Chapter 7
Moving Least Squares
Approximation
An alternative to radial basis function interpolation and approximation is the so-called
moving least squares method. As we will see below, in this method the approximation
P f to f is obtained by solving many
Chapter 8
Some Issues Related to Practical
Implementations
In this chapter we will collect some information about issues that are important for
the practical use of radial basis function and moving least squares methods. These
issues include stability and
Chapter 9
Applications
9.1
Solving Partial Dierential Equations via Collocation
In this section we discuss the numerical solution of elliptic partial dierential equations
using a collocation approach based on radial basis functions. To make the discussion
Chapter 6
Least Squares Approximation
As we saw in Chapter 5 we can interpret radial basis function interpolation as a constrained optimization problem. We now take this point of view again, but start with a
more general formulation. Lets assume we are se
Chapter 5
Error Bounds and the
Variational Approach
In order to estimate the approximation properties of the functions studied thus far we
will introduce the variational approach to scattered data interpolation. This approach
was used rst for radial basis
Chapter 2
Positive Denite and Completely
Monotone Functions
Below we will rst summarize facts about positive denite functions, and closely related
completely and multiply monotone functions. Most of these facts are integral characterizations and were esta
Chapter 3
Scattered Data Interpolation
with Polynomial Precision and
Conditionally Positive Denite
Functions
3.1
Scattered Data Interpolation with Polynomial Precision
Sometimes the assumption on the form (1.1) of the solution to the scattered data
interp
Chapter 4
Compactly Supported Radial
Basis Functions
As we saw earlier, compactly supported functions that are truly strictly conditionally positive denite of order m > 0 do not exist. The compact support automatically
ensures that is strictly positive de
Chapter 1
Introduction
1.1
History and Outline
Originally, the motivation for the basic meshfree approximation methods (radial basis
functions and moving least squares methods) came from applications in geodesy, geophysics, mapping, or meteorology. Later,