rbf{qr radial basis function approximationradial basis function approximation phd student course in...
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RBF–QROutline
Global RBFsRBF limits
Stable methods
Convergence theory
RBF-PUMTheoretical resultsNumerical results
RBF-FD
Radial basis function approximationPhD student course in Approximation Theory
Elisabeth Larsson
2017-09-18
E. Larsson, 2017-09-18 (1 : 77)
RBF–QROutline
Global RBFsRBF limits
Stable methods
Convergence theory
RBF-PUMTheoretical resultsNumerical results
RBF-FD
Outline
Global RBF approximationRBF limits
Stable evaluation methods
Convergence theory
RBF partition of unity methodsTheoretical resultsNumerical results
RBF-FD
E. Larsson, 2017-09-18 (2 : 77)
RBF–QROutline
Global RBFsRBF limits
Stable methods
Convergence theory
RBF-PUMTheoretical resultsNumerical results
RBF-FD
Short introduction to (global) RBF methods
Basis functions: φj(x) = φ(‖x − x j‖). Translates of onesingle function rotated around a center point.
Example: Gaussiansφ(εr) = exp(−ε2r2)
Approximation:sε(x ) =
∑Nj=1 λjφj(x )
Collocation:sε(x i ) = fi ⇒ Aλ = f ε=3ε=1/3ε=1
Advantages:
• Flexibility with respect to geometry.
• As easy in d dimensions.
• Spectral accuracy / exponential convergence.
• Continuosly differentiable approximation.
E. Larsson, 2017-09-18 (3 : 77)
RBF–QROutline
Global RBFsRBF limits
Stable methods
Convergence theory
RBF-PUMTheoretical resultsNumerical results
RBF-FD
Commonly used RBFs
Global infinitely smoothGaussian exp(−ε2r2), ε > 0
(Inverse) multiquadric (1 + ε2r2)β/2, ε > 0, |β| ∈ N
Global piecewise smoothPolyharmonic spline (odd) |r |2m−1, m ∈ NPolyharmonic spline (even) r2m log(r), m ∈ NMatern/Sobolev rνKν(r), ν > 0C 2 Matern (1 + r) exp(−r), v = 3/2
Compactly supported Wendland functionsC 2 and pos def for d ≤ 3, (1− r
ρ)4+(4 r
ρ + 1), ρ > 0
C 2 and pos def for d ≤ 5, (1− rρ)4
+(5 rρ + 1), ρ > 0
E. Larsson, 2017-09-18 (4 : 77)
RBF–QROutline
Global RBFsRBF limits
Stable methods
Convergence theory
RBF-PUMTheoretical resultsNumerical results
RBF-FD
demo1.m(RBF interpolation in 1-D)
E. Larsson, 2017-09-18 (5 : 77)
RBF–QROutline
Global RBFsRBF limits
Stable methods
Convergence theory
RBF-PUMTheoretical resultsNumerical results
RBF-FD
Observations from the results of demo1.m
I As N grows for fixed ε, convergence stagnates.
I As ε decreases for fixed N, the error blows up.
I λmin = −λmax means cancellation.
I Coefficients λ→∞ means that cond(A)→∞.
I For small ε, the RBFs are nearly flat, and almostlinearly dependent. That is, they form a bad basis.
10 20 30
10−5
N
Ma
x e
rro
rε = 1
10 20 30
−5
0
5
N
log
10(m
ax/m
in(λ
))
ε = 1
100
10−5
ε
Ma
x e
rro
r
N = 30
100
−10
0
10
ε
log
10(m
ax/m
in(λ
))
N = 30
E. Larsson, 2017-09-18 (6 : 77)
RBF–QROutline
Global RBFsRBF limits
Stable methods
Convergence theory
RBF-PUMTheoretical resultsNumerical results
RBF-FD
Why is it interesting to use small values of ε?
Driscoll & Fornberg [DF02]
Somewhat surprisingly, in 1-D for small ε
s(x , ε) = PN−1(x) + ε2PN+1(x) + ε4PN+3(x) + · · · ,
where Pj is a polynomial of degree j and PN−1(x) is theLagrange interpolant.
Implications
I It can be shown that cond(A) ∼ O(Nε−2(N−1)), butthe limit interpolant is well behaved.
I It is the intermediate step of computing λ that isill-conditioned.
I By choosing the corresponding nodes, the flat RBFlimit reproduces pseudo-spectral methods.
I This is a good approximation space.E. Larsson, 2017-09-18 (7 : 77)
RBF–QROutline
Global RBFsRBF limits
Stable methods
Convergence theory
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RBF-FD
The multivariate flat RBF limit
Larsson & Fornberg [LF05], Schaback [Sch05]In n-D the flat limit can either be
s(x , ε) = PK (x) + ε2PK+2(x) + ε4PK+4(x) + · · · ,
where
((K − 1) + d
d
)< N ≤
(K + d
d
)and PK is a
polynomial interpolant or
s(x , ε) = ε−2qPM−2q(x) + ε−2q+2PM−2q+2(x) + · · ·+ PM(x) + ε2PM+2(x) + ε4PM+4(x) + · · · .
The questions of uniqueness and existence are connectedwith multivariate polynomial uni-solvency.
Schaback [Sch05]
Gaussian RBF limit interpolants always converge to thede Boor/Ron least polynomial interpolant.
E. Larsson, 2017-09-18 (8 : 77)
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The multivariate flat RBF limit: Divergence
Necessary condition: ∃ Q(x) of degree N0 such thatQ(x j) = 0, j = 1, . . . ,N.Then divergence as ε−2q may occur, whereq = b(M − N0)/2c and M = min non-degenerate degree.
Points Q N0 Basis M qx − y 1 1, x , x2,
x3, x4, x55 2
x2 − y − 1 2 1, x , y , xy ,y2xy2
3 0
x2 + y2 − 1 2 1, x , y , x2, xy ,x3, x2y , x4
4 1
Divergence actually only occurs for the first case as ε−2.
E. Larsson, 2017-09-18 (9 : 77)
RBF–QROutline
Global RBFsRBF limits
Stable methods
Convergence theory
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The multivariate flat RBF limit, contd
Schaback [Sch05], Fornberg & Larsson [LF05]
Example: In two dimensions, the eigenvalues of A followa pattern: µ1 ∼ O(ε0), µ2,3 ∼ O(ε2), µ4,5,6 ∼ O(ε4),. . .
In general, there are
(k + n − 1n − 1
)= (k+1)···(k+n−1)
(n−1)!
eigenvalues µj ∼ O(ε2k) in n dimensions.
Implications
I There is an opportunity for pseudo-spectral-likemethods in n-D.
I There is no amount of variable precision that willsave us.
I For “smooth” functions, a small ε can lead to veryhigh accuracy.
E. Larsson, 2017-09-18 (10 : 77)
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Multivariate interpolation
Theorem (Mairhuber–Curtis)
For a domain Ω ⊂ Rd , d ≥ 2 that has an interior point,there is no basis of continuous functions f1(x ), . . . fN(x ),N ≥ 2 such that an interpolation matrix A = fj(x i )Ni ,j=1
is guaranteed to be non-singular (no Haar basis).
Proof.Let two of the points x i and xk change places along aclosed continuous path in Ω. When the two points havechanged places, two rows in A are interchanged, anddet(A) has changed sign. Then det(A) = 0 somewherealong the path.
I For RBF approximation, A = φ(‖x i − x j‖)Ni ,j=1. Iftwo points change place, two rows and two columnsare swapped. Determinant does not change sign.
E. Larsson, 2017-09-18 (11 : 77)
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Stable methods
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Positive definite functions
Definition (Positive definite function)
A real valued continuous function Φ is positive definite onRd ⇔ it is even and
N∑j=1
N∑k=1
cjckΦ(x j − xk) ≥ 0
for any parwise distinct points x1, . . . , xN ∈ Rd , cj ∈ R.
Theorem (Bochner 1933)
A function Φ ∈ C (Rd) is positive definite on Rd ⇔ it isthe Fourier transform of a finite non-negative Borelmeasure µ on Rd
Φ(x ) =1√
(2π)d
∫Rd
e−ix ·ωdµ(ω).
E. Larsson, 2017-09-18 (12 : 77)
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Bochner Theorem contd
Partial proof
N∑j=1
N∑k=1
cjckΦ(x j − xk) =
=1√
(2π)d
N∑j=1
N∑k=1
(cjck
∫Rd
e−i(x j−xk )·ωdµ(ω)
)
=1√
(2π)d
∫Rd
N∑j=1
cje−ix j ·ω
N∑k=1
ckeixk ·ω
dµ(ω)
=1√
(2π)d
∫Rd
∣∣∣∣∣∣N∑j=1
cje−ix j ·ω
∣∣∣∣∣∣2
dµ(ω) ≥ 0.
E. Larsson, 2017-09-18 (13 : 77)
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Example
The Gaussian is positive definite in any dimension
e−ε2‖x‖2
=1√
(2π)d
∫Rd
1
(√
2ε)de−‖ω‖
2/(4ε2)e ix ·ωdω
Theorem (Schoenberg 1938)
A cont function ϕ : [0,∞)→ R is strictly pos def andradial on Rd for all d ⇔
ϕ(r) =
∫ ∞0
e−r2t2
dµ(t),
where µ is a finite non-negative Borel measure notconcentrated at the origin.
E. Larsson, 2017-09-18 (14 : 77)
RBF–QROutline
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Stable methods
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Results and consequences for RBFapproximation
I Non-singularity of RBF interpolation is guaranteedfor distinct node points and strictly pos def functionssuch as the Gaussian and the inverse multiquadric.
I There are no oscillatory or compactly supportedRBFs that are strictly pos def for all d .Because φ(r0) = 0 breaks theorem, cf. Bessel and Wendland.
I Non-singularity/positive definiteness of interpolationmatrix holds also for conditionally positive definiteRBFs augmented with polynomials.Micchelli [Mic86], cf. multiquadric RBFs
E. Larsson, 2017-09-18 (15 : 77)
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Tensor product vs multivariate basis
Tensor product basis
s(x ) =n∑
i=0
n∑j=0
cijpi (x1)pj(x2)
Number of unknowns NT = (n + 1)d .
Multivariate basisThinking in terms of polynomials, a multivariatepolynomial space of degree n has dimension
NM =
(n + dd
)=
(n + 1) · · · (n + d)
d!
Degrees of freedom for n = 8:
d 1 2 3
NT 9 81 729
NM 9 45 165E. Larsson, 2017-09-18 (16 : 77)
RBF–QROutline
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Stable methods
Convergence theory
RBF-PUMTheoretical resultsNumerical results
RBF-FD
demo2.m(Conditioning and errors)
E. Larsson, 2017-09-18 (17 : 77)
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Stable methods
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Comments on the results of demo2
I Error is small where condition is high and vice versa.
I Interesting region only reachable with stable method.
I Best results for small ε.
171615141312111098765432118
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171615141312111098765432118
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
log10
(ε)
N
RBF−Direct: log10
( cond2(A) )
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
181234567891011121314151617
−2.5 −2 −1.5 −1 −0.5 0 0.5 110
20
30
40
50
60
70
5
10
15
18
log10
(ε)N
RBF−QR: log10
|s(x)−f(x)|
−2.5 −2 −1.5 −1 −0.5 0 0.5 110
20
30
40
50
60
70
−9
−8
−7
−6
−5
−4
−3
−2
−1
Teaser: Conditioning for RBF–QR is perfect...
E. Larsson, 2017-09-18 (18 : 77)
RBF–QROutline
Global RBFsRBF limits
Stable methods
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The Contour-Pade method
Fornberg & Wright [FW04]
I Think of ε as a complex variable.
I The limit ε = 0 is a removable singularity.
I Complex ε for which A is singular lead to poles.
I Pole location only depend on the location of nodes.
Example
I Evaluate f (ε) = 1−cos(ε)ε2
I Numerically unstable.
I Removable singularity at 0.
I Compute f (0) as average off (ε) around “safe path”.
Bad region
Safe path
Target point
Im ε
Re ε
E. Larsson, 2017-09-18 (19 : 77)
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The Contour-Pade method: Algorithm
I Compute s(x , ε) = AeA−1f at M points around a
“safe path” (circle).I Inverse FFT of the M values gives a Laurent
expansion
u(x) = . . .+ s−2(x)ε−4 + s−1(x)ε−2︸ ︷︷ ︸Needs to be converted
+s0(x)+s1(x)ε2+s2(x)ε4+. . .
I Convert the negative power expansion into Padeform and find the correct number of poles and theirlocations
s−1ε−2 + s−2ε
−4 + . . . =p1ε−2 + · · ·+ pmε
−2m
1 + q1ε−2 + · · ·+ qnε−2n.
I Evaluate u(x) using Taylor + Pade for any ε insidethe circle.
E. Larsson, 2017-09-18 (20 : 77)
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The Contour-Pade method: Results
10−3
10−2
10−1
100
10−10
10−5
100
N=50
ε
max |
s(x
)−f(
x)
|
Usual method 1e−9
New method 1e−9
10−3
10−2
10−1
100
10−10
10−5
100
N=75
ε
max |
s(x
)−f(
x)
|
Quad precision 9e−12Usual method 1e− 9New method 1e−10
I Stable computation for all ε with Contour-Pade.
I Limited number of nodes, otherwise general.
I Expensive to compute A−1 at M points.
I Tricky to find poles.
I Modern efficient version RBF-RA [WF17].
E. Larsson, 2017-09-18 (21 : 77)
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Expansions of (Gaussian) RBFs
On the surface of the sphere
Hubbert & Baxter [BH01]
For different RBFs there are expansions
φ(‖x − xk‖) =∞∑j=0
ε2jj∑
m=−jcj ,mY
mj (x )
Cartesian space, polynomial expansion
For Gaussians
φ(‖x − xk‖) = e−ε2(x−xk )·(x−xk )
= e−ε2(x ·x )e−ε
2(xk ·xk )e2ε2(x ·xk )
= e−ε2(x ·x )e−ε
2(xk ·xk )∞∑j=0
ε2j 2j
j!(x · xk)j
E. Larsson, 2017-09-18 (22 : 77)
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Expansions of (Gaussian) RBFs contd
Mercer expansion (Mercer 1909)
For a positive definite kernel K (x , xk) = φ(‖x − xk‖),there is an expansion
φ(‖x − xk‖) =∞∑j=0
λjϕj(x )ϕj(xk),
where λj are positive eigenvalues, and ϕj(x ) areeigenfunctions of an associated compact integraloperator.
E. Larsson, 2017-09-18 (23 : 77)
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The RBF-QR method on the sphere
Fornberg & Piret [FP07]
φ(‖x − xk‖) =∞∑j=0
ε2jj∑
m=−jcj ,mY
mj (x )
The number of SPH functions/power matches the RBFeigenvalue pattern on the sphere.
If we collect RBFs and expansion functions in vectors,and coefficients in the matrix B, we have a relation
Φ(x ) = B · Y = Q · E · R · Y (x )
The new basis Ψ(x ) = R · Y (x ) spans the same space asΦ(x ), but the ill-conditioning has been absorbed in E .
E. Larsson, 2017-09-18 (24 : 77)
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The RBF-QR method in Cartesian space
Fornberg, Larsson, Flyer [FLF11]
The expansion of the Gaussian
φ(‖x − xk‖) = e−ε2(x ·x )e−ε
2(xk ·xk )∞∑j=0
ε2j 2j
j!(x · xk)j
+ The number of expansion functions for each powerof ε matches the eigenvalue pattern in A.
− The expansion functions are the monomials.
Better expansion functions in 2-D
I Change to polar coordinates.
I Trigs in the angular direction are perfect.
I Necessary to preserve powers of ε ⇒Partial conversion to Chebyshev polynomials.
E. Larsson, 2017-09-18 (25 : 77)
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The RBF-QR method in Cartesian space contd
New expansion functions
T cj ,m(x) = e−ε
2r2r2mTj−2m(r) cos((2m + p)θ),
T sj ,m(x) = e−ε
2r2r2mTj−2m(r) sin((2m + p)θ),
Matrix form of factorized expansion
Express Φ(x ) = (φ(‖x − x1‖), . . . , φ(‖x − xN‖))T interms of expansion functions T (x ) = (T c
0,0,Tc1,0, . . .)
T as.
Φ(x ) = C · D · T (x ),
where cij is O(1) and D = diag(O(ε0, ε2, ε2, ε4, . . .)).
Note that C has an infinite number of columns etc.
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The RBF-QR method in Cartesian space contd
The QR part
The coefficient matrix C is QR-factorized so that
Φ(x ) = Q ·[R1 R2
] [ D1 00 D2
]·T (x ), where R1 and
D1 are of size (N × N).
The change of basis
Make the new basis (same space) close to T
Ψ(x ) = D−11 R−1
1 QHΦ(x ) =[I R
]· T (x ).
Analytical scaling of R = D−11 R−1
1 R2D2
Any power of ε in D1 ≤ any power of ε in D2 ⇒Scaling factors O(ε0) or smaller, truncation is possible.
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demo3.m(RBF interpolation in 2-D with and without RBF–QR)
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Stable computation as ε → 0 and N →∞The RBF-QR method allows stable computations forsmall ε. (Fornberg, Larsson, Flyer [FLF11])
Consider a finite non-periodic domain.
Theorem (Platte, Trefethen, and Kuijlaars [PTK11]):
Exponential convergence on equispaced nodes ⇒exponential ill-conditioning.
Solution #1:
Cluster nodes towards the domain boundaries.
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An RBF-QR example with clustered nodes in anon-trivial domain
f (x , y) = exp(−(x − 0.1)2 − 0.5y2)N=793 node pointsCosine-stretching towards each boundaryMaximum error 2.2e-10
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demo4.m(RBF interpolation in 2-D with clustered nodes)
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Brief survey of Mercer based methods
Fasshauer & McCourt [FM12]
Eigenvalues and eigenfunctions in 1-D can be chosen as
λn =
√α2
α2 + δ2 + ε2
(ε2
α2 + δ2 + ε2
)n−1
,
φn = γne−δ2x2
Hn−1(αβx),
where β =(
1 +(
2εα
)2) 1
4
, γn =√
β2n−1Γ(n) , δ2 = α2
2 (β2 − 1).
I Eigenfunctions are orthogonal in a weighted norm.
I The QR-step is similar to that of previous methods.
I Tensor product form is used in higher dimensions ⇒The powers of ε do not match the eigenvalues of A.
I New parameter α to tune.
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Brief survey of Mercer based methods contd
De Marchi & Santin [DMS13]
I Discrete numerical approximation of eigenfunctions.
I W diagonal matrix with cubature weights.Perform SVD
√W · A ·
√W = Q · Σ2 · QT .
The eigenbasis is given by√W−1 · Q · Σ.
I Rapid decay of singular values ⇒ Basis can betruncated ⇒ Low rank approximation of A.
De Marchi & Santin [DMS15]
I Faster: Lanczos algorithm on Krylov space K(A, f ).
I Eigenfunctions through SVD of Hm from Lanczos.
I Computationally efficient.
I Basis depends on f . Potential trouble for f 6∈ NK (X )
For details it is a good idea to ask the authors :-)E. Larsson, 2017-09-18 (33 : 77)
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Differentiation matrices and RBF-QR
Larsson, Lehto, Heryudono, Fornberg [LLHF13]
Let uX be an RBF approximation evaluated at the nodes.
To compute uY evaluated at the set of points Y , we useAλ = uX ⇒ λ = A−1uX to get
uY = AYλ = AYA−1uX
where AY (i , j) = φj(yi ).
To instead evaluate a differential operator applied to u,
uY = ALYA−1uX ,
where ALY (i , j) = Lφj(yi ).
To do the same thing using RBF–QR, replace φj with ψj .
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Solving PDEs with RBFs/RBF-QR
Domain defined by: rb(θ) = 1 + 110 (sin(6θ) + sin(3θ)).
PDE:
∆u=f (x ), x ∈ Ω,u=g(x ), x on ∂Ω,
Solution: u(x ) = sin(x21 + 2x2
2 )− sin(2x21 + (x2 − 0.5)2).
Collocation:(A∆X iA
−1X
I
)(uiXubX
)=
(f iXgbX
)Evaluation:uY = AYA
−1X uX
Domain + nodesE. Larsson, 2017-09-18 (35 : 77)
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demo5.m(Solving the Poisson problem in 2-D using RBFs)
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Reproducing Kernel Hilbert spaces andoptimality
Let N (Ω) be a real Hilbert space of functions u : Ω→ R,where Ω ⊆ Rd with inner product (·, ·)N (Ω).
Consider an RBF as a kernel K (x , y). The followingholds
(i) K (·, x ) ∈ N (Ω) for all x ∈ Ω.
(ii) (u,K (·, x ))N (Ω) = u(x ).
Let I (u) be the interpolant of u ∈ N (Ω). Then
‖I (u)‖N (Ω) ≤ ‖u‖N (Ω)
Consider a finite dimensional subspace N (X ) of thenative space N (Ω). Then
(I (u)− u, v)N (Ω) = 0 for all v ∈ N (X ).E. Larsson, 2017-09-18 (37 : 77)
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Ingredients for exponential convergenceestimates
I Dependence on geometry through interior coneconditions.Approximation quality depends on boundary shape.
I General sampling inequalities based on polynomialapproximation.These tell us how much a smooth error can grow between
nodes.
I Embedding constants relating Native spaces toSobolev spaces.These are needed to go from algebraic to exponential
estimates.
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Interior cone conditions
rθ
R
Definition (Interior cone condition)
A domain Ω ⊂ Rd satisfies an interior cone conditionwith radius r and angle θ if every x ∈ Ω is the vertex ofsuch a cone that is contained entirely within Ω.
Definition (Star shaped)
A domain Ω ⊂ Rd is star shaped with respect to B(xc , r)if for every x ∈ Ω, the convex hull of x and B(xc , r) isentirely enclosed in Ω.
Example
A star shaped domain wrt B(xc , r),enclosed by B(xc ,R) satisfiesan interior cone condition withradius r and angle θ = 2 arcsin( r
2R ).
Narcowich, Ward, Wendland [NWW05]E. Larsson, 2017-09-18 (39 : 77)
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General sampling inequalitites
Rieger & Zwicknagl [RZ10]
Bound derivatives of u through polynomial bounds
|Dαu(x )| ≤ |Dαu(x )− Dαp(x )|+ |Dαp(x )|
Detailed computations with averaged Taylor polys
‖Dαu‖Lq(Ω) ≤C kS δ
k−d( 1p− 1
q)
Ω
(k − |α|)!(δ−|α|Ω + h−|α|)|u|W k
p (Ω)
+ 2δdq
Ωh−|α|‖u‖`∞(X ),
where δΩ is the diameter of Ω, h is the fill distance(largest ball empty of nodes from X ), and the constantCS depends on d , p, and θ, 1 ≤ p <∞, 1 ≤ q ≤ ∞.
Fill distance must be small enough and k large enough.E. Larsson, 2017-09-18 (40 : 77)
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Embedding constants
Rieger & Zwicknagl [RZ10]
Assume there are embedding constants, for all k such that
‖u‖W kp (Ω) ≤ E (k)‖u‖H(Ω)
for some space H(Ω) of smooth functions. Furtherassume that E (k) ≤ C k
Ek(1−ε)k , for ε,CE > 0.
Then, the general sampling inequality can be rewritten as
‖Dαu‖Lq(Ω) ≤ eC log(h)√
h ‖u‖H(Ω) + 2δdq
Ωh−|α|‖u‖`∞(X ),
where C = ε√c0/4 and c0 = min1, r sin θ
4(1+sinθ).We are still considering star shaped domains.
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Lipschitz domains
Rieger & Zwicknagl [RZ10]
For a general Lipschitz domain that satisfies a uniforminterior cone condition, we create a cover of Ω consistingof star shaped subdomains.
This affects the terms in front of the norms, but not theessentials.
For E (k) ≤ C kEk
(1−ε)k
‖Dαu‖Lq(Ω) ≤ eC log(h)√
h ‖u‖H(Ω) + C2h−|α|‖u‖`∞(X ).
For E (k) ≤ C kEk
sk , s ≥ 1
‖Dαu‖Lq(Ω) ≤ eC
h1/(1+s) ‖u‖H(Ω) + C2h−|α|‖u‖`∞(X ).
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Embedding constants for kernel spaces
Fourier characterization of spaces
NK (Rd) =
u ∈ C (Rd) ∩ L2(Rd) : ‖u‖2
NK=
∫Rd
|u(ω)|2
|K (ω)|dω <∞
W k2 (Rd) =
u ∈ L2(Rd) :
∫Rd
|u(ω)|2(1 + ‖ω‖22)kdω <∞
Finding a specific embedding constant
For a particular kernel function K find E (k) such that
(1 + y)k ≤ E (k)2
K (y),
where y = ‖ω‖22.
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Embedding constants for kernel spaces contd
Rieger & Zwicknagl [RZ10]
For the Gaussian K (y) = (2ε2)−d2 e−
y
4ε2 and
E (k) = C kkk2 .
For the inverse multiquadric
K (y) = 21−β
Γ(β)
(√yε
)β(ε√y)−d/2Kd/2−β(
√yε ),
where K is a modified Bessel function of the third kind,leading to E (k) = C kkk .
Using the embedding constants
We finally assume that there is an extension operator Esuch that ‖Eu‖N (Rd ) ≤ ‖u‖N (Ω). Then
‖u‖W k2 (Ω) ≤ ‖Eu‖W k
2 (Rd ) ≤ E (k)‖Eu‖N (Rd ) ≤ E (k)‖u‖N (Ω)
Wendland [Wen05, Theorem 10.46]
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Implications for interpolation errors
Rieger & Zwicknagl [RZ10]
The interpolant I (u) is zero at the node set X (discreteterm goes away). Together with the optimality property‖I (u)‖N (Ω) ≤ ‖u‖N (Ω), we get for the Gaussian
‖Dα(I (u)− u)‖Lq(Ω) ≤ eC log(h)√
h ‖u‖N (Ω),
and for the inverse multiquadric
‖Dα(I (u)− u)‖Lq(Ω) ≤ eC√h ‖u‖N (Ω).
These estimates can be improved, e.g., for a compact cube.
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Cost of global method
Global RBF approximations of smooth functions are veryefficent.
A small number of node points per dimension are needed.
However N = 15 in 1-D becomes N = 50 625 in 4-D.
Up to three dimensions can be handled on a laptop, butnot more.
Furthermore, for less smooth functions, the number ofnodes per dimension grows quickly.
For a dense linear system: Direct solution O(N3), storageO(N2).
⇒ Move to localized methods.
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Motivation for RBF-PUM
Global RBF approximation
+ Ease of implementation in any dimension.
+ Flexibility with respect to geometry.
+ Potentially spectral convergence rates.
− Computationally expensive for large problems.
RBF partition of unity methods
I Local RBF approximations on patches are blendedinto a global solution using a partition of unity.
I Provides spectral or high-order convergence.
I Solves the computational cost issues.
I Allows for local adaptivity.[Wen02, Fas07, HL12, Cav12, CDR14, CDR15, CDRP16, CRP16],
[SVHL15, HLRvS16, SL16, LSH17]
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The RBF partition of unity method
Ωj
Global approximation
u(x) =P∑j=1
wj(x)uj(x)
PU weight functions
Generate weight functions fromcompactly supported C 2 Wendland functions
ψ(ρ) = (4ρ+ 1)(1− ρ)4+
using Shepard’s method wi (x) = ψi (x)∑Mj=1 ψj (x)
.
CoverEach x ∈ Ω must be in the interior of at least one Ωj .Patches that do not contain unique points are pruned.
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Differentiating RBF-PUM approximations
0 100 200 300
0
50
100
150
200
250
300
350
nz = 10801
Applying an operator globally
∆u =M∑i=1
∆wi ui + 2∇wi · ∇ui + wi∆ui
Local differentiation matricesLet ui be the vector of nodal values in patch Ωi , then
ui = Aλi , where Aij = φj(x i ) ⇒
Lui = ALA−1ui , where ALij = Lφj(x i ).
The global differentiation matrix
Local contributions are addedinto the global matrix.
E. Larsson, 2017-09-18 (49 : 77)
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demo6.m(Solving a Poisson problem in 2-D with RBF–PUM)
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An RBF-PUM collocation method
Choices & Implications
I Nodes and evaluation points coincide.Square matrix, iterative solver available (Heryudono, Larsson,
Ramage, von Sydow [HLRvS16]).
I Global node set.Solutions ui (xk) = uj(xk) for xk in overlap regions.
I Patches are cut by the domain boundary.Potentially strange shapes and lowered local order.
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An RBF-PUM least squares method
Choices & Implications
I Each patch has an identical node layout.Computational cost for setup is drastically reduced.
I Evaluation nodes are uniform.Easy to generate both local and global high quality node sets.
I Patches have nodes outside the domain.Good for local order, but requires denser evaluation points.
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The RBF-PUM interpolation error
Eα = Dα(I (u)− u) =M∑j=1
∑|β|≤|α|
(α
β
)DβwjD
α−β(I (uj)− uj)
The weight functions
For C k weight functions and |α| ≤ k
‖Dαwj‖L∞(Ωj ) ≤Cα
H|α|j
, Hj = diam(Ωj).
The local RBF interpolants (Gaussians)
Define the local fill distance hj (Rieger, Zwicknagl [RZ10])
‖Dα(I (uj)− uj)‖L∞(Ωj )≤ cα,jh
mj− d2−|α|
j ‖uj‖N (Ωj ),
‖Dα(I (uj)− uj)‖L∞(Ωj )≤ eγα,j log(hj )/
√hj‖uj‖N (Ωj )
.
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RBF-PUM interpolation error estimates
Algebraic estimate for Hj/hj = c
‖Eα‖L∞(Ω) ≤ K max1≤j≤M CjHmj− d
2−|α|
j ‖u‖N (Ωj )
K — Maximum # of Ωj overlapping at one pointmj — Related to the local # of pointsΩj — Ωj ∩ Ω
Spectral estimate for fixed partitions
‖Eα‖L∞(Ω) ≤ K max1≤j≤M Ceγj log(hj )/√
hj‖u‖N (Ωj )
Implications
I Bad patch reduces global order.
I Two refinement modes.
I Guidelines for adaptive refinement.
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Error estimate for PDE approximation
Larsson, Shcherbakov, Heryudono [LSH17]
The PDE estimate‖u − u‖L∞(Ω) ≤ CPEL + CP‖L·,XL+
Y ,X‖∞ (CMδM + EL),
where CP is a well-posedness constant and CMδM is asmall multiple of the machine precision.
Implications
I Interpolation error EL provides convergence rate.
I Norm of inverse/pseudoinverse can be large.
I Matrix norm better with oversampling.
I Finite precision accuracy limit involves matrix norm.
Follows strategies from Schaback [Sch07, Sch16]
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Does RBF-PUM require stable methods?
In order to achieve convergence we have two options
I Refine patches such that diameter H decreases.
I Increase node numbers such that Nj increases.
I In both cases, theory assumes ε fixed.
The effect of patch refinement
H = 1, ε = 4 H = 0.5, ε = 4 H = 0.25, ε = 4
0 H0
1
0 H0
1
0 H0
1
The RBF–QR method: Stable as ε → 0 for N 1Effectively a change to a stable basis.Fornberg, Piret [FP07], Fornberg, Larsson, Flyer [FLF11], Larsson,
Lehto, Heryudono, Fornberg [LLHF13]
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Effects on the local matrices
10−2
10−1
100
10−10
10−5
100
ε
N=10 N=20 N=40
Relative error in A∆j A−1j
without RBF-QR
Local contribution to a global Laplacian
Lj = (W∆j Aj + 2W∇
j A∇j + WjA∆j )A−1
j .
Typically: Aj ill-conditioned, Lj better conditioned.
RBF-QR for accuracy
I Stable for small RBFshape parameters ε
I Change of basisA = AQR−T1 D−T1
I Same result in theoryALA−1 = ALA−1
I More accurate in practice
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Poisson test problems in 2-D
Domain Ω = [−2, 2]2.
Uniform nodes in the collocation case.
uR(x , y) = 125x2+25y2+1
uT (x , y) = sin(2(x−0.1)2) cos((x−0.3)2)+sin2((y−0.5)2)
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Error results with and without RBF–QR
I Least squares RBF-PUM
I Fixed shape ε = 0.5 or scaled such that εh = c
I Left: 5× 5 patches Right: 55 points per patch
Spectral mode
20 40 60 80 100 120
10−6
10−4
10−2
100
Points per dimension
Err
or
RBF−QR
Direct
Scaled
Algebraic mode
4 8 16 32
10−8
10−6
10−4
10−2
100
Patches per dimensionE
rror
p=1
p=−7
RBF−QR
Direct
Scaled
I With RBF–QR better results for H/h large.
I Scaled approach good until saturation.E. Larsson, 2017-09-18 (59 : 77)
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Convergence as a function of patch size
Runge Trig
0.2 0.3 0.4 0.510
−6
10−4
10−2
100
H
Err
or
0.2 0.3 0.4 0.5
10−8
10−6
10−4
10−2
100
H
Err
or
Collocation (dashed lines) and Least Squares (solid lines).
I Points per patch n = 28, 55, 91.
I Theoretical rates p = 4, 7, 10.
I Numerical rates p ≈ 3.9, 6.9, 9.8.E. Larsson, 2017-09-18 (60 : 77)
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Spectral convergence for fixed patches
Runge Trig
−50 −40 −30 −2010
−8
10−6
10−4
10−2
100
−1/h
Err
or
−40 −30 −2010
−10
10−8
10−6
10−4
10−2
100
−1/h
Err
or
Collocation (dashed lines) and Least Squares (solid lines).
LS-RBF-PU is significantly more accurate due to theconstant number of nodes per patch.
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Robustness and large scale problems
The global error estimate
‖u − u‖L∞(Ω) ≤ CPEL + CP‖L·,XL+Y ,X‖∞ (CMδM + EL)
The dark horse is the ’stability matrix norm’
I The stability norm is related to conditioning.
I In the collocation case, ‖L−1X ,X‖ grows with N.
I How does it behave with least squares?
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Stability norm: Patch size
I Fixed number of points per patch n = 28, 55, 91
I Results as a function of patch diameter H
Stability(H) Error(H)
0.2 0.3 0.4 0.6
103
104
105
106
H
Sta
bili
ty n
orm
0.2 0.3 0.4 0.610
−10
10−8
10−6
10−4
10−2
100
H
Err
or
Collocation (dashed) and LS (solid)
I The norm does not grow for LS-RBF-PUM (!)E. Larsson, 2017-09-18 (63 : 77)
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RBF-generated finite differences RBF–FD
Flyer et al. [FW09, FLB+12]
I Approximate Lu(xc)using the n nearest nodes by
Lu(xc) ≈n∑
k=1
wku(xk)
I Find weights wk by askingexactness for RBF-interpolants.
φ1(x1) φ1(x2) · · · φ1(xn)φ2(x1) φ2(x2) · · · φ2(xn)
......
. . ....
φn(x1) φn(x2) · · · φn(xn)
w1
w2...wn
=
Lφ1(xc)Lφ2(xc)
...Lφn(xc)
.
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Is RBF-QR needed with RBF–FD?Approximation of ∆u with n = 56. Magenta lines arewith added polynomial terms p = 0, . . . , 3.
10−2
10−1
100
10−8
10−6
10−4
10−2
100
102
h
ℓ ∞error
ε = 1.5, direct
ε = 1.5, RBF-QR
ε = 0 (polynomial)
εh= 0.3, direct
I Scaled ε: No ill-conditioning, butsaturation/stagnation. [LLHF13]
I Fixed ε: RBF-QR is needed.
I Added terms: Compromise with partial recovery.E. Larsson, 2017-09-18 (65 : 77)
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RBF-FD with PHS and polynomials
Combine polyharmonic splines, e.g, φ(r) = |r |7 withpolynomial terms 1, x , y , . . . , x2, . . . such that the numberof polynomial terms ≈ the number of nodes.
I Contains both smooth and piecewise smoothcomponents, that have different roles in theapproximation.
I No shape parameter to tune.
I Heuristically, skewed stencils seem to behave wellnear boundaries.
Bayona, Flyer, Fornberg, Barnett [FFBB16, BFFB17]
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References I
Victor Bayona, Natasha Flyer, Bengt Fornberg, andGregory A. Barnett, On the role of polynomials inRBF-FD approximations: II. Numerical solution ofelliptic PDEs, J. Comput. Phys. 332 (2017),257–273.
Brad J. C. Baxter and Simon Hubbert, Radial basisfunctions for the sphere, Recent progress inmultivariate approximation (Witten-Bommerholz,2000), Internat. Ser. Numer. Math., vol. 137,Birkhauser, Basel, 2001, pp. 33–47. MR 1877496
Roberto Cavoretto, Partition of unity algorithm fortwo-dimensional interpolation using compactlysupported radial basis functions, Commun. Appl. Ind.Math. 3 (2012), no. 2, e–431, 13. MR 3063253
E. Larsson, 2017-09-18 (67 : 77)
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References II
Roberto Cavoretto and Alessandra De Rossi, Ameshless interpolation algorithm using a cell-basedsearching procedure, Comput. Math. Appl. 67(2014), no. 5, 1024–1038. MR 3166529
, A trivariate interpolation algorithm using acube-partition searching procedure, SIAM J. Sci.Comput. 37 (2015), no. 4, A1891–A1908. MR3376134
R. Cavoretto, A. De Rossi, and E. Perracchione,Efficient computation of partition of unityinterpolants through a block-based searchingtechnique, Comput. Math. Appl. 71 (2016), no. 12,2568–2584. MR 3504167
E. Larsson, 2017-09-18 (68 : 77)
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References III
Roberto Cavoretto, Alessandra De Rossi, and EmmaPerracchione, RBF-PU interpolation with variablesubdomain sizes and shape parameters, AIPConference Proceedings 1776 (2016), no. 1, 070003.
T. A. Driscoll and B. Fornberg, Interpolation in thelimit of increasingly flat radial basis functions,Comput. Math. Appl. 43 (2002), no. 3-5, 413–422,Radial basis functions and partial differentialequations. MR 1883576
Stefano De Marchi and Gabriele Santin, A new stablebasis for radial basis function interpolation, J.Comput. Appl. Math. 253 (2013), 1–13. MR3056591
E. Larsson, 2017-09-18 (69 : 77)
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References IV
, Fast computation of orthonormal basis forRBF spaces through Krylov space methods, BIT 55(2015), no. 4, 949–966. MR 3434025
Gregory E. Fasshauer, Meshfree approximationmethods with MATLAB, InterdisciplinaryMathematical Sciences, vol. 6, World ScientificPublishing Co. Pte. Ltd., Hackensack, NJ, 2007. MR2357267
Natasha Flyer, Bengt Fornberg, Victor Bayona, andGregory A. Barnett, On the role of polynomials inRBF-FD approximations: I. Interpolation andaccuracy, J. Comput. Phys. 321 (2016), 21–38. MR3527556
E. Larsson, 2017-09-18 (70 : 77)
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References V
Natasha Flyer, Erik Lehto, Sebastien Blaise, Grady B.Wright, and Amik St-Cyr, A guide to RBF-generatedfinite differences for nonlinear transport: Shallowwater simulations on a sphere, J. Comput. Phys. 231(2012), no. 11, 4078–4095. MR 2911786
Bengt Fornberg, Elisabeth Larsson, and NatashaFlyer, Stable computations with Gaussian radial basisfunctions, SIAM J. Sci. Comput. 33 (2011), no. 2,869–892. MR 2801193
Gregory E. Fasshauer and Michael J. McCourt,Stable evaluation of Gaussian radial basis functioninterpolants, SIAM J. Sci. Comput. 34 (2012), no. 2,A737–A762. MR 2914302
E. Larsson, 2017-09-18 (71 : 77)
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References VI
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