1. Introduction - IUST Personal Webpages

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Transcript 1. Introduction - IUST Personal Webpages

Interpolation By
Radial Basis Function
By: Reihane Khajepiri , Narges Gorji
Supervisor: Dr.Rabiei
1. Introduction
โ€ข Our problem is to interpolate the following tabular
Where the nodes ๐‘ฅ๐‘– โˆˆ ๐‘‹and ๐‘ข(๐‘ฅ๐‘– ) โˆˆ โ„›.
โ€ข The interpolation function has the form
๐‘ข(๐‘ฅ) โ‰ƒ ๐‘  ๐‘ฅ =
๐›ผ๐‘— ๐œ‘๐‘— (๐‘ฅ) ,
such that ๐œ‘๐‘— (๐‘ฅ)โ€™s are the basis of a prescribed ndimensional vector space of functions on ๐‘‹, i.e.,
๐›ค =< ๐œ‘1 , ๐œ‘2 , โ€ฆ , ๐œ‘๐‘š >.
โ€ข This is a system of ๐‘š linear equations in ๐‘š
unknowns. It can be written in matrix form as ๐ด๐œถ
= ๐’ƒ, or in details as
๐œ‘1 (๐‘ฅ1 ) ๐œ‘2 (๐‘ฅ1 )
๐œ‘๐‘š (๐‘ฅ1 ) ๐›ผ1
๐‘ข(๐‘ฅ1 )
๐œ‘1 (๐‘ฅ2 ) ๐œ‘2 (๐‘ฅ2 )
๐œ‘๐‘š (๐‘ฅ2 ) ๐›ผ2
๐‘ข(๐‘ฅ2 )
๐œ‘1 (๐‘ฅ๐‘š ) ๐œ‘2 (๐‘ฅ๐‘š ) โ‹ฏ ๐œ‘๐‘š (๐‘ฅ๐‘š ) ๐›ผ๐‘š
๐‘ข(๐‘ฅ๐‘š )
โ€ข The ๐‘š × ๐‘š 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‘š distinct nodes ๐‘ฅ๐‘– .
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.
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
achieve exponential accuracy where traditional
methods either have difficulties or fail.
โ€ข RBF methods are generalization of Multi Quadric RBF
, MQ RBF have a rich theoretical development.
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.
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
presented by Nelson &Madych.
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
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
Remaking Images By RBF
โ€ข It should be pointed out that meshfree local
independently in statistics for more than 100
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
3. RBF Method
3.1 Radial Function
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
radial. Radial functions are invariant under all Euclidean
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œ‘: [0, โˆž) โ†’ โ„› such that
๐›ท(๐‘ฅ) = ๐œ‘ (๐‘Ÿ) where r = ๐‘ฅ
And . is some norm on โ„› ๐‘  , usually the Euclidean norm.
โ€ข For a radial function ๐›ท :
๐‘ฅ1 =
๐‘ฅ2 ๏ƒ ๐›ท(๐‘ฅ1 ) = ๐›ท(๐‘ฅ2 )
๐‘ฅ1 , ๐‘ฅ2 โˆˆ 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œ‘for all choices of s.
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‘—=1 ๐‘˜=1 ๐‘๐‘— ๐‘๐‘˜ ๐ด๐‘—๐‘˜
[๐‘1 , โ€ฆ , ๐‘๐‘ ]๐‘‡ โˆˆ ๐‘…๐‘
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).
โ€ข Def: A real-valued continuous function ๐›ท is positive
definite on ๐‘… ๐‘  if & only if
๐‘˜=1 ๐‘๐‘— ๐‘๐‘˜ ๐›ท(๐‘ฅ๐‘—
โˆ’ ๐‘ฅ๐‘˜ ) โ‰ฅ 0
For any N pairwise different points ๐‘ฅ1 , ๐‘ฅ2 , โ€ฆ , ๐‘ฅ๐‘ โˆˆ 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.
3.3 Interpolation of scattered data
โ€ข One dimensional data polynomial &fourier interpolation
has the form:
S(x) = ๐œ†๐‘— ๐œ“๐‘— (๐‘ฅ)
๐œ“๐‘— : basis function
{๐‘ฅ๐‘— } : are
๐‘†(๐‘ฅ๐‘— ) = ๐‘“๐‘—
๐‘‘๐‘’๐‘ก๐‘’๐‘Ÿ๐‘š๐‘–๐‘›๐‘’ ๐œ†๐‘—
โ€ข For any set of basis function ๐œ“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.
โ€ข Theorem1. Let ฮ“ have the basis {๐œ‘1 , ๐œ‘2 , โ€ฆ , ๐œ‘๐‘š }. These
properties are equivalent:
a) ฮ“ is a Haar space.
b) det(๐œ‘๐‘— (๐‘ฅ๐‘– )) โ‰  0 for any set of distinct points
๐‘ฅ1 , ๐‘ฅ2 , โ€ฆ , ๐‘ฅ๐‘š โˆˆ 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‘š
2)๐‘’ ๐œ†1 ๐‘ฅ , ๐‘’ ๐œ†2 ๐‘ฅ , โ€ฆ , ๐‘’ ๐œ†๐‘š ๐‘ฅ ,
(๐œ†1 < ๐œ†2 < โ‹ฏ < ๐œ†๐‘š )
3)1, ๐‘๐‘œ๐‘ โ„Ž๐‘ฅ, ๐‘ ๐‘–๐‘›โ„Ž๐‘ฅ, โ€ฆ , ๐‘โ„Ž๐‘ โ„Ž๐‘š๐‘ฅ, ๐‘ ๐‘–๐‘›โ„Ž๐‘š๐‘ฅ
4)(๐‘ฅ + ๐œ†1 )โˆ’1 , (๐‘ฅ + ๐œ†2 )โˆ’1 , โ€ฆ ,(๐‘ฅ + ๐œ†๐‘› )โˆ’1 , 0 โ‰ค ๐œ†1 โ‰ค โ‹ฏ โ‰ค ๐œ†๐‘›
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œ†๐‘— |๐‘ฅ โˆ’ ๐‘ฅ๐‘— |
S (๐‘ฅ๐‘— ) = ๐‘“๐‘— ๏ƒ  ๐œ†๐‘— is determined
โ€ข Problems in Continuously differentiability of S ๐‘ฅ result
to the following form:
S ๐‘ฅ =
+ ๐‘ฅ โˆ’ ๐‘ฅ๐‘—
cโ‰ 0
Which is a linear combination of translate of the
hyperbola basis function.
Accurate representation of a topographic profile in
more than one dimensional space
S ๐‘ฅ, ๐‘ฆ =
๐‘ฅ โˆ’ ๐‘ฅ๐‘—
+ ๐‘ฆ โˆ’ ๐‘ฆ๐‘—
S ๐‘ฅ๐‘— , ๐‘ฆ๐‘— = ๐‘“๐‘—
S ๐‘ฅ, ๐‘ฆ =
S ๐‘ฅ, ๐‘ฆ =
+ ๐‘ฅ โˆ’ ๐‘ฅ๐‘—
+ ๐‘ฆ โˆ’ ๐‘ฆ๐‘—
๐œ†๐‘— ๐‘ 2 + ||๐ฑ โˆ’ ๐ฑ๐‘— ||2
S ๐‘ฅ๐‘— , ๐‘ฆ๐‘— = ๐‘“๐‘— ๏ƒ  ๐ด๐œ† = ๐‘“
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œ†๐‘— ๐œ‘ ๐‘ฅ โˆ’ ๐‘ฅ๐‘—
๐ด๐œ† = ๐‘“
= ๐œ‘( ๐‘ฅ๐‘— โˆ’ ๐‘ฅ๐‘˜ )
โ€ข Micchelli gave sufficient conditions for ฯ†(r) to guarantee that the
matrix A is unconditionally nonsingular.
๏ƒ  RBF method is uniquely solvable.
Def: A function ๐œ‘ โˆถ 0, โˆž โ†’ โ„› is completely
monotone on[0,โˆž] if :
1. ๐œ‘๐œ–๐ถ[0, โˆž)
2. ๐œ‘๐œ–๐ถ โˆž (0, โˆž)
3. โˆ’1 ๐‘™ ๐œ‘๐‘™ (๐‘Ÿ) โ‰ฅ 0; r>0 ;
Types of basis function:
โ€ข Infinitely smooth RBF
โ€ข Piecewise smooth RBF