MATH 590: Meshfree Methods“Flat” Limits of Kernel Interpolants

Greg Fasshauer

Department of Applied MathematicsIllinois Institute of Technology

Fall 2014

fasshauer@iit.edu MATH 590 1

Outline

1 Introduction

2 Infinitely Smooth RBFs

3 RBFs with Finite Smoothness

fasshauer@iit.edu MATH 590 2

Introduction

Outline

1 Introduction

2 Infinitely Smooth RBFs

3 RBFs with Finite Smoothness

fasshauer@iit.edu MATH 590 3

Introduction

A “flat” limit seems to exist for interpolation with C2 Wendlandkernels on [0,1] based on N uniformly spaced points with εvarying from 10−3 to 100.

Does this happen for other kernels?Does it happen for all kernels?What is this limit?

fasshauer@iit.edu MATH 590 4

Introduction

A “flat” limit seems to exist for interpolation with C2 Wendlandkernels on [0,1] based on N uniformly spaced points with εvarying from 10−3 to 100.Does this happen for other kernels?

Does it happen for all kernels?What is this limit?

fasshauer@iit.edu MATH 590 4

Introduction

A “flat” limit seems to exist for interpolation with C2 Wendlandkernels on [0,1] based on N uniformly spaced points with εvarying from 10−3 to 100.Does this happen for other kernels?Does it happen for all kernels?

What is this limit?

fasshauer@iit.edu MATH 590 4

Introduction

A “flat” limit seems to exist for interpolation with C2 Wendlandkernels on [0,1] based on N uniformly spaced points with εvarying from 10−3 to 100.Does this happen for other kernels?Does it happen for all kernels?What is this limit?

fasshauer@iit.edu MATH 590 4

Introduction

About 10 years ago an interesting connection was discovered betweeninterpolants based on infinitely smooth RBFs such as Gaussians,generalized (inverse) multiquadrics and the oscillatory kernels1

K (x , z) =Jβ/2−1(ε‖x − z‖)ε‖x − z‖dβ/2−1 , β ≥ d

In many cases the limiting (“flat”) RBF interpolants were identicalto polynomial interpolants — especially in 1D experiments.

(see, e.g., [DF02, FF05, FW04, LF03, LF05])

1For fixed z these are fundamental solutions, bounded at the origin, of thed-dimensional Helmholtz operator in spherical coordinates (see [FLW06], where theyare called Bessel kernels, see also [Fas07, Ch.4], where they are referred to asPoisson functions).

fasshauer@iit.edu MATH 590 5

Introduction

About 10 years ago an interesting connection was discovered betweeninterpolants based on infinitely smooth RBFs such as Gaussians,generalized (inverse) multiquadrics and the oscillatory kernels1

K (x , z) =Jβ/2−1(ε‖x − z‖)ε‖x − z‖dβ/2−1 , β ≥ d

In many cases the limiting (“flat”) RBF interpolants were identicalto polynomial interpolants — especially in 1D experiments.

(see, e.g., [DF02, FF05, FW04, LF03, LF05])

1For fixed z these are fundamental solutions, bounded at the origin, of thed-dimensional Helmholtz operator in spherical coordinates (see [FLW06], where theyare called Bessel kernels, see also [Fas07, Ch.4], where they are referred to asPoisson functions).

fasshauer@iit.edu MATH 590 5

Infinitely Smooth RBFs

Outline

1 Introduction

2 Infinitely Smooth RBFs

3 RBFs with Finite Smoothness

fasshauer@iit.edu MATH 590 6

Infinitely Smooth RBFs

In [DF02] univariate (d = 1) interpolation with ε-scaled infinitelysmooth radial kernels was studied.

Driscoll and Fornberg show that the RBF interpolant

sε(x) =N∑

j=1

cjκ(‖ε(x − xj)‖), x ∈ [a,b] ⊂ R,

to function values at N distinct data sites tends to the Lagrangeinterpolating polynomial of f as ε→ 0.

Run FlatGaussian.cdf

RemarkIn order to fully exploit this relationship it will be necessary to developstable evaluation algorithms for “flat” kernels.

fasshauer@iit.edu MATH 590 7

Infinitely Smooth RBFs

In [DF02] univariate (d = 1) interpolation with ε-scaled infinitelysmooth radial kernels was studied.

Driscoll and Fornberg show that the RBF interpolant

sε(x) =N∑

j=1

cjκ(‖ε(x − xj)‖), x ∈ [a,b] ⊂ R,

to function values at N distinct data sites tends to the Lagrangeinterpolating polynomial of f as ε→ 0.

Run FlatGaussian.cdf

RemarkIn order to fully exploit this relationship it will be necessary to developstable evaluation algorithms for “flat” kernels.

fasshauer@iit.edu MATH 590 7

Infinitely Smooth RBFs

In [DF02] univariate (d = 1) interpolation with ε-scaled infinitelysmooth radial kernels was studied.

Driscoll and Fornberg show that the RBF interpolant

sε(x) =N∑

j=1

cjκ(‖ε(x − xj)‖), x ∈ [a,b] ⊂ R,

to function values at N distinct data sites tends to the Lagrangeinterpolating polynomial of f as ε→ 0.

Run FlatGaussian.cdf

RemarkIn order to fully exploit this relationship it will be necessary to developstable evaluation algorithms for “flat” kernels.

fasshauer@iit.edu MATH 590 7

Infinitely Smooth RBFs

In [DF02] univariate (d = 1) interpolation with ε-scaled infinitelysmooth radial kernels was studied.

Driscoll and Fornberg show that the RBF interpolant

sε(x) =N∑

j=1

cjκ(‖ε(x − xj)‖), x ∈ [a,b] ⊂ R,

to function values at N distinct data sites tends to the Lagrangeinterpolating polynomial of f as ε→ 0.

Run FlatGaussian.cdf

RemarkIn order to fully exploit this relationship it will be necessary to developstable evaluation algorithms for “flat” kernels.

fasshauer@iit.edu MATH 590 7

Infinitely Smooth RBFs

The “flat” polynomial limit for Gaussian interpolation, stably computedwith the algorithm from [FM12].

fasshauer@iit.edu MATH 590 8

Infinitely Smooth RBFs

The multivariate case is more complicated.

RemarkNote that most of the following results are limited to radial kernels, i.e.,radial basis functions (RBFs).

The limiting RBF interpolant is given by a low-degree multivariatepolynomial (see [Boo06, LF05, LYY07, Sch05, Sch08]).

For example, if the data sites are located such that they guaranteea unique polynomial interpolant, then the limiting RBF interpolant isgiven by this polynomial.If polynomial interpolation is not unique, then the RBF limit is still apolynomial whose form depends on the choice of basic function.

Lee and Micchelli [LM13] recently showed that in the multivariatesetting, when the interpolation points that are unisolvent ford-variate polynomials of total degree `, there is also a uniquepolynomial limiting interpolant for a given (not necessarily radial)kernel, provided it is analytic.

fasshauer@iit.edu MATH 590 9

Infinitely Smooth RBFs

The multivariate case is more complicated.

RemarkNote that most of the following results are limited to radial kernels, i.e.,radial basis functions (RBFs).

The limiting RBF interpolant is given by a low-degree multivariatepolynomial (see [Boo06, LF05, LYY07, Sch05, Sch08]).

For example, if the data sites are located such that they guaranteea unique polynomial interpolant, then the limiting RBF interpolant isgiven by this polynomial.If polynomial interpolation is not unique, then the RBF limit is still apolynomial whose form depends on the choice of basic function.

Lee and Micchelli [LM13] recently showed that in the multivariatesetting, when the interpolation points that are unisolvent ford-variate polynomials of total degree `, there is also a uniquepolynomial limiting interpolant for a given (not necessarily radial)kernel, provided it is analytic.

fasshauer@iit.edu MATH 590 9

Infinitely Smooth RBFs

The multivariate case is more complicated.

RemarkNote that most of the following results are limited to radial kernels, i.e.,radial basis functions (RBFs).

The limiting RBF interpolant is given by a low-degree multivariatepolynomial (see [Boo06, LF05, LYY07, Sch05, Sch08]).

For example, if the data sites are located such that they guaranteea unique polynomial interpolant, then the limiting RBF interpolant isgiven by this polynomial.

If polynomial interpolation is not unique, then the RBF limit is still apolynomial whose form depends on the choice of basic function.

Lee and Micchelli [LM13] recently showed that in the multivariatesetting, when the interpolation points that are unisolvent ford-variate polynomials of total degree `, there is also a uniquepolynomial limiting interpolant for a given (not necessarily radial)kernel, provided it is analytic.

fasshauer@iit.edu MATH 590 9

Infinitely Smooth RBFs

The multivariate case is more complicated.

RemarkNote that most of the following results are limited to radial kernels, i.e.,radial basis functions (RBFs).

The limiting RBF interpolant is given by a low-degree multivariatepolynomial (see [Boo06, LF05, LYY07, Sch05, Sch08]).

For example, if the data sites are located such that they guaranteea unique polynomial interpolant, then the limiting RBF interpolant isgiven by this polynomial.If polynomial interpolation is not unique, then the RBF limit is still apolynomial whose form depends on the choice of basic function.

Lee and Micchelli [LM13] recently showed that in the multivariatesetting, when the interpolation points that are unisolvent ford-variate polynomials of total degree `, there is also a uniquepolynomial limiting interpolant for a given (not necessarily radial)kernel, provided it is analytic.

fasshauer@iit.edu MATH 590 9

Infinitely Smooth RBFs

The multivariate case is more complicated.

RemarkNote that most of the following results are limited to radial kernels, i.e.,radial basis functions (RBFs).

The limiting RBF interpolant is given by a low-degree multivariatepolynomial (see [Boo06, LF05, LYY07, Sch05, Sch08]).

For example, if the data sites are located such that they guaranteea unique polynomial interpolant, then the limiting RBF interpolant isgiven by this polynomial.If polynomial interpolation is not unique, then the RBF limit is still apolynomial whose form depends on the choice of basic function.

Lee and Micchelli [LM13] recently showed that in the multivariatesetting, when the interpolation points that are unisolvent ford-variate polynomials of total degree `, there is also a uniquepolynomial limiting interpolant for a given (not necessarily radial)kernel, provided it is analytic.

fasshauer@iit.edu MATH 590 9

Infinitely Smooth RBFs

Theorem (Driscoll, Fornberg, Larsson, Schaback, Yoon [2002-08])

Assume the strictly positive definite radial kernel κ has an expansion

κ(r) =∞∑

j=0

aj r2j

into even powers of r (i.e., κ is infinitely smooth), and that the data Xare unisolvent with respect to any set of N linearly independentpolynomials of degree at most m. Then

limε→0

sε(x) = pm(x), x ∈ Rs,

where pm is determined as follows:If interpolation with polynomials of degree at most m is unique,then pm is that unique polynomial interpolant.If interpolation with polynomials of degree at most m is not unique,then pm is a polynomial interpolant whose form depends on thechoice of RBF.

fasshauer@iit.edu MATH 590 10

Infinitely Smooth RBFs

Some Examples

Inverse quadratic

κ(εr) =1

1 + ε2r2 = 1− (εr)2 + (εr)4 − (εr)6 + (εr)8 + · · ·

Gaussian

κ(εr) = e−ε2r2

= 1− (εr)2 +(εr)4

2− (εr)6

6+

(εr)8

24+ · · ·

Inverse MQ

κ(εr) =1√

1 + ε2r2= 1− (εr)2

2+

3(εr)4

8− 5(εr)6

16+

35(εr)8

128+ · · ·

Poisson, β = d = 2

κ(εr) = J0(εr) = 1− (εr)2

4+

(εr)4

64− (εr)6

2304+

(εr)8

147456+ · · ·

fasshauer@iit.edu MATH 590 11

Infinitely Smooth RBFs

RemarkThe statements in the theorem require the RBFs to satisfy acondition on certain coefficient matrices Ap,J . This condition wasleft unproven in [LF05] and verified in [LYY07].

For the special case of Gaussians [Sch05] shows that as ε→ 0the RBF interpolant converges to the de Boor and Ron leastpolynomial interpolant (see [Boo92, BR90, BR92] and also[Boo06]).In [LF05] the authors use Taylor expansions to also provide anexplanation for the error behavior for small values of the shapeparameter, and for the existence of an optimal (positive) value of εgiving rise to a global minimum of the error function.

fasshauer@iit.edu MATH 590 12

Infinitely Smooth RBFs

RemarkThe statements in the theorem require the RBFs to satisfy acondition on certain coefficient matrices Ap,J . This condition wasleft unproven in [LF05] and verified in [LYY07].For the special case of Gaussians [Sch05] shows that as ε→ 0the RBF interpolant converges to the de Boor and Ron leastpolynomial interpolant (see [Boo92, BR90, BR92] and also[Boo06]).

In [LF05] the authors use Taylor expansions to also provide anexplanation for the error behavior for small values of the shapeparameter, and for the existence of an optimal (positive) value of εgiving rise to a global minimum of the error function.

fasshauer@iit.edu MATH 590 12

Infinitely Smooth RBFs

RemarkThe statements in the theorem require the RBFs to satisfy acondition on certain coefficient matrices Ap,J . This condition wasleft unproven in [LF05] and verified in [LYY07].For the special case of Gaussians [Sch05] shows that as ε→ 0the RBF interpolant converges to the de Boor and Ron leastpolynomial interpolant (see [Boo92, BR90, BR92] and also[Boo06]).In [LF05] the authors use Taylor expansions to also provide anexplanation for the error behavior for small values of the shapeparameter, and for the existence of an optimal (positive) value of εgiving rise to a global minimum of the error function.

fasshauer@iit.edu MATH 590 12

Infinitely Smooth RBFs

RemarkThe work by Narayan and Xiu [NX12] suggests that a connectionmay be found between other unique multivariate polynomialinterpolants (determined by different families of orthogonalpolynomials) and corresponding RBF interpolants.

In [NX12] it is shown that their orthogonal polynomial interpolantwith Hermite polynomials corresponds to the de Boor and Ron leastpolynomial interpolant.In [FM12] it was shown that the flat limit of Gaussians is given byHermite polynomials.

fasshauer@iit.edu MATH 590 13

Infinitely Smooth RBFs

RemarkIn [FW04] the authors describe a so-called Contour-Padéalgorithm that makes it possible (for data sets of relatively modestsize) to compute the RBF interpolant for all values of the shapeparameter ε including the limiting case ε→ 0.

Some numerical result obtained with Grady Wright’s MATLAB

toolbox are included in [Fas07, Ch. 17].

Other recent work obtaining RBF interpolants close to thepolynomial limit, i.e., for small ε, is [FM12, FLF11].

We will discuss the Hilbert–Schmidt stable evaluation algorithm of[FM12] soon.

fasshauer@iit.edu MATH 590 14

Infinitely Smooth RBFs

RemarkIn [FW04] the authors describe a so-called Contour-Padéalgorithm that makes it possible (for data sets of relatively modestsize) to compute the RBF interpolant for all values of the shapeparameter ε including the limiting case ε→ 0.

Some numerical result obtained with Grady Wright’s MATLAB

toolbox are included in [Fas07, Ch. 17].

Other recent work obtaining RBF interpolants close to thepolynomial limit, i.e., for small ε, is [FM12, FLF11].

We will discuss the Hilbert–Schmidt stable evaluation algorithm of[FM12] soon.

fasshauer@iit.edu MATH 590 14

Infinitely Smooth RBFs

RemarkIn [FW04] the authors describe a so-called Contour-Padéalgorithm that makes it possible (for data sets of relatively modestsize) to compute the RBF interpolant for all values of the shapeparameter ε including the limiting case ε→ 0.

Some numerical result obtained with Grady Wright’s MATLAB

toolbox are included in [Fas07, Ch. 17].

Other recent work obtaining RBF interpolants close to thepolynomial limit, i.e., for small ε, is [FM12, FLF11].

We will discuss the Hilbert–Schmidt stable evaluation algorithm of[FM12] soon.

fasshauer@iit.edu MATH 590 14

Infinitely Smooth RBFs

RemarkIn [FW04] the authors describe a so-called Contour-Padéalgorithm that makes it possible (for data sets of relatively modestsize) to compute the RBF interpolant for all values of the shapeparameter ε including the limiting case ε→ 0.

Some numerical result obtained with Grady Wright’s MATLAB

toolbox are included in [Fas07, Ch. 17].

Other recent work obtaining RBF interpolants close to thepolynomial limit, i.e., for small ε, is [FM12, FLF11].

We will discuss the Hilbert–Schmidt stable evaluation algorithm of[FM12] soon.

fasshauer@iit.edu MATH 590 14

Infinitely Smooth RBFs

Big Deal: “Flat” Limits, So What?

One of the most intriguing aspects associated with the polynomiallimit of RBF interpolants is the fact that RBF interpolants seem tobe most accurate (for a fixed number N of given samples) for apositive value of the shape parameter ε.

The following figure clearly exhibits a minimum in the interpolationerror distinctly away from zero.

Figure: f (x) = sin(x/2)− 2 cos(x) + 4 sin(πx), x ∈ [−4,4]

fasshauer@iit.edu MATH 590 15

Infinitely Smooth RBFs

Big Deal: “Flat” Limits, So What?

One of the most intriguing aspects associated with the polynomiallimit of RBF interpolants is the fact that RBF interpolants seem tobe most accurate (for a fixed number N of given samples) for apositive value of the shape parameter ε.

The following figure clearly exhibits a minimum in the interpolationerror distinctly away from zero.

Figure: f (x) = sin(x/2)− 2 cos(x) + 4 sin(πx), x ∈ [−4,4]fasshauer@iit.edu MATH 590 15

Infinitely Smooth RBFs

The observations of this section imply that RBF interpolants are(more flexible) generalizations of polynomial interpolants, andtherefore must be at least as accurate as (and often quite a bitmore than) polynomial interpolants.

However, polynomials are the basis of traditional algorithms(usually referred to a spectral methods) for the numerical solutionof equations whose solution is known to be smooth.

Message from this section:

RBFs, using stable evaluation algorithms and good pre-dictors of optimal shape parameters, should be able to dobetter than polynomials.

fasshauer@iit.edu MATH 590 16

Infinitely Smooth RBFs

The observations of this section imply that RBF interpolants are(more flexible) generalizations of polynomial interpolants, andtherefore must be at least as accurate as (and often quite a bitmore than) polynomial interpolants.

However, polynomials are the basis of traditional algorithms(usually referred to a spectral methods) for the numerical solutionof equations whose solution is known to be smooth.

Message from this section:

RBFs, using stable evaluation algorithms and good pre-dictors of optimal shape parameters, should be able to dobetter than polynomials.

fasshauer@iit.edu MATH 590 16

Infinitely Smooth RBFs

The observations of this section imply that RBF interpolants are(more flexible) generalizations of polynomial interpolants, andtherefore must be at least as accurate as (and often quite a bitmore than) polynomial interpolants.

However, polynomials are the basis of traditional algorithms(usually referred to a spectral methods) for the numerical solutionof equations whose solution is known to be smooth.

Message from this section:

RBFs, using stable evaluation algorithms and good pre-dictors of optimal shape parameters, should be able to dobetter than polynomials.

fasshauer@iit.edu MATH 590 16

RBFs with Finite Smoothness

Outline

1 Introduction

2 Infinitely Smooth RBFs

3 RBFs with Finite Smoothness

fasshauer@iit.edu MATH 590 17

RBFs with Finite Smoothness

The flat limit of RBFs with finite smoothness was not studied untilthe recent paper [SRFH12] in which interpolation on Rd wasinvestigated.

The thesis [BS10] contains similar investigations donesimultaneously and independently.

Before we explain the results obtained in [SRFH12], we recall afew finitely smooth radial kernels and their interpretation as fullspace Green’s functions.

fasshauer@iit.edu MATH 590 18

RBFs with Finite Smoothness

Example (Radial kernels with finite smoothness)

The univariate C0 Matérn kernel K (x , z).

= e−ε|x−z| is thefull-space Green’s function for the differential operator

L = − d2

dx2 + ε2I.

On the other hand, it is well-known that univariate C0 piecewiselinear splines may be expressed in terms of kernels of the formK (x , z)

.= |x − z|. The corresponding differential operator is

L = − d2

dx2 .

Note that the differential operator for the Matérn kernel“converges” to that of the piecewise linear splines as ε→ 0.

fasshauer@iit.edu MATH 590 19

RBFs with Finite Smoothness

Example (Radial kernels with finite smoothness)

The univariate C0 Matérn kernel K (x , z).

= e−ε|x−z| is thefull-space Green’s function for the differential operator

L = − d2

dx2 + ε2I.

On the other hand, it is well-known that univariate C0 piecewiselinear splines may be expressed in terms of kernels of the formK (x , z)

.= |x − z|. The corresponding differential operator is

L = − d2

dx2 .

Note that the differential operator for the Matérn kernel“converges” to that of the piecewise linear splines as ε→ 0.

fasshauer@iit.edu MATH 590 19

RBFs with Finite Smoothness

Example (Radial kernels with finite smoothness)

The univariate C0 Matérn kernel K (x , z).

= e−ε|x−z| is thefull-space Green’s function for the differential operator

L = − d2

dx2 + ε2I.

On the other hand, it is well-known that univariate C0 piecewiselinear splines may be expressed in terms of kernels of the formK (x , z)

.= |x − z|. The corresponding differential operator is

L = − d2

dx2 .

Note that the differential operator for the Matérn kernel“converges” to that of the piecewise linear splines as ε→ 0.

fasshauer@iit.edu MATH 590 19

RBFs with Finite Smoothness

Example (cont.)

The univariate C2 tension spline kernel [Sch66, Ren87]K (x , z)

.= e−ε|x−z| + ε|x − z| is the Green’s kernel of

L = − d4

dx4 + ε2 d2

dx2 ,

while the univariate C2 cubic spline kernel K (x , z).

= |x − z|3corresponds to

L = − d4

dx4 .

Again, the differential operator for the tension spline “converges”to that of the cubic spline as ε→ 0.

fasshauer@iit.edu MATH 590 20

RBFs with Finite Smoothness

Example (cont.)

The univariate C2 tension spline kernel [Sch66, Ren87]K (x , z)

.= e−ε|x−z| + ε|x − z| is the Green’s kernel of

L = − d4

dx4 + ε2 d2

dx2 ,

while the univariate C2 cubic spline kernel K (x , z).

= |x − z|3corresponds to

L = − d4

dx4 .

Again, the differential operator for the tension spline “converges”to that of the cubic spline as ε→ 0.

fasshauer@iit.edu MATH 590 20

RBFs with Finite Smoothness

Example (cont.)

The univariate C2 tension spline kernel [Sch66, Ren87]K (x , z)

.= e−ε|x−z| + ε|x − z| is the Green’s kernel of

L = − d4

dx4 + ε2 d2

dx2 ,

while the univariate C2 cubic spline kernel K (x , z).

= |x − z|3corresponds to

L = − d4

dx4 .

Again, the differential operator for the tension spline “converges”to that of the cubic spline as ε→ 0.

fasshauer@iit.edu MATH 590 20

RBFs with Finite Smoothness

Example (cont.)In [BTA04] we find a so-called univariate Sobolev kernel of theform K (x , z)

.= e−ε|x−z| sin

(ε|x − z|+ π

4

)which is associated with

L = − d4

dx4 − ε2I.

The operator for this kernel also “converges” to that of the cubicspline kernel, but the effect of the scale parameter is different thanfor the tension spline.

RemarkNote that this Sobolev kernel is different from the Sobolev splines(Matérn functions) discussed earlier — terminology . . .

fasshauer@iit.edu MATH 590 21

RBFs with Finite Smoothness

Example (cont.)The general multivariate Matérn kernels are of the form

K (x , z).

= Kβ−d/2 (ε‖x − z‖) (ε‖x − z‖)β−d/2 , β >d2,

and can be obtained as Green’s kernels of (see [FY11])

L =(ε2I −∆

)β, β >

d2.

We contrast this with the polyharmonic spline kernels

K (x , z).

=

{‖x − z‖2β−d , d odd,‖x − z‖2β−d log ‖x − z‖, d even,

andL = (−1)β∆β, β >

d2.

fasshauer@iit.edu MATH 590 22

RBFs with Finite Smoothness

Example (cont.)The general multivariate Matérn kernels are of the form

K (x , z).

= Kβ−d/2 (ε‖x − z‖) (ε‖x − z‖)β−d/2 , β >d2,

and can be obtained as Green’s kernels of (see [FY11])

L =(ε2I −∆

)β, β >

d2.

We contrast this with the polyharmonic spline kernels

K (x , z).

=

{‖x − z‖2β−d , d odd,‖x − z‖2β−d log ‖x − z‖, d even,

andL = (−1)β∆β, β >

d2.

fasshauer@iit.edu MATH 590 22

RBFs with Finite Smoothness

All examples above show that the differential operators associatedwith finitely smooth RBF kernels “converge” to those of apiecewise polynomial or polyharmonic spline kernel as ε→ 0.

We therefore ask if RBF interpolants based on finitely smoothkernels converge to (polyharmonic) spline interpolants for ε→ 0as is the case for infinitely smooth radial kernels and polynomials.

As mentioned above, infinitely smooth radial kernels can beexpanded into an infinite series of even powers of r .

Finitely smooth radial kernels can also be expanded into aninfinite series of powers of r .

In this case there always exists some minimal odd power of r withnonzero coefficient indicating the smoothness of the kernel.

fasshauer@iit.edu MATH 590 23

RBFs with Finite Smoothness

All examples above show that the differential operators associatedwith finitely smooth RBF kernels “converge” to those of apiecewise polynomial or polyharmonic spline kernel as ε→ 0.

We therefore ask if RBF interpolants based on finitely smoothkernels converge to (polyharmonic) spline interpolants for ε→ 0as is the case for infinitely smooth radial kernels and polynomials.

As mentioned above, infinitely smooth radial kernels can beexpanded into an infinite series of even powers of r .

Finitely smooth radial kernels can also be expanded into aninfinite series of powers of r .

In this case there always exists some minimal odd power of r withnonzero coefficient indicating the smoothness of the kernel.

fasshauer@iit.edu MATH 590 23

RBFs with Finite Smoothness

All examples above show that the differential operators associatedwith finitely smooth RBF kernels “converge” to those of apiecewise polynomial or polyharmonic spline kernel as ε→ 0.

We therefore ask if RBF interpolants based on finitely smoothkernels converge to (polyharmonic) spline interpolants for ε→ 0as is the case for infinitely smooth radial kernels and polynomials.

As mentioned above, infinitely smooth radial kernels can beexpanded into an infinite series of even powers of r .

Finitely smooth radial kernels can also be expanded into aninfinite series of powers of r .

In this case there always exists some minimal odd power of r withnonzero coefficient indicating the smoothness of the kernel.

fasshauer@iit.edu MATH 590 23

RBFs with Finite Smoothness

All examples above show that the differential operators associatedwith finitely smooth RBF kernels “converge” to those of apiecewise polynomial or polyharmonic spline kernel as ε→ 0.

We therefore ask if RBF interpolants based on finitely smoothkernels converge to (polyharmonic) spline interpolants for ε→ 0as is the case for infinitely smooth radial kernels and polynomials.

As mentioned above, infinitely smooth radial kernels can beexpanded into an infinite series of even powers of r .

Finitely smooth radial kernels can also be expanded into aninfinite series of powers of r .

In this case there always exists some minimal odd power of r withnonzero coefficient indicating the smoothness of the kernel.

fasshauer@iit.edu MATH 590 23

RBFs with Finite Smoothness

Example

For univariate C0, C2 and C4 Matérn kernels, respectively, we have

κ(εr).

= e−εr

= 1− εr +12

(εr)2 − 16

(εr)3 + · · · ,

κ(εr).

= (1 + εr)e−εr

= 1− 12

(εr)2 +13

(εr)3 − 18

(εr)4 + · · · ,

κ(εr).

=(

3 + 3εr + (εr)2)

e−εr

= 3− 12

(εr)2 +18

(εr)4 − 115

(εr)5 +1

48(εr)6 + · · · .

fasshauer@iit.edu MATH 590 24

RBFs with Finite Smoothness

Theorem ([SRFH12])

Suppose κ is radial and conditionally positive definite of order m ≤ nwith an expansion of the form

κ(r) = a0 + a2r2 + . . .+ a2nr2n + a2n+1r2n+1 + a2n+2r2n+2 + . . . ,

where 2n + 1 denotes the smallest odd power of r present in theexpansion (i.e., κ is finitely smooth). Also assume that the data Xcontain a unisolvent set with respect to the space Πd

2n of d-variatepolynomials of degree less than 2n. Then

limε→0

sε(x) =N∑

j=1

cj‖x − x j‖2n+1 +M∑

k=1

dkpk (x), x ∈ Rd ,

where {pk : k = 1, . . . ,M} denotes a basis of Πdn .

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RBFs with Finite Smoothness

RemarkThe previous theorem does not cover Matérn kernels with odd-ordersmoothness. However, all other examples listed above are covered.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

dataε=2ε=1ε=0.1piecewise linear

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

dataε=2ε=1ε=0.1cubic spline

Figure: Convergence of C0 (left) and C2 (right) Matérn interpolants topiecewise linear (left) and cubic (right) spline interpolants.

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RBFs with Finite Smoothness

RemarkThe previous theorem does not cover Matérn kernels with odd-ordersmoothness. However, all other examples listed above are covered.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

dataε=2ε=1ε=0.1piecewise linear

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

dataε=2ε=1ε=0.1cubic spline

Figure: Convergence of C0 (left) and C2 (right) Matérn interpolants topiecewise linear (left) and cubic (right) spline interpolants.

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RBFs with Finite Smoothness

On bounded intervals, interpolants with iterated Brownian bridgekernels (or tension splines) converge to piecewise polynomial splineinterpolants as discussed earlier.For β = 1 this means

K1,ε(x , z) =

{sinh(εx) sinh(ε(1−z))ε sinh(ε) , 0 ≤ x ≤ z ≤ 1,

sinh(εz) sinh(ε(1−x))ε sinh(ε) , 0 ≤ z ≤ x ≤ 1.

ε→0−→ K1(x , z) = min(x , z)− xz.

Figure: Brownian bridge (left) and tension spline (right) kernels for 15 equallyspaced values of z in [0,1].

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RBFs with Finite Smoothness

The following “relaxation spline” kernel also converges to the Brownianbridge kernel:

Krelax(x , z) =

{ sin(εx) sin(ε(1−z))ε sin(ε) , 0 ≤ x ≤ z ≤ 1,

sin(εz) sin(ε(1−x))ε sin(ε) , 0 ≤ z ≤ x ≤ 1.

−→ K1(x , z) = min(x , z)− xz.

Figure: Brownian bridge (left) and relaxation spline (right) kernels for 15equally spaced values of z in [0,1].

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RBFs with Finite Smoothness

RemarkLee and Micchelli [LM13] show that in the univariate setting not only“flat” smooth RBF interpolants converge to polynomial interpolants, butthat the same holds for interpolants based on “flat” smooth translationinvariant kernels, and even for general smooth kernels.In the multivariate setting the authors consider interpolation points thatare unisolvent for d-variate polynomials of total degree `. In this casethey also obtain a unique polynomial limiting interpolant for a given(not necessarily radial) kernel, provided it is analytic.

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Appendix References

References I

[Boo92] C. de Boor, On the error in multivariate polynomial interpolation, AppliedNumerical Mathematics 10 (1992), 297–305.

[Boo06] , On interpolation by radial polynomials, Advances in ComputationalMathematics 24 (2006), no. 1-4, 143–153.

[BR90] C. de Boor and A. Ron, On multivariate polynomial interpolation, Constr.Approx. 6 (1990), 287–302.

[BR92] , The least solution for the polynomial interpolation problem, Math.Z. 210 (1992), 347–378.

[BS10] Rebekka Brink-Spalink, Flat Limits von radialen Basisfunktionen,Diplomarbeit, Universität Göttingen, 2010.

[BTA04] A. Berlinet and C. Thomas-Agnan, Reproducing Kernel Hilbert Spaces inProbability and Statistics, Kluwer Academic Publishers, Dordrecht, 2004.

[DF02] T. A. Driscoll and B. Fornberg, Interpolation in the limit of increasingly flatradial basis functions, Comput. Math. Appl. 43 (2002), no. 3–5, 413–422.

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Appendix References

References II

[Fas07] G. E. Fasshauer, Meshfree Approximation Methods with MATLAB,Interdisciplinary Mathematical Sciences, vol. 6, World Scientific PublishingCo., Singapore, 2007.

[FF05] B. Fornberg and N. Flyer, Accuracy of radial basis function interpolationand derivative approximations on 1-D infinite grids, Adv. Comput. Math. 23(2005), no. 1-2, 5–20.

[FLF11] Bengt Fornberg, Elisabeth Larsson, and Natasha Flyer, Stablecomputations with Gaussian radial basis functions, SIAM Journal onScientific Computing 33 (2011), no. 2, 869–892.

[FLW06] B. Fornberg, E. Larsson, and G. Wright, A new class of oscillatory radialbasis functions, Computers & Mathematics with Applications 51 (2006),no. 8, 1209–1222.

[FM12] G. E. Fasshauer and M. J. McCourt, Stable evaluation of Gaussian radialbasis function interpolants, SIAM J. Sci. Comput. 34 (2012), no. 2,A737—A762.

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Appendix References

References III

[FW04] B. Fornberg and G. Wright, Stable computation of multiquadric interpolantsfor all values of the shape parameter, Comput. Math. Appl. 48 (2004),no. 5–6, 853–867.

[FY11] G. E. Fasshauer and Qi Ye, Reproducing kernels of generalized Sobolevspaces via a Green function approach with distributional operators,Numerische Mathematik 119 (2011), 585–611.

[LF03] E. Larsson and B. Fornberg, A numerical study of some radial basisfunction based solution methods for elliptic PDEs, Comput. Math. Appl. 46(2003), no. 5–6, 891–902.

[LF05] , Theoretical and computational aspects of multivariate interpolationwith increasingly flat radial basis functions, Comput. Math. Appl. 49 (2005),no. 1, 103–130.

[LM13] Yeon Ju Lee and Charles A. Micchelli, On collocation matrices forinterpolation and approximation, Journal of Approximation Theory 174(2013), 148–181.

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Appendix References

References IV

[LYY07] Y. J. Lee, G. J. Yoon, and J. Yoon, Convergence of increasingly flat radialbasis interpolants to polynomial interpolants, SIAM Journal onMathematical Analysis 39 (2007), no. 2, 537–553.

[NX12] Akil Narayan and Dongbin Xiu, Stochastic collocation methods onunstructured grids in high dimensions via interpolation, SIAM Journal onScientific Computing 34 (2012), no. 3, A1729–A1752.

[Ren87] R. J. Renka, Interpolatory tension splines with automatic selection oftension factors, SIAM Journal on Scientific and Statistical Computing 8(1987), no. 3, 393–415.

[Sch66] Daniel G. Schweikert, An interpolation curve using a spline in tension, J.Math. Phys. 45 (1966), 312–317.

[Sch05] Robert Schaback, Multivariate interpolation by polynomials and radialbasis functions, Constructive Approximation 21 (2005), 293–317.

[Sch08] R. Schaback, Limit problems for interpolation by analytic radial basisfunctions, J. Comput. Appl. Math. 212 (2008), 127–149.

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Appendix References

References V

[SRFH12] G. Song, J. Riddle, G. E. Fasshauer, and F. J. Hickernell, Multivariateinterpolation with increasingly flat radial basis functionsof finitesmoothness, Adv. Comput. Math. 36 (2012), no. 3, 485–501.

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