numericalsolutionofsurfacepdes with radialbasisfunctionsandriy sokolov institut fur angewandte...
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![Page 1: NumericalsolutionofsurfacePDEs with RadialBasisFunctionsAndriy Sokolov Institut fur Angewandte Mathematik (LS3) TU Dortmund andriy.sokolov@math.tu-dortmund.de TU Dortmund June 1, 2017](https://reader036.vdocument.in/reader036/viewer/2022071302/60adbc553881f004ef126645/html5/thumbnails/1.jpg)
Numerical solution of surface PDEs
with
Radial Basis Functions
Andriy Sokolov
Institut fur Angewandte Mathematik (LS3)TU Dortmund
TU DortmundJune 1, 2017
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Outline
1 Radial Basis Functions (RBFs)
2 FD-RBF approximation of differential operators(Finite Difference Radial Basis Functions)
3 FD-RBF level set method for surface PDEs
4 Outlook
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Outline
1 Radial Basis Functions (RBFs)
2 FD-RBF approximation of differential operators(Finite Difference Radial Basis Functions)
3 FD-RBF level set method for surface PDEs
4 Outlook
![Page 4: NumericalsolutionofsurfacePDEs with RadialBasisFunctionsAndriy Sokolov Institut fur Angewandte Mathematik (LS3) TU Dortmund andriy.sokolov@math.tu-dortmund.de TU Dortmund June 1, 2017](https://reader036.vdocument.in/reader036/viewer/2022071302/60adbc553881f004ef126645/html5/thumbnails/4.jpg)
Outline
1 Radial Basis Functions (RBFs)
2 FD-RBF approximation of differential operators(Finite Difference Radial Basis Functions)
3 FD-RBF level set method for surface PDEs
4 Outlook
![Page 5: NumericalsolutionofsurfacePDEs with RadialBasisFunctionsAndriy Sokolov Institut fur Angewandte Mathematik (LS3) TU Dortmund andriy.sokolov@math.tu-dortmund.de TU Dortmund June 1, 2017](https://reader036.vdocument.in/reader036/viewer/2022071302/60adbc553881f004ef126645/html5/thumbnails/5.jpg)
Outline
1 Radial Basis Functions (RBFs)
2 FD-RBF approximation of differential operators(Finite Difference Radial Basis Functions)
3 FD-RBF level set method for surface PDEs
4 Outlook
![Page 6: NumericalsolutionofsurfacePDEs with RadialBasisFunctionsAndriy Sokolov Institut fur Angewandte Mathematik (LS3) TU Dortmund andriy.sokolov@math.tu-dortmund.de TU Dortmund June 1, 2017](https://reader036.vdocument.in/reader036/viewer/2022071302/60adbc553881f004ef126645/html5/thumbnails/6.jpg)
Problem
Figure: Scattered data interpolation
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Idea 1: polynomial interpolation
Given points a < x1,x2, . . . ,xN < b
and data values f1, f2, . . . , fNconstruct a polynomial p of degree N − 1 s.t. p(xi) = fi.
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Idea 1: polynomial interpolation
Given points a < x1,x2, . . . ,xN < b
and data values f1, f2, . . . , fNconstruct a polynomial p of degree N − 1 s.t. p(xi) = fi.
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Idea 1: polynomial interpolation
Given points a < x1,x2, . . . ,xN < b
and data values f1, f2, . . . , fNconstruct a polynomial p of degree N − 1 s.t. p(xi) = fi.
Figure: Runge phenomenon
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Idea 2: splines
Cubic splines:
S3(X) = s ∈ C2[a, b] : s|[xi,xi+1]
∈ P3(R), 0 ≤ i ≤ N.
dimS3(X) = N + 4, but N interpolation conditions s(xi) = fi.
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Idea 2: splines
Cubic splines:
S3(X) = s ∈ C2[a, b] : s|[xi,xi+1]
∈ P3(R), 0 ≤ i ≤ N.
dimS3(X) = N + 4, but N interpolation conditions s(xi) = fi.
Natural splines:
N3(X) = s ∈ S3(X) : s|[a,x1], s|[xN,b]
∈ P1(R).
Here, the initial interpolation problem has a unique solution
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Idea 2: splines, 2D case
Figure: Stanford Bunny
In the subdivision of a region by triangles, the dimension of the splinespace is in general unknown
Even if great progresses have been made in the 2D setting, the methodis not suited for general dimensions
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Idea 2: splines, 2D case
Figure: Stanford Bunny
In the subdivision of a region by triangles, the dimension of the splinespace is in general unknown
Even if great progresses have been made in the 2D setting, the methodis not suited for general dimensions
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Idea 2: splines, 2D case
Figure: Stanford Bunny
In the subdivision of a region by triangles, the dimension of the splinespace is in general unknown
Even if great progresses have been made in the 2D setting, the methodis not suited for general dimensions
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Interpolation with RBFs
Idea:
Let us search interpolant as a linear combination of basis functions ϕjNj=1:
s(x) =
N∑
j=1
λjϕj(x).
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Interpolation with RBFs
Idea:
Let us search interpolant as a linear combination of basis functions ϕjNj=1:
s(x) =
N∑
j=1
λjϕj(x).
Applying interpolation conditions
s(xi) = fi, i = 1, . . . , N
we obtain a system of linear equations
ϕ1(x1) . . . ϕN (x1)... · · ·
...ϕ1(xN) . . . ϕN (xN)
︸ ︷︷ ︸
=A
λ1
. . .
λN
=
f1. . .
fN
.
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Interpolation with RBFs
Idea:
Let us search interpolant as a linear combination of basis functions ϕjNj=1:
s(x) =
N∑
j=1
λjϕj(x).
Applying interpolation conditions
s(xi) = fi, i = 1, . . . , N
we obtain a system of linear equations
ϕ1(x1) . . . ϕN (x1)... · · ·
...ϕ1(xN) . . . ϕN (xN)
︸ ︷︷ ︸
=A
λ1
. . .
λN
=
f1. . .
fN
.
Is this SLE solvable?
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Interpolation with RBFs
The answer: in general NO
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Interpolation with RBFs
The answer: in general NO
Definition
Let the finite-dimensional linear function space Φ ⊂ C(Ω), have a basisφ1, φ2, . . . , φN. Then Φ is a Haar space on Ω if
det(A) 6= 0
for any set of distinct points x1, . . . , xN ∈ Ω. Here, A = φj(xi)ij=1,N .
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Interpolation with RBFs
The answer: in general NO
Definition
Let the finite-dimensional linear function space Φ ⊂ C(Ω), have a basisφ1, φ2, . . . , φN. Then Φ is a Haar space on Ω if
det(A) 6= 0
for any set of distinct points x1, . . . , xN ∈ Ω. Here, A = φj(xi)ij=1,N .
Theorem (Mairhuber-Curtis)
Let Ω ⊆ Rd, d ≥ 2 contains an interior point, thenthere exist no Haar space
of continuous functions except for the 1-dimensional case.
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Interpolation with RBFs
The answer: in general NO
Definition
Let the finite-dimensional linear function space Φ ⊂ C(Ω), have a basisφ1, φ2, . . . , φN. Then Φ is a Haar space on Ω if
det(A) 6= 0
for any set of distinct points x1, . . . , xN ∈ Ω. Here, A = φj(xi)ij=1,N .
Theorem (Mairhuber-Curtis)
Let Ω ⊆ Rd, d ≥ 2 contains an interior point, thenthere exist no Haar space
of continuous functions except for the 1-dimensional case.
Let φi be dependent on xi!
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Interpolation with RBFs
Approach
shift the function ϕj(x) to the point xj
ϕj(‖x− xj‖)
Choose a radially symmetric function
ϕ(r) = ϕ(‖x‖)
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Interpolation with RBFs
Approach
shift the function ϕj(x) to the point xj
ϕj(‖x− xj‖)
Choose a radially symmetric function
ϕ(r) = ϕ(‖x‖)
Remark 1: Every natural spline s has the representation
s(x) =
N∑
j=1
ajϕ(‖x− xj‖) + p(x), x ∈ R
where ϕ(r) = r3, r ≥ 0 and p ∈ P1(R).
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Interpolation with RBFs
Radial Basis Functions (RBFs)
s(x) =
N∑
j=1
λjϕ(‖x− xj‖),
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Interpolation with RBFs
Radial Basis Functions (RBFs)
s(x) =
N∑
j=1
λjϕ(‖x− xj‖),
where λj ’s are calculated to satisfy s(xi) = fi:
ϕ(‖x1 − x1‖) ϕ(‖x1 − x2‖) . . . ϕ(‖x1 − xN‖)ϕ(‖x2 − x1‖) ϕ(‖x2 − x2‖) . . . ϕ(‖x2 − xN‖)
...... · · ·
...ϕ(‖xN − x1‖) ϕ(‖xN − x2‖) . . . ϕ(‖xN − xN‖)
︸ ︷︷ ︸
=A distance matrix
λ1
λ2
. . .
λN
=
f1f2. . .
fN
.
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Interpolation with RBFs
Radial Basis Functions (RBFs)
s(x) =
n∑
j=1
λjϕ(‖x− xj‖),
dimension independent,
sufficiently smooth,
easy to find (multiple-) derivatives,
the corresponding SLE is well-posed, i.e. det(A) 6= 0,
meshfree,
other advantages to discuss later.
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Interpolation with RBFs
Linear vs Multiquadratic RBF:
s(x) =
N∑
j=1
λj‖x− xj‖ s(x) =
N∑
j=1
λj
√
c2 + ‖x− xj‖2
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Interpolation with RBFs
Types of RBFs:
Infinitely smooth RBFs ϕ(r) (r ≥ 0)
Gaussian (GA) e−(εr)2
Inverse quadratic (IQ)1
1 + (εr)2
Inverse multiquadratic (IMQ)1
√1 + (εr)2
Multiquadratic (MQ)√
1 + (εr)2
Piecewise smooth RBFs
Linear r
Cubic r3
Thin plate spline (TPS) r2 log r
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Interpolation with RBFs
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Interpolation with RBFs
Solvability of the SLE :
ϕ(‖x1 − x1‖) ϕ(‖x1 − x2‖) . . . ϕ(‖x1 − xN‖)ϕ(‖x2 − x1‖) ϕ(‖x2 − x2‖) . . . ϕ(‖x2 − xN‖)
...... · · ·
...ϕ(‖xN − x1‖) ϕ(‖xN − x2‖) . . . ϕ(‖xN − xN‖)
︸ ︷︷ ︸
=A distance matrix
λ1
λ2
. . .
λN
=
f1f2. . .
fN
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Interpolation with RBFs
Solvability of the SLE :
ϕε(‖x1 − x1‖) ϕε(‖x1 − x2‖) . . . ϕε(‖x1 − xN‖)ϕε(‖x2 − x1‖) ϕε(‖x2 − x2‖) . . . ϕε(‖x2 − xN‖)
...... · · ·
...ϕε(‖xN − x1‖) ϕε(‖xN − x2‖) . . . ϕε(‖xN − xN‖)
︸ ︷︷ ︸
=A(ε) distance matrix
λ1
λ2
. . .
λN
=
f1f2. . .
fN
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Interpolation with RBFs
Solvability of the SLE :
ϕε(‖x1 − x1‖) ϕε(‖x1 − x2‖) . . . ϕε(‖x1 − xN‖)ϕε(‖x2 − x1‖) ϕε(‖x2 − x2‖) . . . ϕε(‖x2 − xN‖)
...... · · ·
...ϕε(‖xN − x1‖) ϕε(‖xN − x2‖) . . . ϕε(‖xN − xN‖)
︸ ︷︷ ︸
=A(ε) distance matrix
λ1
λ2
. . .
λN
=
f1f2. . .
fN
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Interpolation with RBFs
Solvability of the SLE :
ϕε(‖x− x1‖)ϕε(‖x− x2‖)
...
...ϕε(‖x− xN‖)
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Interpolation with RBFs
Solvability of the SLE :
ϕε(‖x− x1‖)ϕε(‖x− x2‖)
...
...ϕε(‖x− xN‖)
=
· · · · · ·· · · · · ·· · C · · ·· · · · · ·· · · · · ·· · · · · ·
ε0
ε2
. . .
ε4
. . .
Y 00 (x)
Y −11 (x).........
,
where C is a matrix with entries of size O(1).
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Interpolation with RBFs
Solvability of the SLE :
ϕε(‖x− x1‖)ϕε(‖x− x2‖)
...
...ϕε(‖x− xN‖)
=
· · · · · ·· · · · · ·· · C · · ·· · · · · ·· · · · · ·· · · · · ·
ε0
ε2
. . .
ε4
. . .
Y 00 (x)
Y −11 (x).........
,
where C is a matrix with entries of size O(1).
Remedy: special QR-decomposition ⇒ work of Markus Verkely
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Interpolation with RBFs
Error estimation for the RBF interpolation:(Gaussian and multiquadratic-like functions)
if |f (l)| ≤ l!M l ⇒ ‖f − sf,X‖L∞(Ω) ≤ e−c/hX,Ω |f |N (Ω),
if |f (l)| ≤ Ml ⇒ ‖f − sf,X‖L∞(Ω) ≤ e
c log hX,Ω/hX,Ω |f |N (Ω),
where
hX,Ω = supx∈Ω
minxj∈X
‖x− xj‖2
is the fill distance.
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Interpolation with RBFs
0 0.1 0.2 0.3 0.4 0.50
0.05
0.1
0.15
0.2
0.25
exp(log(x)./x)
x 2
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Outline
1 Radial Basis Functions (RBFs)
2 FD-RBF approximation of differential operators(Finite Difference Radial Basis Functions)
3 FD-RBF level set method for surface PDEs
4 Outlook
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Finite Difference Radial Basis Function (FD-RBF)
Finite difference (notation u(xi) = ui):
Forward: (ux)i =ui+1 − ui
h+RF (h, ·)
Backward: (ux)i =ui − ui−1
h+RB(h, ·)
Central: (ux)i =ui+1 − ui−1
2h+RC(h, ·)
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Finite Difference Radial Basis Function (FD-RBF)
Finite difference (notation u(xi) = ui):
Forward: (ux)i =1
hui+1 +
−1
hui +RF (h, ·)
Backward: (ux)i =1
hui +
−1
hui−1 +RB(h, ·)
Backward: (ux)i =1
2hui+1 +
−1
2hui−1 +RC(h, ·)
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Finite Difference Radial Basis Function (FD-RBF)
Finite difference (notation u(xi) = ui):
Forward: (ux)i = ωi+1ui+1 + ωiui +RF (h, ·)
Backward: (ux)i = ωiui + ωi−1ui−1 +RB(h, ·)
Central: (ux)i = ωi+1ui+1 + ωi−1ui−1 +RC(h, ·)
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Finite Difference Radial Basis Function (FD-RBF)
Finite difference (notation u(xi) = ui):
Forward: (ux)i =∑
i∈ΞF
ωiui +RF (h, ·)
Backward: (ux)i =∑
i∈ΞB
ωiui +RB(h, ·)
Central: (ux)i =∑
i∈ΞC
ωiui +RC(h, ·)
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Finite Difference Radial Basis Function (FD-RBF)
Approximation of the Laplace operator:
∆u = f, Ω ⊂ Rd
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Finite Difference Radial Basis Function (FD-RBF)
Approximation of the Laplace operator:
∆u = f, Ω ⊂ Rd
u(x) ≈ s(x) =
N∑
i=1
ciϕ(‖x− xi‖)
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Finite Difference Radial Basis Function (FD-RBF)
Approximation of the Laplace operator:
∆u = f, Ω ⊂ Rd
∆s(ζ) ≈∑
ξ∈Ξζ
ωξu(ξ), Ξζ = ζ, ξ1, ξ2, . . . , ξK.
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Finite Difference Radial Basis Function (FD-RBF)
Approximation of the Laplace operator:
∆u = f, Ω ⊂ Rd
∆s(ζ) =
K∑
j=0
ωju(ξj),
∆N∑
i=0
ciϕ(‖ζ − ξi‖) =K∑
j=0
ωju(ξj),
N∑
i=0
ci∆ϕ(‖ζ − ξi‖) =N∑
i=0
ci
K∑
j=0
ωjϕ(‖ξj − ξi‖)
∆ϕ(‖ζ − ξi‖)︸ ︷︷ ︸
rhs
=K∑
j=0
ωj︸︷︷︸
ω
ϕ(‖ξj − ξi‖)︸ ︷︷ ︸
Φ
Φω = rhs(∆)
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Finite Difference Radial Basis Function (FD-RBF)
Approximation of a differential operator:
Lu = f, Ω ⊂ Rd
∆s(ζ) =
K∑
j=0
ωju(ξj),
LN∑
i=0
ciϕ(‖ζ − ξi‖) =K∑
j=0
ωju(ξj),
N∑
i=0
ciLϕ(‖ζ − ξi‖) =N∑
i=0
ci
K∑
j=0
ωjϕ(‖ξj − ξi‖)
Lϕ(‖ζ − ξi‖)︸ ︷︷ ︸
rhs
=K∑
j=0
ωj︸︷︷︸
ω
ϕ(‖ξj − ξi‖)︸ ︷︷ ︸
Φ
Φω = rhs(L)
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Finite Difference Radial Basis Function (FD-RBF)
Derivative(s) of the Gaussian RBF:
ϕ(‖x− ξ‖︸ ︷︷ ︸
r(x)
) = ϕ(r(x)) = e−ε2r2(x)
,
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Finite Difference Radial Basis Function (FD-RBF)
Derivative(s) of the Gaussian RBF:
ϕ(‖x− ξ‖︸ ︷︷ ︸
r(x)
) = ϕ(r(x)) = e−ε2r2(x)
,
s(x) =r2(x)
2⇒ ϕ(s(x)) = e
−2 ε2s(x),
ϕxi(s(x)) = ϕs
︸︷︷︸
−2ε2ϕ
sxi︸︷︷︸
(xi−ξi)
= −2ε2ϕ [xi − ξi] ,
ϕxixi(s(x)) = ϕss sxi
sxi︸ ︷︷ ︸
(xi−ξi)2
+ϕs sxixi︸ ︷︷ ︸
1
ϕxixj(s(x)) = ϕss(xi − ξi)(xj − ξj), since sxixj
= 0.
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Finite Difference Radial Basis Function (FD-RBF)
Consistency + Stability ⇒ convergence
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Finite Difference Radial Basis Function (FD-RBF)
Consistency + Stability ⇒ convergence
Consistency: ”Error Bounds for Kernel-Based Numerical Differentiation” by O.Davydov and R. Schaback, Numer. Math., 132 (2016), 243–269.
Stability: ”Error Analysis of Nodal Meshless Methods” by R. Schaback,Numerical Analysis, arXiv:1612.07550 (2016) ... + ???
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Finite Difference Radial Basis Function (FD-RBF)
Poisson equation:
−∇ · (∇u(x)) = f(x) in Ω = [0, 1]2,
uanalyt = sin(πx) sin(πy).
![Page 53: NumericalsolutionofsurfacePDEs with RadialBasisFunctionsAndriy Sokolov Institut fur Angewandte Mathematik (LS3) TU Dortmund andriy.sokolov@math.tu-dortmund.de TU Dortmund June 1, 2017](https://reader036.vdocument.in/reader036/viewer/2022071302/60adbc553881f004ef126645/html5/thumbnails/53.jpg)
Finite Difference Radial Basis Function (FD-RBF)
Poisson equation:
−∇ · (∇u(x)) = f(x) in Ω = [0, 1]2,
uanalyt = sin(πx) sin(πy).
01
0.2
0.4
1
0.6
0.8
0.8
0.5 0.6
1
0.40.2
0 0
Figure: Numericalsolution, h=1/20.
El2 EOC(l2) Emax EOC(max)
h=1/5 13019 – 22401 –
h=1/10 330 4,967 726 4,947
h=1/20 89 1,891 188 1,949
h=1/40 23 1,952 47 2,000
h=1/80 5,87 1,970 11,89 1,983
Table: 5-points stencil (including the main node).
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Finite Difference Radial Basis Function (FD-RBF)
Anisotropic diffusion:
−∇ (A∇u(x)) = f(x) in Ω = [0, 1]2,
where
A =
(
cosφ − sinφsinφ cosφ
)(
1 00 ε
)(
cosφ sinφ− sinφ cos φ
)
.
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Finite Difference Radial Basis Function (FD-RBF)
Anisotropic diffusion:
−∇ (A∇u(x)) = f(x) in Ω = [0, 1]2,
where
A =
(
cosφ − sinφsinφ cosφ
)(
1 00 ε
)(
cosφ sinφ− sinφ cos φ
)
.
01
0.2
0.4
1
0.6
0.8
0.8
0.5 0.6
1
0.40.2
0 0
Figure: Numericalsolution, h=1/20.
El2EOC(l2) Emax EOC(max)
Stencil=5, ε = 10−6, φ = π/6h=1/5 58180 – 143297 –
h=1/10 71389 diverges 152334 diverges
h=1/20 76663 diverges 155059 diverges
Stencil=9, ε = 10−6, φ = π/6h=1/5 4895 – 10783 –
h=1/10 1882 1.379 4025 1.421
h=1/20 560 1.748 1150 1.807
Stencil=25, ε = 10−6, φ = π/6h=1/5 609 – 1529 –
h=1/10 47 3.695 85 4.168
h=1/20 3 3.969 12 2.824
Table: Numerical experiments.
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Finite Difference Radial Basis Function (FD-RBF)
Transport equation (the solid body rotation):
ut + v · ∇u = 0 in Ω = [0, 1]2.
01
0.2
0.4
1
0.6
0.8
Initial solution
0.8
0.5 0.6
1
0.40.2
0 0
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Finite Difference Radial Basis Function (FD-RBF)
Transport equation (the solid body rotation):
ut + v · ∇u−0.0008∆u = 0 in Ω = [0, 1]2.
01
0.2
0.4
1
0.6
0.8
Initial solution
0.8
0.5 0.6
1
0.40.2
0 0
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Finite Difference Radial Basis Function (FD-RBF)
Transport equation (the solid body rotation):
ut + v · ∇u−0.0008∆u = 0 in Ω = [0, 1]2.
Stabilization: ”Stabilization of RBF-generated finite difference methods for convectivePDEs” by Bengt Fornberg and Erik Lehto, Journal of Computational Physics, 2010.
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Finite Difference Radial Basis Function (FD-RBF)
Transport equation (the solid body rotation):
ut + v · ∇u−0.0008∆u = 0 in Ω = [0, 1]2.
Stabilization: ”Stabilization of RBF-generated finite difference methods for convectivePDEs” by Bengt Fornberg and Erik Lehto, Journal of Computational Physics, 2010.
... or ???
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Finite Difference Radial Basis Function (FD-RBF)
Reaction-Convection-Diffusion Equation:
ut + v · ∇u−∇ (A(x)∇u(x)) = f in Ω ⊂ R2.
![Page 61: NumericalsolutionofsurfacePDEs with RadialBasisFunctionsAndriy Sokolov Institut fur Angewandte Mathematik (LS3) TU Dortmund andriy.sokolov@math.tu-dortmund.de TU Dortmund June 1, 2017](https://reader036.vdocument.in/reader036/viewer/2022071302/60adbc553881f004ef126645/html5/thumbnails/61.jpg)
Finite Difference Radial Basis Function (FD-RBF)
Reaction-Convection-Diffusion Equation:
ut + v · ∇u−∇ (A(x)∇u(x)) = f in Ω ⊂ R2.
More? ⇒ work of Ufuk Erkul
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Outline
1 Radial Basis Functions (RBFs)
2 FD-RBF approximation of differential operators(Finite Difference Radial Basis Functions)
3 FD-RBF method for surface PDEs
4 Outlook
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FD-RBF for surface PDEs
Laplace-Beltrami:
∆Γu = u(x) + f on Γ ⊂ Rd,
where Γ is sufficiently smooth, closed hypersurface.
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FD-RBF for surface PDEs
Laplace-Beltrami:
∇Γ · (∇Γu(x)) = u(x) + f on Γ ⊂ Rd,
where Γ is sufficiently smooth, closed hypersurface.
How to treat the surface?
How to treat ∇Γ·?
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FD-RBF for surface PDEs
Laplace-Beltrami:
∇Γ · (∇Γu(x)) = u(x) + f on Γ ⊂ Rd,
where Γ is sufficiently smooth, closed hypersurface.
How to treat the surface?
How to treat ∇Γ·?
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FD-RBF for surface PDEs
Laplace-Beltrami:
∇Γ · (∇Γu(x)) = u(x) + f on Γ ⊂ Rd,
where Γ is sufficiently smooth, closed hypersurface.
How to treat the surface?
How to treat ∇Γ·?
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FD-RBF for surface PDEs
The phase-field method:
ut −∇ · (∇Γu(x)) = u(x) + f on Γ ⊂ Rd,
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FD-RBF for surface PDEs
The phase-field method:
B(φ) ut −∇ · (B(φ)∇u(x)) = B(φ) (u(x) + f) in Ωε ⊂ Rd,
01
0.2
0.4
1
0.6
0.8
Initial solution
0.8
0.5 0.6
1
0.40.2
0 0
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FD-RBF for surface PDEs
The level set based method:
−D∇Γ(t) ·(∇Γ(t)u
)= f(u), on Γ(t)
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FD-RBF for surface PDEs
The level set based method:
−D∇Γ(t) ·(∇Γ(t)u
)= f(u), on Γ(t)
Introducing the level-set function
φ(x) =
−dist(x,Γ) if x is inside Γ
0 if x ∈ Γ
dist(x,Γ) if x is outside Γ
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FD-RBF for surface PDEs
The level set based method:
−D∇Γ(t) ·(∇Γ(t)u
)= f(u), on Γ(t)
Introducing the level-set function
φ(x) =
−dist(x,Γ) if x is inside Γ
0 if x ∈ Γ
dist(x,Γ) if x is outside Γ
we obtain
PΓ u =
(
I −∇φ
‖∇φ‖⊗
∇φ
‖∇φ‖
)
∇u.
![Page 72: NumericalsolutionofsurfacePDEs with RadialBasisFunctionsAndriy Sokolov Institut fur Angewandte Mathematik (LS3) TU Dortmund andriy.sokolov@math.tu-dortmund.de TU Dortmund June 1, 2017](https://reader036.vdocument.in/reader036/viewer/2022071302/60adbc553881f004ef126645/html5/thumbnails/72.jpg)
FD-RBF for surface PDEs
The level set based method:
∇Γ(t)u =
(ex − nxn) · ∇
(ey − nyn) · ∇
(ez − nzn) · ∇
u =
px · ∇
py · ∇
pz · ∇
u =
Gx
Gy
Gz
u
![Page 73: NumericalsolutionofsurfacePDEs with RadialBasisFunctionsAndriy Sokolov Institut fur Angewandte Mathematik (LS3) TU Dortmund andriy.sokolov@math.tu-dortmund.de TU Dortmund June 1, 2017](https://reader036.vdocument.in/reader036/viewer/2022071302/60adbc553881f004ef126645/html5/thumbnails/73.jpg)
FD-RBF for surface PDEs
The level set based method:
∇Γ(t)u =
(ex − nxn) · ∇
(ey − nyn) · ∇
(ez − nzn) · ∇
u =
px · ∇
py · ∇
pz · ∇
u =
Gx
Gy
Gz
u
∆Γ(t)u = ∇Γ(t) · ∇Γ(t)u = ∇Γ(t) ·
Gx
Gy
Gz
u
=(Gx Gy Gz
)
Gx
Gy
Gz
u =(GxGx + GyGy + GzGz
)u
![Page 74: NumericalsolutionofsurfacePDEs with RadialBasisFunctionsAndriy Sokolov Institut fur Angewandte Mathematik (LS3) TU Dortmund andriy.sokolov@math.tu-dortmund.de TU Dortmund June 1, 2017](https://reader036.vdocument.in/reader036/viewer/2022071302/60adbc553881f004ef126645/html5/thumbnails/74.jpg)
FD-RBF for surface PDEs
The level set based method:
(Gx Iφu(x)) |x=xi=
N∑
j=1
cj (Gx φ(rj(x))) |x=xi
=
N∑
j=1
cj [ ((1 − nxi n
xi )(xi − xj)−
− nyi n
yi (yi − yj)
−nzin
zi (zi − zj))
φ′
r(rj(xi))
rj(xi)] .
![Page 75: NumericalsolutionofsurfacePDEs with RadialBasisFunctionsAndriy Sokolov Institut fur Angewandte Mathematik (LS3) TU Dortmund andriy.sokolov@math.tu-dortmund.de TU Dortmund June 1, 2017](https://reader036.vdocument.in/reader036/viewer/2022071302/60adbc553881f004ef126645/html5/thumbnails/75.jpg)
FD-RBF for surface PDEs
The level set based method:
u(x)−∆Γu(x) = f(x) on Γ = x : |x| = 1.
uanalyt =1
|x|526|x|2
|x|2 + 25(x5
1 − 10x31x
22 + 5x1x
42)
f =26
|x|5(x5
1 − 10x31x
22 + 5x1x
42).
![Page 76: NumericalsolutionofsurfacePDEs with RadialBasisFunctionsAndriy Sokolov Institut fur Angewandte Mathematik (LS3) TU Dortmund andriy.sokolov@math.tu-dortmund.de TU Dortmund June 1, 2017](https://reader036.vdocument.in/reader036/viewer/2022071302/60adbc553881f004ef126645/html5/thumbnails/76.jpg)
FD-RBF for surface PDEs
The level set based method:
u(x)−∆Γu(x) = f(x) on Γ = x : |x| = 1.
-0.6
1.5
-0.4
1
-0.2
1.5
0
0.51
0.2
solutions in band
0 0.5
0.4
0
0.6
-0.5-0.5
-1-1
-1.5 -1.5
![Page 77: NumericalsolutionofsurfacePDEs with RadialBasisFunctionsAndriy Sokolov Institut fur Angewandte Mathematik (LS3) TU Dortmund andriy.sokolov@math.tu-dortmund.de TU Dortmund June 1, 2017](https://reader036.vdocument.in/reader036/viewer/2022071302/60adbc553881f004ef126645/html5/thumbnails/77.jpg)
FD-RBF for surface PDEs
The level set based method:
u(x)−∆Γu(x) = f(x) on Γ = x : |x| = 1.
![Page 78: NumericalsolutionofsurfacePDEs with RadialBasisFunctionsAndriy Sokolov Institut fur Angewandte Mathematik (LS3) TU Dortmund andriy.sokolov@math.tu-dortmund.de TU Dortmund June 1, 2017](https://reader036.vdocument.in/reader036/viewer/2022071302/60adbc553881f004ef126645/html5/thumbnails/78.jpg)
FD-RBF for surface PDEs
The level set based method:
u(x)−∆Γu(x) = f(x) on Γ = x : |x| = 1.
E(ε− band) EOC(ε− band)
h=3/10 0.003001 –
h=3/20 0.001026 1.5484
h=3/40 0.000242 2.0840
h=3/80 0.000051 2.2464
h=3/160 0.000013 1.9720
Table: 9-points stencil (including the main node), ε-band = 0.1.
E(ε− band) = ‖unum − uanalyt‖l2(ε−band)/√
#nodes
![Page 79: NumericalsolutionofsurfacePDEs with RadialBasisFunctionsAndriy Sokolov Institut fur Angewandte Mathematik (LS3) TU Dortmund andriy.sokolov@math.tu-dortmund.de TU Dortmund June 1, 2017](https://reader036.vdocument.in/reader036/viewer/2022071302/60adbc553881f004ef126645/html5/thumbnails/79.jpg)
FD-RBF for surface PDEs
The level set based method:
u(x)−∆Γu(x) = f(x) on Γ = x : |x| = 1.
E(ε− band) EOC(ε− band)
h=3/10 0.003001 –
h=3/20 0.001026 1.5484
h=3/40 0.000242 2.0840
h=3/80 0.000051 2.2464
h=3/160 0.000013 1.9720
Table: 9-points stencil (including the main node), ε-band = 0.1.
E(ε− band) = ‖unum − uanalyt‖l2(ε−band)/√
#nodes
More? ⇒ work of David Borringo
![Page 80: NumericalsolutionofsurfacePDEs with RadialBasisFunctionsAndriy Sokolov Institut fur Angewandte Mathematik (LS3) TU Dortmund andriy.sokolov@math.tu-dortmund.de TU Dortmund June 1, 2017](https://reader036.vdocument.in/reader036/viewer/2022071302/60adbc553881f004ef126645/html5/thumbnails/80.jpg)
FD-RBF for surface PDEs
The level set based method for Γ(t):
∂∗t u = D∆Γ(t)u(x, t) + f(x, t) on Γ = Γ(t),
where Γ(t) = x : φ(t,x) = 0 and
φ(t,x) = |x| − 0.75 + sin(4 t)(|x| − 0.5)(1− |x|).
![Page 81: NumericalsolutionofsurfacePDEs with RadialBasisFunctionsAndriy Sokolov Institut fur Angewandte Mathematik (LS3) TU Dortmund andriy.sokolov@math.tu-dortmund.de TU Dortmund June 1, 2017](https://reader036.vdocument.in/reader036/viewer/2022071302/60adbc553881f004ef126645/html5/thumbnails/81.jpg)
FD-RBF for surface PDEs
The level set based method for Γ(t):
∂•
t u︷ ︸︸ ︷
ut(x, t) + v · ∇u+u∇Γ · v︸ ︷︷ ︸
∂∗
t u
= D∇Γ(t)·(∇Γ(t)u(x, t)
)+f(x, t) on Γ = Γ(t),
where v = V n+ vS and
V = v · n = −φt
|∇φ|.
![Page 82: NumericalsolutionofsurfacePDEs with RadialBasisFunctionsAndriy Sokolov Institut fur Angewandte Mathematik (LS3) TU Dortmund andriy.sokolov@math.tu-dortmund.de TU Dortmund June 1, 2017](https://reader036.vdocument.in/reader036/viewer/2022071302/60adbc553881f004ef126645/html5/thumbnails/82.jpg)
FD-RBF for surface PDEs
The level set based method for Γ(t):
∂•
t u︷ ︸︸ ︷
ut(x, t) + v · ∇u+u∇Γ · v︸ ︷︷ ︸
∂∗
t u
= D∇Γ(t)·(∇Γ(t)u(x, t)
)+f(x, t) on Γ = Γ(t),
where v = V n+ vS and
V = v · n = −φt
|∇φ|.
The analytical solution is chosen to be
u(x, t) = e−t/|x|2 x1
|x|.
![Page 83: NumericalsolutionofsurfacePDEs with RadialBasisFunctionsAndriy Sokolov Institut fur Angewandte Mathematik (LS3) TU Dortmund andriy.sokolov@math.tu-dortmund.de TU Dortmund June 1, 2017](https://reader036.vdocument.in/reader036/viewer/2022071302/60adbc553881f004ef126645/html5/thumbnails/83.jpg)
FD-RBF for surface PDEs
The level set based method for Γ(t):
∂∗t u = D∆Γ(t)u(x, t) + f(x, t) on Γ = Γ(t).
-1 -0.5 0 0.5 1-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
![Page 84: NumericalsolutionofsurfacePDEs with RadialBasisFunctionsAndriy Sokolov Institut fur Angewandte Mathematik (LS3) TU Dortmund andriy.sokolov@math.tu-dortmund.de TU Dortmund June 1, 2017](https://reader036.vdocument.in/reader036/viewer/2022071302/60adbc553881f004ef126645/html5/thumbnails/84.jpg)
FD-RBF for surface PDEs
The level set based method for Γ(t):
∂∗t u = D∆Γ(t)u(x, t) + f(x, t) on Γ = Γ(t).
-11
-0.5
0.5 1
0
solutions in band
0.5
0
1
0-0.5
-1 -1
numericalanalytical
![Page 85: NumericalsolutionofsurfacePDEs with RadialBasisFunctionsAndriy Sokolov Institut fur Angewandte Mathematik (LS3) TU Dortmund andriy.sokolov@math.tu-dortmund.de TU Dortmund June 1, 2017](https://reader036.vdocument.in/reader036/viewer/2022071302/60adbc553881f004ef126645/html5/thumbnails/85.jpg)
FD-RBF for surface PDEs
The level set based method for Γ(t):
∂∗t u = D∆Γ(t)u(x, t) + f(x, t) on Γ = Γ(t).
01
0.5
1
0.5 1
1.5
×10 -3
2
error in Omega
0.50
2.5
3
0-0.5 -0.5
-1 -1
![Page 86: NumericalsolutionofsurfacePDEs with RadialBasisFunctionsAndriy Sokolov Institut fur Angewandte Mathematik (LS3) TU Dortmund andriy.sokolov@math.tu-dortmund.de TU Dortmund June 1, 2017](https://reader036.vdocument.in/reader036/viewer/2022071302/60adbc553881f004ef126645/html5/thumbnails/86.jpg)
FD-RBF for surface PDEs
The level set based method for Γ(t):
∂∗t u = D∆Γ(t)u(x, t) + f(x, t) on Γ = Γ(t).
E(ε− band) EOC(ε − band)
N=20 0.010179 –
N=40 0.003626 1.4891
N=80 0.002065 0.8122
N=160 0.001584 0.3875
Table: 9-points stencil (including the main node). ε-band = 0.1
![Page 87: NumericalsolutionofsurfacePDEs with RadialBasisFunctionsAndriy Sokolov Institut fur Angewandte Mathematik (LS3) TU Dortmund andriy.sokolov@math.tu-dortmund.de TU Dortmund June 1, 2017](https://reader036.vdocument.in/reader036/viewer/2022071302/60adbc553881f004ef126645/html5/thumbnails/87.jpg)
FD-RBF for surface PDEs
The level set based method for Γ(t):
∂∗t u = D∆Γ(t)u(x, t) + f(x, t) on Γ = Γ(t).
E(ε− band) EOC(ε − band)
N=20 0.010179 –
N=40 0.003626 1.4891
N=80 0.002065 0.8122
N=160 0.001584 0.3875
Table: 9-points stencil (including the main node). ε-band = 0.1
More? ⇒ work in progress...
![Page 88: NumericalsolutionofsurfacePDEs with RadialBasisFunctionsAndriy Sokolov Institut fur Angewandte Mathematik (LS3) TU Dortmund andriy.sokolov@math.tu-dortmund.de TU Dortmund June 1, 2017](https://reader036.vdocument.in/reader036/viewer/2022071302/60adbc553881f004ef126645/html5/thumbnails/88.jpg)
Conclusion
(FD)RBF is a new rapidly developing approach
It it possible to use it for real-world applications
Mesh adaptivity
![Page 89: NumericalsolutionofsurfacePDEs with RadialBasisFunctionsAndriy Sokolov Institut fur Angewandte Mathematik (LS3) TU Dortmund andriy.sokolov@math.tu-dortmund.de TU Dortmund June 1, 2017](https://reader036.vdocument.in/reader036/viewer/2022071302/60adbc553881f004ef126645/html5/thumbnails/89.jpg)
Conclusion
(FD)RBF is a new rapidly developing approach
It it possible to use it for real-world applications
Mesh adaptivity
![Page 90: NumericalsolutionofsurfacePDEs with RadialBasisFunctionsAndriy Sokolov Institut fur Angewandte Mathematik (LS3) TU Dortmund andriy.sokolov@math.tu-dortmund.de TU Dortmund June 1, 2017](https://reader036.vdocument.in/reader036/viewer/2022071302/60adbc553881f004ef126645/html5/thumbnails/90.jpg)
Conclusion
(FD)RBF is a new rapidly developing approach
It it possible to use it for real-world applications
Mesh adaptivity
![Page 91: NumericalsolutionofsurfacePDEs with RadialBasisFunctionsAndriy Sokolov Institut fur Angewandte Mathematik (LS3) TU Dortmund andriy.sokolov@math.tu-dortmund.de TU Dortmund June 1, 2017](https://reader036.vdocument.in/reader036/viewer/2022071302/60adbc553881f004ef126645/html5/thumbnails/91.jpg)
Conclusion
(FD)RBF is a new rapidly developing approach
It it possible to use it for real-world applications
Mesh adaptivity
Lots of new and unexplored fields
![Page 92: NumericalsolutionofsurfacePDEs with RadialBasisFunctionsAndriy Sokolov Institut fur Angewandte Mathematik (LS3) TU Dortmund andriy.sokolov@math.tu-dortmund.de TU Dortmund June 1, 2017](https://reader036.vdocument.in/reader036/viewer/2022071302/60adbc553881f004ef126645/html5/thumbnails/92.jpg)
Conclusion
(FD)RBF is a new rapidly developing approach
It it possible to use it for real-world applications
Mesh adaptivity
Lots of new and unexplored fields
![Page 93: NumericalsolutionofsurfacePDEs with RadialBasisFunctionsAndriy Sokolov Institut fur Angewandte Mathematik (LS3) TU Dortmund andriy.sokolov@math.tu-dortmund.de TU Dortmund June 1, 2017](https://reader036.vdocument.in/reader036/viewer/2022071302/60adbc553881f004ef126645/html5/thumbnails/93.jpg)
Acknowledgements
Prof. Dr. Oleg Davydov, University of Giessen
Prof. Dr. Dmitri Kuzmin, TU Dortmund
Prof. Dr. Stefan Turek, TU Dortmund
![Page 94: NumericalsolutionofsurfacePDEs with RadialBasisFunctionsAndriy Sokolov Institut fur Angewandte Mathematik (LS3) TU Dortmund andriy.sokolov@math.tu-dortmund.de TU Dortmund June 1, 2017](https://reader036.vdocument.in/reader036/viewer/2022071302/60adbc553881f004ef126645/html5/thumbnails/94.jpg)
Thank you very much
for your attention!