Approximation and interpolation employing radial basis functions has found important applications since the early 1980's in areas such as signal processing, medical imaging, as well as neural networks. Several applications demand that certain physical properties be fulfilled, such as a function being divergence free. No such class of radial basis functions that reflects these physical properties was known until 1994, when Narcowich and Ward introduced a family of matrix-valued radial basis functions that are divergence free. They also obtained error bounds and stability estimates for interpolation by means of these functions. These divergence-free functions are very smooth, and have unbounded support. In this thesis we introduce a new class of matrix-valued radial basis functions that are divergence free as well as compactly supported. This leads to the possibility of applying fast solvers for inverting interpolation matrices, as these matrices are not only symmetric and positive definite, but also sparse because of this compact support. We develop error bounds and stability estimates which hold for a broad class of functions. We conclude with applications to the numerical solution of the Navier-Stokes equation for certain incompressible fluid flows.
Approximation and interpolation employing radial basis functions has
found important applications since the early 1980's in areas such
as signal processing, medical imaging, as well as neural networks.
Several applications demand that certain physical properties be
fulfilled, such as a function being divergence free. No such class
of radial basis functions that reflects these physical properties
was known until 1994, when Narcowich and Ward introduced a family of
matrix-valued radial basis functions that are divergence free. They
also obtained error bounds and stability estimates for interpolation
by means of these functions. These divergence-free functions are
very smooth, and have unbounded support. In this thesis we
introduce a new class of matrix-valued radial basis functions that are
divergence free as well as compactly supported. This leads to the
possibility of applying fast solvers for inverting interpolation
matrices, as these matrices are not only symmetric and positive
definite, but also sparse because of this compact support. We develop
error bounds and stability estimates which hold for a broad class of
functions. We conclude with applications to the numerical solution of
the Navier-Stokes equation for certain incompressible fluid flows.
