#### Transcript 1. Introduction - IUST Personal Webpages

```Interpolation By
(RBF)
By: Reihane Khajepiri , Narges Gorji
Supervisor: Dr.Rabiei
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1. Introduction
โข Our problem is to interpolate the following tabular
function:
Where the nodes ๐ฅ๐ โ ๐and ๐ข(๐ฅ๐ ) โ โ.
โข The interpolation function has the form
๐
๐ข(๐ฅ) โ ๐  ๐ฅ =
๐ผ๐ ๐๐ (๐ฅ) ,
๐=1
such that ๐๐ (๐ฅ)โs are the basis of a prescribed ndimensional vector space of functions on ๐, i.e.,
๐ค =< ๐1 , ๐2 , โฆ , ๐๐ >.
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โข This is a system of ๐ linear equations in ๐
unknowns. It can be written in matrix form as ๐ด๐ถ
= ๐, or in details as
๐1 (๐ฅ1 ) ๐2 (๐ฅ1 )
๐๐ (๐ฅ1 ) ๐ผ1
๐ข(๐ฅ1 )
โฏ
๐1 (๐ฅ2 ) ๐2 (๐ฅ2 )
๐๐ (๐ฅ2 ) ๐ผ2
๐ข(๐ฅ2 )
=
โฎ
โฎ
โฑ
โฎ
โฎ
๐1 (๐ฅ๐ ) ๐2 (๐ฅ๐ ) โฏ ๐๐ (๐ฅ๐ ) ๐ผ๐
๐ข(๐ฅ๐ )
โข The ๐ × ๐ matrix ๐ด appearing here is called the
interpolation matrix.
โข In order that our problem be solvable for any choice
of arbitrary ๐ข(๐ฅ๐ ), it is necessary and sufficient that
the interpolation matrix be nonsingular.
โข The ideal situation is that this matrix be nonsingular
for all choices of ๐ distinct nodes ๐ฅ๐ .
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โข Classical methods for numerical solution of PDEs
such as finite difference, finite elements, finite
volume, pseudo-spectral methods are base on
polynomial interpolation.
โข Local polynomial based methods (finite
difference, finite elements and finite volume) are
limited by their algebraic convergence rate.
โข MQ collection methods in comparison to finite
element method have superior accuracy.
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โข 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.
โข 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
methods either have difficulties or fail.
โข RBF methods are generalization of Multi Quadric RBF
, MQ RBF have a rich theoretical development.
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2. Literature Review
โข 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.
โข 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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โข 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.
โข In 1990 the first use of MQ to solve PDE was
presented by Edward Kansa.
โข In 1992 spectral convergence rate of MQ interpolation
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โข 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.
โข 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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โข It should be pointed out that meshfree local
regression
methods
have
been
used
independently in statistics for more than 100
years.
โข 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
โข 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
transformations (translations, reflections & rotations)
โข Def: A function ๐ท: โ ๐  โ โ is called radial provided there exist
a univariate function ๐: [0, โ) โ โ such that
๐ท(๐ฅ) = ๐ (๐) where r = ๐ฅ
And . is some norm on โ ๐  , usually the Euclidean norm.
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โข For a radial function ๐ท :
๐ฅ1 =
๐ฅ2 ๏ ๐ท(๐ฅ1 ) = ๐ท(๐ฅ2 )
๐ฅ1 , ๐ฅ2 โ โ๐ .
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 ๐ท (whose complexity will increase with
increasing space dimension s) we can work with
the same univariate function ๐for all choices of s.
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3.2 Positive Definite Matrices & Functions
โข Def : A real symmetric matrix A is called positive semidefinite if its associated quadratic form is non negative
For c=
๐
๐
๐=1 ๐=1 ๐๐ ๐๐ ๐ด๐๐
[๐1 , โฆ , ๐๐ ]๐ โ ๐๐
โฅ0
(1)
โข If the only vector c that turns (1) into an equality is
the zero vector , then A is called positive definite.
โข 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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โข Def: A real-valued continuous function ๐ท is positive
definite on ๐ ๐  if & only if
๐
๐=1
๐
๐=1 ๐๐ ๐๐ ๐ท(๐ฅ๐
โ ๐ฅ๐ ) โฅ 0
(2)
For any N pairwise different points ๐ฅ1 , ๐ฅ2 , โฆ , ๐ฅ๐ โ ๐ ๐  &
c = [๐1 , โฆ , ๐๐ ]๐
โข The function ๐ท is strictly positive definite on ๐ ๐  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
โข One dimensional data polynomial &fourier interpolation
has the form:
S(x) = ๐๐ ๐๐ (๐ฅ)
๐๐ : basis function
{๐ฅ๐ } : are
distinct
๐(๐ฅ๐ ) = ๐๐
โ
๐๐๐ก๐๐๐๐๐๐ ๐๐
โข For any set of basis function ๐๐ (๐ฅ) independent of data
points & sets of distinct data points {๐ฅ๐ } such that the linear
system of equation for determining the expansion
coefficient become non-singular {Haar theorem}
โข Def: An n-dimensional vector space ฮ of functions on a
domain ๐ is said to be a Haar space if the only element of
ฮwhich has more than n-1 roots in ๐ is the zero element.
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โข Theorem1. Let ฮ have the basis {๐1 , ๐2 , โฆ , ๐๐ }. These
properties are equivalent:
a) ฮ is a Haar space.
b) det(๐๐ (๐ฅ๐ )) โ  0 for any set of distinct points
๐ฅ1 , ๐ฅ2 , โฆ , ๐ฅ๐ โ ๐.
Any basis for a Haar space is called Chebyshev system.
โข A Haar space is a space of functions that guarantees
invertibility of the interpolation matrix.
โข Some examples of Chebyshev systems on โ:
1)1, ๐ฅ, ๐ฅ 2 , โฆ , ๐ฅ ๐
2)๐ ๐1 ๐ฅ , ๐ ๐2 ๐ฅ , โฆ , ๐ ๐๐ ๐ฅ ,
(๐1 < ๐2 < โฏ < ๐๐ )
3)1, ๐๐๐ โ๐ฅ, ๐ ๐๐โ๐ฅ, โฆ , ๐โ๐ โ๐๐ฅ, ๐ ๐๐โ๐๐ฅ
4)(๐ฅ + ๐1 )โ1 , (๐ฅ + ๐2 )โ1 , โฆ ,(๐ฅ + ๐๐ )โ1 , 0 โค ๐1 โค โฏ โค ๐๐
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3-4 development of RBF method
โข Hardy present RBF which is linear combination of
translate of a single basis function that is radially
symmetric a bout its center.
๐(๐ฅ) =
๐๐ |๐ฅ โ ๐ฅ๐ |
S (๐ฅ๐ ) = ๐๐ ๏  ๐๐ is determined
โข Problems in Continuously differentiability of S ๐ฅ result
to the following form:
S ๐ฅ =
๐๐
๐2
+ ๐ฅ โ ๐ฅ๐
2
cโ 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 ๐ฅ, ๐ฆ =
๐๐
๐ฅ โ ๐ฅ๐
2
+ ๐ฆ โ ๐ฆ๐
2
S ๐ฅ๐ , ๐ฆ๐ = ๐๐
S ๐ฅ, ๐ฆ =
๐๐
S ๐ฅ, ๐ฆ =
๐2
+ ๐ฅ โ ๐ฅ๐
2
+ ๐ฆ โ ๐ฆ๐
2
๐๐ ๐ 2 + ||๐ฑ โ ๐ฑ๐ ||2
S ๐ฅ๐ , ๐ฆ๐ = ๐๐ ๏  ๐ด๐ = ๐
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3.5 RBF categories:
โข We have two kinds of RBF methods
1. Basis RBF
2. Augmented RBF
โข RBF basic methods:
n distinct data points {๐ฅ๐ } & corresponding data values {๐๐ },
โข The basic RBF interpolant is given by
๐ ๐ฅ =
๐๐๐
๐๐ ๐ ๐ฅ โ ๐ฅ๐
๐ด๐ = ๐
= ๐( ๐ฅ๐ โ ๐ฅ๐ )
โข Micchelli gave sufficient conditions for ฯ(r) to guarantee that the
matrix A is unconditionally nonsingular.
๏  RBF method is uniquely solvable.
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Def: A function ๐ โถ 0, โ โ โ is completely
monotone on[0,โ] if :
1. ๐๐๐ถ[0, โ)
2. ๐๐๐ถ โ (0, โ)
3. โ1 ๐ ๐๐ (๐) โฅ 0; r>0 ;
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Types of basis function:
โข Infinitely smooth RBF
โข Piecewise smooth RBF
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