FunctionsDefinition
Φs,β(x) =Kβ− s
2(‖x‖)‖x‖β−
s2
2β−1Γ(β), β >
s2
Kν : modified Bessel function of the second kind of order ν.
Properties:Φs,β strictly positive definite on Rs for all s < 2β since
Φs,β(ω) =(
1 + ‖ω‖2)−β
> 0
κ(x ,y) = Φs,β(x − y) are reproducing kernels of Sobolev spacesW β
2 (Ω)
‖f − Pf‖W kq (Ω) ≤ Chβ−k−s(1/2−1/q)+‖f‖Cβ(Ω), k ≤ β
[Narcowich, Ward & Wendland (2005)]Kν > 0 =⇒ Φs,β > 0
[email protected] Lecture VI Dolomites 2008
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Examples of RBFs and MATLAB code Matérn RBFs
Examples
Let r = ε‖x‖, t = ‖ω‖ε
β Φ3,β(r)/√
2π ε3Φ3,β(t)
3 (1 + r) e−r
16
(1 + t2)−3
4(3 + 3r + r2) e−r
96
(1 + t2)−4
5(15 + 15r + 6r2 + r3) e−r
768
(1 + t2)−5
6(105 + 105r + 45r2 + 10r3 + r4) e−r
7680
(1 + t2)−6
Table: Matérn functions and their Fourier transforms for s = 3 and variouschoices of β.
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Examples of RBFs and MATLAB code Matérn RBFs
Figure: Matérn functions and Fourier transforms for Φ3,3 (top) and Φ3,6(bottom) centered at the origin in R2 (ε = 10 scaling used).
[email protected] Lecture VI Dolomites 2008
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Examples of RBFs and MATLAB code Matérn RBFs
Implicit Smoothing [F. (1999), Beatson & Bui (2007)]
Crucial property of Matérn RBFs
Φs,β ∗ Φs,α = Φs,α+β , α, β > 0
Therefore with
u(x) =N∑
j=1
cjΦs,β(x − x j)
we get
u ∗ Φs,α =
N∑j=1
cjΦs,β(· − x j)
∗ Φs,α
=N∑
j=1
cjΦs,α+β(· − x j)
=: SαuReturn
[email protected] Lecture VI Dolomites 2008
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Examples of RBFs and MATLAB code Matérn RBFs
Implicit Smoothing [F. (1999), Beatson & Bui (2007)]
Crucial property of Matérn RBFs
Φs,β ∗ Φs,α = Φs,α+β , α, β > 0
Therefore with
u(x) =N∑
j=1
cjΦs,β(x − x j)
we get
u ∗ Φs,α =
N∑j=1
cjΦs,β(· − x j)
∗ Φs,α
=N∑
j=1
cjΦs,α+β(· − x j)
=: SαuReturn
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Examples of RBFs and MATLAB code Matérn RBFs
Noisy and smoothed interpolants
Figure: Solved and evaluated with Φ3,3 (left), evaluated with Φ3,4 (right).
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Examples of RBFs and MATLAB code Matérn RBFs
Noisy and smoothed interpolants
Figure: Solved and evaluated with Φ3,3 (left), evaluated with Φ3,4 (right).
[email protected] Lecture VI Dolomites 2008
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Examples of RBFs and MATLAB code Matérn RBFs
Noisy and smoothed interpolants
Figure: Solved and evaluated with Φ3,3 (left), evaluated with Φ3,3.2 (right).
[email protected] Lecture VI Dolomites 2008
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Operator Newton Method Practical Newton Iteration for Lu = f
Algorithm (Approximate Newton Iteration)
[F. & Jerome (1999), F., Gartland & Jerome (2000), F. (2002), Bernal & Kindelan (2007)]
Create computational “grids” X1 ⊆ · · · ⊆ XK ⊂ Ω, and chooseinitial guess u0
For k = 1,2, . . . ,K1 Solve the linearized problem
Luk−1v = f − Luk−1 on Xk
2 Perform optional smoothing of Newton correction
v ← Sθk v
3 Perform Newton update of k -th iterate
uk = uk−1 + v
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Operator Newton Method Practical Newton Iteration for Lu = f
Algorithm (Nash Iteration)
[F. & Jerome (1999), F., Gartland & Jerome (2000), F. (2002)]
Create computational “grids” X1 ⊆ · · · ⊆ XK ⊂ Ω, and chooseinitial guess u0
For k = 1,2, . . . ,K1 Solve the linearized problem
Luk−1v = f − Luk−1 on Xk
2 Perform optional smoothing of Newton correction
v ← Sθk v
3 Perform Newton update of k -th iterate
uk = uk−1 + v
[email protected] Lecture VI Dolomites 2008
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Smoothing Loss of Derivatives
Why Do We Need Smoothing?
Approximate Newton method based on approximation of (F ′)−1 bynumerical inversion Th, i.e., for u, v in appropriate Banach spaces
‖[F ′(u)Th(u)− I
]v‖ ≤ τ(h)‖v‖
for some continuous monotone increasing function τ(usually τ(h) = O(hp) for some p)
Differentiation reduces the order of approximation, i.e., introducesa loss of derivatives[Jerome (1985)] used Newton-Kantorovich theory to show anappropriate smoothing of the Newton update will yield globalsuperlinear convergence for approximate Newton iteration
[email protected] Lecture VI Dolomites 2008
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Smoothing Loss of Derivatives
Why Do We Need Smoothing?
Approximate Newton method based on approximation of (F ′)−1 bynumerical inversion Th, i.e., for u, v in appropriate Banach spaces
‖[F ′(u)Th(u)− I
]v‖ ≤ τ(h)‖v‖
for some continuous monotone increasing function τ(usually τ(h) = O(hp) for some p)Differentiation reduces the order of approximation, i.e., introducesa loss of derivatives
[Jerome (1985)] used Newton-Kantorovich theory to show anappropriate smoothing of the Newton update will yield globalsuperlinear convergence for approximate Newton iteration
[email protected] Lecture VI Dolomites 2008
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Smoothing Loss of Derivatives
Why Do We Need Smoothing?
Approximate Newton method based on approximation of (F ′)−1 bynumerical inversion Th, i.e., for u, v in appropriate Banach spaces
‖[F ′(u)Th(u)− I
]v‖ ≤ τ(h)‖v‖
for some continuous monotone increasing function τ(usually τ(h) = O(hp) for some p)Differentiation reduces the order of approximation, i.e., introducesa loss of derivatives[Jerome (1985)] used Newton-Kantorovich theory to show anappropriate smoothing of the Newton update will yield globalsuperlinear convergence for approximate Newton iteration
[email protected] Lecture VI Dolomites 2008
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Smoothing Hörmander’s Smoothing
Hörmander’s SmoothingTheorem ([Hörmander (1976), F. & Jerome (1999)])
Let 0 ≤ ` ≤ k and p be integers. In Sobolev spaces W kp (Ω) there exist
smoothings Sθ satisfying1 Semigroup property: ‖Sθu − u‖Lp → 0 as θ →∞2 Bernstein inequality: ‖Sθu‖W k
p≤ Cθk−`‖u‖W `
p
3 Jackson inequality: ‖Sθu − u‖W `p≤ Cθ`−k‖u‖W k
p
Remark: Also true in intermediate Besov spaces Bσp,∞(Ω)
Hörmander defined Sθ by convolution
Sθu = φθ ∗ u, φθ = θsφ(θ·)
New: Use φθ = Φs,α Matérn RBFs
Note: Jackson and Bernstein theorems known for interpolation withMatérn functions, but not for smoothing [Beatson & Bui (2007)]
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Smoothing Hörmander’s Smoothing
Hörmander’s SmoothingTheorem ([Hörmander (1976), F. & Jerome (1999)])
Let 0 ≤ ` ≤ k and p be integers. In Sobolev spaces W kp (Ω) there exist
smoothings Sθ satisfying1 Semigroup property: ‖Sθu − u‖Lp → 0 as θ →∞2 Bernstein inequality: ‖Sθu‖W k
p≤ Cθk−`‖u‖W `
p
3 Jackson inequality: ‖Sθu − u‖W `p≤ Cθ`−k‖u‖W k
p
Remark: Also true in intermediate Besov spaces Bσp,∞(Ω)
Hörmander defined Sθ by convolution
Sθu = φθ ∗ u, φθ = θsφ(θ·)
New: Use φθ = Φs,α Matérn RBFs
Note: Jackson and Bernstein theorems known for interpolation withMatérn functions, but not for smoothing [Beatson & Bui (2007)]
[email protected] Lecture VI Dolomites 2008
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Smoothing Hörmander’s Smoothing
Hörmander’s SmoothingTheorem ([Hörmander (1976), F. & Jerome (1999)])
Let 0 ≤ ` ≤ k and p be integers. In Sobolev spaces W kp (Ω) there exist
smoothings Sθ satisfying1 Semigroup property: ‖Sθu − u‖Lp → 0 as θ →∞2 Bernstein inequality: ‖Sθu‖W k
p≤ Cθk−`‖u‖W `
p
3 Jackson inequality: ‖Sθu − u‖W `p≤ Cθ`−k‖u‖W k
p
Remark: Also true in intermediate Besov spaces Bσp,∞(Ω)
Hörmander defined Sθ by convolution
Sθu = φθ ∗ u, φθ = θsφ(θ·)
New: Use φθ = Φs,α Matérn RBFs
Note: Jackson and Bernstein theorems known for interpolation withMatérn functions, but not for smoothing [Beatson & Bui (2007)]
[email protected] Lecture VI Dolomites 2008
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Smoothing Hörmander’s Smoothing
Hörmander’s SmoothingTheorem ([Hörmander (1976), F. & Jerome (1999)])
Let 0 ≤ ` ≤ k and p be integers. In Sobolev spaces W kp (Ω) there exist
smoothings Sθ satisfying1 Semigroup property: ‖Sθu − u‖Lp → 0 as θ →∞2 Bernstein inequality: ‖Sθu‖W k
p≤ Cθk−`‖u‖W `
p
3 Jackson inequality: ‖Sθu − u‖W `p≤ Cθ`−k‖u‖W k
p
Remark: Also true in intermediate Besov spaces Bσp,∞(Ω)
Hörmander defined Sθ by convolution
Sθu = φθ ∗ u, φθ = θsφ(θ·)
New: Use φθ = Φs,α Matérn RBFs
Note: Jackson and Bernstein theorems known for interpolation withMatérn functions, but not for smoothing [Beatson & Bui (2007)]
[email protected] Lecture VI Dolomites 2008
Page 30
Smoothing Hörmander’s Smoothing
Hörmander’s SmoothingTheorem ([Hörmander (1976), F. & Jerome (1999)])
Let 0 ≤ ` ≤ k and p be integers. In Sobolev spaces W kp (Ω) there exist
smoothings Sθ satisfying1 Semigroup property: ‖Sθu − u‖Lp → 0 as θ →∞2 Bernstein inequality: ‖Sθu‖W k
p≤ Cθk−`‖u‖W `
p
3 Jackson inequality: ‖Sθu − u‖W `p≤ Cθ`−k‖u‖W k
p
Remark: Also true in intermediate Besov spaces Bσp,∞(Ω)
Hörmander defined Sθ by convolution
Sθu = φθ ∗ u, φθ = θsφ(θ·)
New: Use φθ = Φs,α Matérn RBFs
Note: Jackson and Bernstein theorems known for interpolation withMatérn functions, but not for smoothing [Beatson & Bui (2007)]
[email protected] Lecture VI Dolomites 2008
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RBF-Collocation Kansa’s Method
Non-symmetric RBF Collocation
Linear(ized) BVP
Lu(x) = f (x), x ∈ Ω ⊂ Rs
Bu(x) = g(x), x ∈ ∂Ω
Use Ansatz u(x) =N∑
j=1
cjϕ(‖x − x j‖) [Kansa (1990)]
Collocation at x1, . . . ,x I︸ ︷︷ ︸∈Ω
,x I+1, . . . ,xN︸ ︷︷ ︸∈∂Ω
leads to linear system
Ac = y
with
A =
[ALAB
], y =
[fg
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RBF-Collocation Kansa’s Method
Computational Grids for N = 289
Figure: Uniform (left), Chebyshev (center), and Halton (right) collocationpoints.
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Numerical Illustration Nonlinear 2D-BVP
Numerical Illustration
Nonlinear PDE: Lu = f
−µ2∇2u − u + u3 = f , in Ω = (0,1)× (0,1)
u = 0, on ∂Ω
Linearized equation: Luv = f − Lu
−µ2∇2v + (3u2 − 1)v = f + µ2∇2u + u − u3
Computational grids: uniformly spaced, Chebyshev, or Haltonpoints in [0,1]× [0,1]
Use µ = 0.1 for all examples
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Numerical Illustration Nonlinear 2D-BVP
Numerical Illustration (cont.)
RBFs used: Matérn functions
Φs,β(x) =Kβ− s
2(‖εx‖)‖εx‖β−
s2
2β−1Γ(β), β >
s2
Φs,β(0) =Γ(β − s
2)√
2sΓ(β)
with
∇2Φs,β(x) =[(‖εx‖2 + 4(β − s
2)2)
Kβ− s2(‖εx‖)
−2(β − s2
)‖εx‖Kβ− s2 +1(‖εx‖)
] ε2‖εx‖β−s2−2
2β−1Γ(β)
∇2Φs,β(0) =ε2Γ(β − s
2 − 1)√
2sΓ(β)
Fixed shape parameter ε =√
N/2
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Numerical Illustration Nonlinear 2D-BVP
function rbf_definitionMaternglobal rbf Lrbfrbf = @(ep,r,s,b) matern(ep,r,s,b); % Matern functionsLrbf = @(ep,r,s,b) Lmatern(ep,r,s,b); % Laplacian
function rbf = matern(ep,r,s,b)scale = gamma(b-s/2)*2^(-s/2)/gamma(b);rbf = scale*ones(size(r));nz = find(r~=0);rbf(nz) = 1/(2^(b-1)*gamma(b))*besselk(b-s/2,ep*r(nz))...
.*(ep*r(nz)).^(b-s/2);
function Lrbf = Lmatern(ep,r,s,b)scale = -ep^2*gamma(b-s/2-1) / (2^(s/2)*gamma(b));Lrbf = scale*ones(size(r));nz = find(r~=0);Lrbf(nz) = ep^2/(2^(b-1)*gamma(b))*(ep*r(nz)).^(b-s/2-2).*...
(((ep*r(nz)).^2+4*(b-s/2)^2).* besselk(b-s/2,ep*r(nz))...-2*(b-s/2)*(ep*r(nz)).*besselk(b-s/2+1,ep*r(nz)));
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Numerical Illustration Nonlinear 2D-BVP
Exact solution and initial guess
Figure: Solution u (left), initial guess u(x , y) = 16x(1− x)y(1− y) (right).
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Numerical Illustration Newton and Nash Iteration on Single Grid
Newton and Nash Iteration on Single Uniform Grid
Newton NashN RMS-error K RMS-error K ρ
25(41) 1.356070 10−1 7 1.064151 10−1 5 0.32881(113) 2.404571 10−2 9 2.183223 10−2 10 0.527289(353) 4.237178 10−3 9 2.276646 10−3 20 0.953
1089(1217) 8.982388 10−4 9 3.450676 10−4 37 0.9994225(4481) 1.855711 10−4 10 7.780351 10−5 32 0.999
Matérn parameters: s = 3, β = 4, uniform points
Nash smoothing: α = ρθbkwith θ = 1.1435, b = 1.2446
Sample MATLAB calls: Newton_NLPDE(289,’u’,3,4,0),Newton_NLPDE(289,’u’,3,4,0.953)
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Page 38
Numerical Illustration Newton and Nash Iteration on Single Grid
Newton approximations and updates for N = 289
Figure: Approximate solution (left), and updates (right).
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Numerical Illustration Newton and Nash Iteration on Single Grid
Newton approximations and updates for N = 289
Figure: Approximate solution (left), and updates (right).
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Page 40
Numerical Illustration Newton and Nash Iteration on Single Grid
Newton approximations and updates for N = 289
Figure: Approximate solution (left), and updates (right).
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Page 41
Numerical Illustration Newton and Nash Iteration on Single Grid
Newton approximations and updates for N = 289
Figure: Approximate solution (left), and updates (right).
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Numerical Illustration Newton and Nash Iteration on Single Grid
Newton approximations and updates for N = 289
Figure: Approximate solution (left), and updates (right).
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Numerical Illustration Newton and Nash Iteration on Single Grid
Newton approximations and updates for N = 289
Figure: Approximate solution (left), and updates (right).
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Page 44
Numerical Illustration Newton and Nash Iteration on Single Grid
Newton approximations and updates for N = 289
Figure: Approximate solution (left), and updates (right).
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Page 45
Numerical Illustration Newton and Nash Iteration on Single Grid
Newton approximations and updates for N = 289
Figure: Approximate solution (left), and updates (right).
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Page 46
Numerical Illustration Newton and Nash Iteration on Single Grid
Nash approximations and updates for N = 289
Figure: Approximate solution (left), and updates (right).
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Page 47
Numerical Illustration Newton and Nash Iteration on Single Grid
Nash approximations and updates for N = 289
Figure: Approximate solution (left), and updates (right).
[email protected] Lecture VI Dolomites 2008
Page 48
Numerical Illustration Newton and Nash Iteration on Single Grid
Nash approximations and updates for N = 289
Figure: Approximate solution (left), and updates (right).
[email protected] Lecture VI Dolomites 2008
Page 49
Numerical Illustration Newton and Nash Iteration on Single Grid
Nash approximations and updates for N = 289
Figure: Approximate solution (left), and updates (right).
[email protected] Lecture VI Dolomites 2008
Page 50
Numerical Illustration Newton and Nash Iteration on Single Grid
Nash approximations and updates for N = 289
Figure: Approximate solution (left), and updates (right).
[email protected] Lecture VI Dolomites 2008
Page 51
Numerical Illustration Newton and Nash Iteration on Single Grid
Nash approximations and updates for N = 289
Figure: Approximate solution (left), and updates (right).
[email protected] Lecture VI Dolomites 2008
Page 52
Numerical Illustration Newton and Nash Iteration on Single Grid
Nash approximations and updates for N = 289
Figure: Approximate solution (left), and updates (right).
[email protected] Lecture VI Dolomites 2008
Page 53
Numerical Illustration Newton and Nash Iteration on Single Grid
Nash approximations and updates for N = 289
Figure: Approximate solution (left), and updates (right).
[email protected] Lecture VI Dolomites 2008
Page 54
Numerical Illustration Newton and Nash Iteration on Single Grid
Nash approximations and updates for N = 289
Figure: Approximate solution (left), and updates (right).
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Numerical Illustration Newton and Nash Iteration on Single Grid
Nash approximations and updates for N = 289
Figure: Approximate solution (left), and updates (right).
[email protected] Lecture VI Dolomites 2008
Page 56
Numerical Illustration Newton and Nash Iteration on Single Grid
Nash approximations and updates for N = 289
Figure: Approximate solution (left), and updates (right).
[email protected] Lecture VI Dolomites 2008
Page 57
Numerical Illustration Newton and Nash Iteration on Single Grid
Nash approximations and updates for N = 289
Figure: Approximate solution (left), and updates (right).
[email protected] Lecture VI Dolomites 2008
Page 58
Numerical Illustration Newton and Nash Iteration on Single Grid
Nash approximations and updates for N = 289
Figure: Approximate solution (left), and updates (right).
[email protected] Lecture VI Dolomites 2008
Page 59
Numerical Illustration Newton and Nash Iteration on Single Grid
Nash approximations and updates for N = 289
Figure: Approximate solution (left), and updates (right).
[email protected] Lecture VI Dolomites 2008
Page 60
Numerical Illustration Newton and Nash Iteration on Single Grid
Nash approximations and updates for N = 289
Figure: Approximate solution (left), and updates (right).
[email protected] Lecture VI Dolomites 2008
Page 61
Numerical Illustration Newton and Nash Iteration on Single Grid
Nash approximations and updates for N = 289
Figure: Approximate solution (left), and updates (right).
[email protected] Lecture VI Dolomites 2008
Page 62
Numerical Illustration Newton and Nash Iteration on Single Grid
Nash approximations and updates for N = 289
Figure: Approximate solution (left), and updates (right).
[email protected] Lecture VI Dolomites 2008
Page 63
Numerical Illustration Newton and Nash Iteration on Single Grid
Nash approximations and updates for N = 289
Figure: Approximate solution (left), and updates (right).
[email protected] Lecture VI Dolomites 2008
Page 64
Numerical Illustration Newton and Nash Iteration on Single Grid
Nash approximations and updates for N = 289
Figure: Approximate solution (left), and updates (right).
[email protected] Lecture VI Dolomites 2008
Page 65
Numerical Illustration Newton and Nash Iteration on Single Grid
Error drops and smoothing parameters for N = 289
Figure: Drop of RMS error (left), and smoothing parameter α (right).
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Numerical Illustration Newton and Nash Iteration on Single Grid
Newton and Nash Iteration on Single Chebyshev Grid
Newton NashN RMS-error K RMS-error K ρ
25(41) 8.809920 10−2 8 7.825548 10−2 8 0.29981(113) 3.546179 10−3 9 3.277817 10−3 8 0.541289(353) 6.198255 10−4 9 8.420461 10−5 35 0.999
1089(1217) 1.495895 10−4 8 5.470357 10−6 37 0.9994225(4481) 3.734340 10−4 7 7.790757 10−6 35 0.999
Matérn parameters: s = 3, β = 4, Chebyshev points
Nash smoothing: α = ρθbkwith θ = 1.1435, b = 1.2446
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Numerical Illustration Newton and Nash Iteration on Single Grid
Newton and Nash Iteration on Single Halton Grid
Newton NashN RMS-error K RMS-error K ρ
25(41) 3.160062 10−2 7 2.597881 10−2 7 0.38981(113) 9.828342 10−3 9 8.125240 10−3 13 0.791289(353) 2.896087 10−3 9 1.981563 10−3 15 0.953
1089(1217) 9.480208 10−4 9 3.305680 10−4 36 0.9994225(4481) 3.563199 10−4 8 1.330167 10−4 37 0.999
Matérn parameters: s = 3, β = 4, Halton points
Nash smoothing: α = ρθbkwith θ = 1.1435, b = 1.2446
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Numerical Illustration Newton and Nash Iteration on Single Grid
Convergence for Different Collocation Point Sets
Figure: Convergence of Newton and Nash iteration for different choices ofcollocation points.
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Numerical Illustration Newton and Nash Iteration on Single Grid
Newton and Nash Iteration on Single Chebyshev Grid
Newton Nashβ RMS-error K RMS-error K ρ
3 4.022065 10−3 7 9.757401 10−4 38 0.9994 6.198255 10−4 9 8.420461 10−5 35 0.9995 1.803903 10−4 9 9.620937 10−5 8 0.4476 2.715679 10−4 8 1.259029 10−4 8 0.3767 2.279834 10−4 8 1.237608 10−4 9 0.320
Matérn parameters: N = 289, s = 3, Chebyshev points
Nash smoothing: α = ρθbkwith θ = 1.1435, b = 1.2446
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Numerical Illustration Newton and Nash Iteration on Single Grid
Convergence for Different Matérn Functions
Figure: Convergence of Newton and Nash iteration for different Matérnfunctions (β).
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Conclusions and Future Work
Conclusions and Future Work
ConclusionsImplicit smoothing improves convergence of non-symmetric RBFcollocation for nonlinear test caseImplicit smoothing easy and cheap to implement for RBF collocationSmoothing with Matérn kernels recovers some of the “loss ofderivative” of numerical inversion. Can’t really work since saturated.More accurate results than earlier with MQ-RBFsRequired more than 20002 points with earlier FD experiments[F., Gartland & Jerome (2000)] (without smoothing) for sameaccuracy as 1089 points here
Future WorkTry mesh refinement within Newton algorithm via adaptivecollocationFurther investigate use of different Matérn parametersCouple smoothing parameter to current residualsDo smoothing with an approximate smoothing kernelApply similar ideas in RBF-PS framework
[email protected] Lecture VI Dolomites 2008
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Conclusions and Future Work
Conclusions and Future Work
ConclusionsImplicit smoothing improves convergence of non-symmetric RBFcollocation for nonlinear test caseImplicit smoothing easy and cheap to implement for RBF collocationSmoothing with Matérn kernels recovers some of the “loss ofderivative” of numerical inversion. Can’t really work since saturated.More accurate results than earlier with MQ-RBFsRequired more than 20002 points with earlier FD experiments[F., Gartland & Jerome (2000)] (without smoothing) for sameaccuracy as 1089 points here
Future WorkTry mesh refinement within Newton algorithm via adaptivecollocationFurther investigate use of different Matérn parametersCouple smoothing parameter to current residualsDo smoothing with an approximate smoothing kernelApply similar ideas in RBF-PS framework
[email protected] Lecture VI Dolomites 2008
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Appendix References
References I
Buhmann, M. D. (2003).Radial Basis Functions: Theory and Implementations.Cambridge University Press.
Fasshauer, G. E. (2007).Meshfree Approximation Methods with MATLAB.World Scientific Publishers.
Higham, D. J. and Higham, N. J. (2005).MATLAB Guide.SIAM (2nd ed.), Philadelphia.
Wendland, H. (2005).Scattered Data Approximation.Cambridge University Press.
Beatson, R. K. and Bui, H.-Q. (2007).Mollification formulas and implicit smoothing.Adv. in Comput. Math., 27, pp. 125–149.
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Appendix References
References II
Bernal, F. and Kindelan, M. (2007).Meshless solution of isothermal Hele-Shaw flow.In Meshless Methods 2007, A. Ferreira, E. Kansa, G. Fasshauer, and V. Leitão(eds.), INGENI Edições Porto, pp. 41–49.
Fasshauer, G. E. (1999).On smoothing for multilevel approximation with radial basis functions.In Approximation Theory XI, Vol.II: Computational Aspects, C. K. Chui andL. L. Schumaker (eds.), Vanderbilt University Press, pp. 55–62.
Fasshauer, G. E. (2002).Newton iteration with multiquadrics for the solution of nonlinear PDEs.Comput. Math. Applic. 43, pp. 423–438.
Fasshauer, G. E., Gartland, E. C. and Jerome, J. W. (2000).Newton iteration for partial differential equations and the approximation of theidentity.Numerical Algorithms 25, pp. 181–195.
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Appendix References
References III
Fasshauer, G. E. and Jerome, J. W. (1999).Multistep approximation algorithms: Improved convergence rates throughpostconditioning with smoothing kernels.Adv. in Comput. Math. 10, pp. 1–27.
Hörmander, L. (1976).The boundary problems of physical geodesy.Arch. Ration. Mech. Anal. 62, pp. 1–52.
Jerome, J. W. (1985).An adaptive Newton algorithm based on numerical inversion: regularization aspostconditioner.Numer. Math. 47, pp. 123–138.
Kansa, E. J. (1990).Multiquadrics — A scattered data approximation scheme with applications tocomputational fluid-dynamics — II: Solutions to parabolic, hyperbolic and ellipticpartial differential equations.Comput. Math. Applic. 19, pp. 147–161.
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Appendix References
References IV
Moser, J. (1966).A rapidly convergent iteration method and nonlinear partial differential equationsI.Ann. Scoula Norm. Pisa XX, pp. 265–315.
Narcowich, F. J., Ward, J. D. and Wendland, H. (2005).Sobolev bounds on functions with scattered zeros, with applications to radialbasis function surface fitting.Math. Comp. 74, pp. 743–763.
Narcowich, F. J., Ward, J. D. and Wendland, H. (2006).Sobolev error estimates and a Bernstein inequality for scattered datainterpolation via radial basis functions.Constr. Approx. 24, pp. 175–186.
Nash, J. (1956).The imbedding problem for Riemannian manifolds.Ann. Math. 63, pp. 20–63.
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Appendix References
References V
Schaback, R. and Wendland, H. (2002).Inverse and saturation theorems for radial basis function interpolation.Math. Comp. 71, pp. 669–681.
Wu, Z. and Schaback, R. (1993).Local error estimates for radial basis function interpolation of scattered data.IMA J. Numer. Anal. 13, pp. 13–27.
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