The RBF-QR method and its applications—A tutorial in two parts

56 Pages·2015·7.91 MB·English

The RBF-QR method and its
RBF–QR applications—A tutorial in two parts
Outline
Intro and motivation
RBF limits
Elisabeth Larsson
Contour-Pad´e
Expansions with thanks to numerous co-investigators and
RBF–QR methods colleagues
RBF-QR and PDEs
RBF–PUM
Division of Scientific Computing
Department of Information Technology
Uppsala University
Dolomites Research Week on Approximation 2015
E. Larsson, DRWA15 (1 : 56)
Inofficial competition
Produce the most beautiful picture by modifying the RBF-QR
demo MATLAB codes. The winner can get a copy of the
English version of this book or eternal glory. . .
RBF–QR
Outline
Intro and motivation
RBF limits
Contour-Pad´e
Expansions
RBF–QR methods
RBF-QR and PDEs
RBF–PUM
Head over Heels—Seventeen women scientist’s thoughts on
shoes.
E. Larsson, DRWA15 (2 : 56)
Outline
Introduction and motivation
RBF–QR
Outline RBF limits
Intro and motivation
RBF limits
The Contour-Pad´e method
Contour-Pad´e
Expansions
RBF–QR methods
Expansions
RBF-QR and PDEs
RBF–PUM
RBF–QR methods
RBF-QR and PDEs
RBF partition of unity methods for PDEs
E. Larsson, DRWA15 (3 : 56)
ε=1 ε=1/3 ε=3
Short introduction to (global) RBF methods
Basis functions: φj (x) = φ(‖x − xj‖). Translates of one
single function rotated around a center point.
RBF–QR
Example: Gaussians
Outline 2 2
φ(εr) = exp(−ε r )
Intro and motivation
RBF limits
Approximation:
Contour-Pad´e ∑
N
Expansions sε(x ) = j=1 λjφj (x )
RBF–QR methods
RBF-QR and PDEs Collocation:
RBF–PUM sε(x
i ) = fi ⇒ Aλ = f
Advantages:
• Flexibility with respect to geometry.
• As easy in d dimensions.
• Spectral accuracy / exponential convergence.
• Continuosly differentiable approximation.
E. Larsson, DRWA15 (4 : 56)
RBF–QR
Outline
Intro and motivation
RBF limits demo1.m
Contour-Pad´e (RBF interpolation in 1-D)
Expansions
RBF–QR methods
RBF-QR and PDEs
RBF–PUM
E. Larsson, DRWA15 (5 : 56)
ε = 1 ε = 1
5
−5
10
0
−5
Observations from the results of demo1.m
10 20 30 10 20 30
◮ As N grows for fixed ε, convergence stagnates.
N N
◮ As ε decreases for fixed N, the error blows up.
N = 30 N = 30
RBF–QR ◮ λmin = −λmax means cancellation.
Outline
◮ Coefficients λ → ∞ means that cond(A) → ∞.
Intro and motivation
RBF limits 10
Contour-Pad´e ◮
For small ε, the RBFs are nearly flat, and almost
Expansions
RBF–QR methods linearly dependent. That is, they form a bad basis.
RBF-QR and PDEs
RBF–PUM −5 0
10
−10
0 0
10 10
ε ε
E. Larsson, DRWA15 (6 : 56)
Max error Max error
log (max/min(λ))
10 log (max/min(λ))
10
Why is it interesting to use small values of ε?
Driscoll & Fornberg 2002
Somewhat surprisingly, in 1-D for small ε
RBF–QR 2 4
s(x, ε) = PN−1(x) + ε PN+1(x) + ε PN+3(x) + · · · ,
Outline
Intro and motivation
RBF limits where Pj is a polynomial of degree j and PN−1(x) is the
Contour-Pad´e
Lagrange interpolant.
Expansions
RBF–QR methods Implications
RBF-QR and PDEs
RBF–PUM
◮ It can be shown that cond(A) ∼ O(Nε−2(N−1)), but
the limit interpolant is well behaved.
◮ It is the intermediate step of computing λ that is
ill-conditioned.
◮ By choosing the corresponding nodes, the flat RBF
limit reproduces pseudo-spectral methods.
◮ This is a good approximation space.
E. Larsson, DRWA15 (7 : 56)
The multivariate flat RBF limit
Larsson & Fornberg 2005, Schaback 2005
In n-D the flat limit can either be
RBF–QR 2 4
s(x, ε) = PK(x) + ε PK+2(x) + ε PK+4(x) + · · · ,
Outline
Intro and motivation ( ) ( )
(K − 1) + d K + d
RBF limits where < N ≤ and P
K is a
Contour-Pad´e d d
Expansions polynomial interpolant or
RBF–QR methods
RBF-QR and PDEs s(x, ε) = ε−2qP
M−2q(x) + ε−2q+2PM−2q+2(x) + · · ·
RBF–PUM
2 4
+ PM(x) + ε PM+2(x) + ε PM+4(x) + · · · .
The questions of uniqueness and existence are connected
with multivariate polynomial uni-solvency.
Schaback 2005
Gaussian RBF limit interpolants always converge to the
de Boor/Ron least polynomial interpolant.
E. Larsson, DRWA15 (8 : 56)
The multivariate flat RBF limit: Divergence
Necessary condition: ∃ Q(x) of degree N0 such that
Q(xj ) = 0, j = 1, . . . , N.
−2q
Then divergence as ε may occur, where
RBF–QR
Outline q = ⌊(M − N0)/2⌋ and M = min non-degenerate degree.
Intro and motivation
RBF limits Points Q N0 Basis M q
2
Contour-Pad´e x − y 1 1, x, x , 5 2
Expansions x3, x4, x5
RBF–QR methods
RBF-QR and PDEs
2
RBF–PUM x − y − 1 2 1, x, y , xy , 3 0
2 2
y xy
2 2 2
x + y − 1 2 1, x, y , x , xy , 4 1
3 2 4
x , x y , x
−2
Divergence actually only occurs for the first case as ε .
E. Larsson, DRWA15 (9 : 56)
The multivariate flat RBF limit, contd
Schaback 2005, Fornberg & Larsson 2005
Example: In two dimensions, the eigenvalues of A follow
0 2 4
RBF–QR a pattern: µ1 ∼ O(ε ), µ2,3 ∼ O(ε ), µ4,5,6 ∼ O(ε ),. . .
Outline ( )
Intro and motivation k + n − 1 (k+1)···(k+n−1)
In general, there are =
RBF limits n − 1 (n−1)!
Contour-Pad´e 2k
eigenvalues µj ∼ O(ε ) in n dimensions.
Expansions
RBF–QR methods
RBF-QR and PDEs
Implications
RBF–PUM
◮ There is an opportunity for pseudo-spectral-like
methods in n-D.
◮ There is no amount of variable precision that will
save us.
◮ For “smooth” functions, a small ε can lead to very
high accuracy.
E. Larsson, DRWA15 (10 : 56)