Transcript 1. Introduction - IUST Personal Webpages

Interpolation By
Radial Basis Function
(RBF)
By: Reihane Khajepiri , Narges Gorji
Supervisor: Dr.Rabiei
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1. Introduction
• Our problem is to interpolate the following tabular
function:
Where the nodes 𝑥𝑖 ∈ 𝑋and 𝑢(𝑥𝑖 ) ∈ ℛ.
• The interpolation function has the form
𝑚
𝑢(𝑥) ≃ 𝑠 𝑥 =
𝛼𝑗 𝜑𝑗 (𝑥) ,
𝑗=1
such that 𝜑𝑗 (𝑥)’s are the basis of a prescribed ndimensional vector space of functions on 𝑋, i.e.,
𝛤 =< 𝜑1 , 𝜑2 , … , 𝜑𝑚 >.
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• This is a system of 𝑚 linear equations in 𝑚
unknowns. It can be written in matrix form as 𝐴𝜶
= 𝒃, or in details as
𝜑1 (𝑥1 ) 𝜑2 (𝑥1 )
𝜑𝑚 (𝑥1 ) 𝛼1
𝑢(𝑥1 )
⋯
𝜑1 (𝑥2 ) 𝜑2 (𝑥2 )
𝜑𝑚 (𝑥2 ) 𝛼2
𝑢(𝑥2 )
=
⋮
⋮
⋱
⋮
⋮
𝜑1 (𝑥𝑚 ) 𝜑2 (𝑥𝑚 ) ⋯ 𝜑𝑚 (𝑥𝑚 ) 𝛼𝑚
𝑢(𝑥𝑚 )
• The 𝑚 × 𝑚 matrix 𝐴 appearing here is called the
interpolation matrix.
• In order that our problem be solvable for any choice
of arbitrary 𝑢(𝑥𝑖 ), it is necessary and sufficient that
the interpolation matrix be nonsingular.
• The ideal situation is that this matrix be nonsingular
for all choices of 𝑚 distinct nodes 𝑥𝑖 .
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• Classical methods for numerical solution of PDEs
such as finite difference, finite elements, finite
volume, pseudo-spectral methods are base on
polynomial interpolation.
• Local polynomial based methods (finite
difference, finite elements and finite volume) are
limited by their algebraic convergence rate.
• MQ collection methods in comparison to finite
element method have superior accuracy.
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• Global polynomial methods such as spectral methods
have exponential convergence rate but are limited by
being tied to a fixed grid. Standard" multivariate
approximation methods (splines or finite elements)
require an underlying mesh (e.g., a triangulation) for
the definition of basis functions or elements. This is
usually rather difficult to accomplish in space
dimensions > 2.
• RBF method are not tied to a grid but to a category of
methods called meshless methods. The global non
polynomial RBF methods are successfully applied to
achieve exponential accuracy where traditional
methods either have difficulties or fail.
• RBF methods are generalization of Multi Quadric RBF
, MQ RBF have a rich theoretical development.
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2. Literature Review
• RBF developed by Iowa State university, Rolland
Hardy in 1968 for scattered data be easily used in
computations which polynomial interpolation has
failed in some cases. RBF present a topological
surface as well as other three dimensional shapes.
• In 1979 at Naval post graduate school Richard
Franke compared different methods to solve
scattered data interpolation problem and he
applied Hardy's MQ method and shows it is the
best approximation & also the matrix is invertible.
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• In 1986 Charles Micchelli a mathematician with IBM
proved the system matrix for MQ is invertible and
the theoretical basis began to develop. His approach
is based on conditionally positive definite functions.
• In 1990 the first use of MQ to solve PDE was
presented by Edward Kansa.
• In 1992 spectral convergence rate of MQ interpolation
presented by Nelson &Madych.
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• Originally, the motivation for two of the most common
basis meshfree approximation methods (radial basis
functions and moving least squares methods) came
from applications in geodesy, geophysics, mapping, or
meteorology.
• Later, applications were found in many areas such as
in the numerical solution of PDEs, artificial
intelligence, learning theory, neural networks, signal
processing, sampling theory, statistics ,finance, and
optimization.
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Remaking Images By RBF
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• It should be pointed out that meshfree local
regression
methods
have
been
used
independently in statistics for more than 100
years.
• Radial Basis Function "RBF" interpolate a multi
dimensional scattered data which easily
generalized to several space dimension &
provide spectral accuracy. So it is so popular
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3. RBF Method
3.1 Radial Function
• In many applications it is desirable to have invariance not
only under translation, but also under rotation & reflection.
This leads to positive definite functions which are also
radial. Radial functions are invariant under all Euclidean
transformations (translations, reflections & rotations)
• Def: A function 𝛷: ℛ 𝑠 → ℛ is called radial provided there exist
a univariate function 𝜑: [0, ∞) → ℛ such that
𝛷(𝑥) = 𝜑 (𝑟) where r = 𝑥
And . is some norm on ℛ 𝑠 , usually the Euclidean norm.
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• For a radial function 𝛷 :
𝑥1 =
𝑥2 𝛷(𝑥1 ) = 𝛷(𝑥2 )
𝑥1 , 𝑥2 ∈ ℛ𝑑 .
By radial functions the interpolation problem
becomes insensitive to the dimension s of the
space in which the data sites lie.
Instead of having to deal with a multivariate
function 𝛷 (whose complexity will increase with
increasing space dimension s) we can work with
the same univariate function 𝜑for all choices of s.
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3.2 Positive Definite Matrices & Functions
• Def : A real symmetric matrix A is called positive semidefinite if its associated quadratic form is non negative
For c=
𝑁
𝑁
𝑗=1 𝑘=1 𝑐𝑗 𝑐𝑘 𝐴𝑗𝑘
[𝑐1 , … , 𝑐𝑁 ]𝑇 ∈ 𝑅𝑁
≥0
(1)
• If the only vector c that turns (1) into an equality is
the zero vector , then A is called positive definite.
• An important property of positive definite matrices is
that all their eigenvalues are positive, and therefore a
positive definite matrix is non-singular (but certainly
not vice versa).
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• Def: A real-valued continuous function 𝛷 is positive
definite on 𝑅 𝑠 if & only if
𝑁
𝑗=1
𝑁
𝑘=1 𝑐𝑗 𝑐𝑘 𝛷(𝑥𝑗
− 𝑥𝑘 ) ≥ 0
(2)
For any N pairwise different points 𝑥1 , 𝑥2 , … , 𝑥𝑁 ∈ 𝑅 𝑠 &
c = [𝑐1 , … , 𝑐𝑁 ]𝑇
• The function 𝛷 is strictly positive definite on 𝑅 𝑠 if the
only vector c that turns (2) into an equality is the
zero vector.
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3.3 Interpolation of scattered data
• One dimensional data polynomial &fourier interpolation
has the form:
S(x) = 𝜆𝑗 𝜓𝑗 (𝑥)
𝜓𝑗 : basis function
{𝑥𝑗 } : are
distinct
𝑆(𝑥𝑗 ) = 𝑓𝑗
→
𝑑𝑒𝑡𝑒𝑟𝑚𝑖𝑛𝑒 𝜆𝑗
• For any set of basis function 𝜓𝑗 (𝑥) independent of data
points & sets of distinct data points {𝑥𝑗 } such that the linear
system of equation for determining the expansion
coefficient become non-singular {Haar theorem}
• Def: An n-dimensional vector space Γ of functions on a
domain 𝑋 is said to be a Haar space if the only element of
Γwhich has more than n-1 roots in 𝑋 is the zero element.
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• Theorem1. Let Γ have the basis {𝜑1 , 𝜑2 , … , 𝜑𝑚 }. These
properties are equivalent:
a) Γ is a Haar space.
b) det(𝜑𝑗 (𝑥𝑖 )) ≠ 0 for any set of distinct points
𝑥1 , 𝑥2 , … , 𝑥𝑚 ∈ 𝑋.
Any basis for a Haar space is called Chebyshev system.
• A Haar space is a space of functions that guarantees
invertibility of the interpolation matrix.
• Some examples of Chebyshev systems on ℝ:
1)1, 𝑥, 𝑥 2 , … , 𝑥 𝑚
2)𝑒 𝜆1 𝑥 , 𝑒 𝜆2 𝑥 , … , 𝑒 𝜆𝑚 𝑥 ,
(𝜆1 < 𝜆2 < ⋯ < 𝜆𝑚 )
3)1, 𝑐𝑜𝑠ℎ𝑥, 𝑠𝑖𝑛ℎ𝑥, … , 𝑐ℎ𝑠ℎ𝑚𝑥, 𝑠𝑖𝑛ℎ𝑚𝑥
4)(𝑥 + 𝜆1 )−1 , (𝑥 + 𝜆2 )−1 , … ,(𝑥 + 𝜆𝑛 )−1 , 0 ≤ 𝜆1 ≤ ⋯ ≤ 𝜆𝑛
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3-4 development of RBF method
• Hardy present RBF which is linear combination of
translate of a single basis function that is radially
symmetric a bout its center.
𝑆(𝑥) =
𝜆𝑗 |𝑥 − 𝑥𝑗 |
S (𝑥𝑗 ) = 𝑓𝑗  𝜆𝑗 is determined
• Problems in Continuously differentiability of S 𝑥 result
to the following form:
S 𝑥 =
𝜆𝑗
𝑐2
+ 𝑥 − 𝑥𝑗
2
c≠0
Which is a linear combination of translate of the
hyperbola basis function.
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Accurate representation of a topographic profile in
more than one dimensional space
S 𝑥, 𝑦 =
𝜆𝑗
𝑥 − 𝑥𝑗
2
+ 𝑦 − 𝑦𝑗
2
S 𝑥𝑗 , 𝑦𝑗 = 𝑓𝑗
S 𝑥, 𝑦 =
𝜆𝑗
S 𝑥, 𝑦 =
𝑐2
+ 𝑥 − 𝑥𝑗
2
+ 𝑦 − 𝑦𝑗
2
𝜆𝑗 𝑐 2 + ||𝐱 − 𝐱𝑗 ||2
S 𝑥𝑗 , 𝑦𝑗 = 𝑓𝑗  𝐴𝜆 = 𝑓
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3.5 RBF categories:
• We have two kinds of RBF methods
1. Basis RBF
2. Augmented RBF
• RBF basic methods:
n distinct data points {𝑥𝑗 } & corresponding data values {𝑓𝑗 },
• The basic RBF interpolant is given by
𝑆 𝑥 =
𝑎𝑗𝑘
𝜆𝑗 𝜑 𝑥 − 𝑥𝑗
𝐴𝜆 = 𝑓
= 𝜑( 𝑥𝑗 − 𝑥𝑘 )
• Micchelli gave sufficient conditions for φ(r) to guarantee that the
matrix A is unconditionally nonsingular.
 RBF method is uniquely solvable.
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Def: A function 𝜑 ∶ 0, ∞ → ℛ is completely
monotone on[0,∞] if :
1. 𝜑𝜖𝐶[0, ∞)
2. 𝜑𝜖𝐶 ∞ (0, ∞)
3. −1 𝑙 𝜑𝑙 (𝑟) ≥ 0; r>0 ;
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Types of basis function:
• Infinitely smooth RBF
• Piecewise smooth RBF
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