30,850 research outputs found
Approximation of Eigenfunctions in Kernel-based Spaces
Kernel-based methods in Numerical Analysis have the advantage of yielding
optimal recovery processes in the "native" Hilbert space \calh in which they
are reproducing. Continuous kernels on compact domains have an expansion into
eigenfunctions that are both -orthonormal and orthogonal in \calh
(Mercer expansion). This paper examines the corresponding eigenspaces and
proves that they have optimality properties among all other subspaces of
\calh. These results have strong connections to -widths in Approximation
Theory, and they establish that errors of optimal approximations are closely
related to the decay of the eigenvalues.
Though the eigenspaces and eigenvalues are not readily available, they can be
well approximated using the standard -dimensional subspaces spanned by
translates of the kernel with respect to nodes or centers. We give error
bounds for the numerical approximation of the eigensystem via such subspaces. A
series of examples shows that our numerical technique via a greedy point
selection strategy allows to calculate the eigensystems with good accuracy
Approximation Error Bounds via Rademacher's Complexity
Approximation properties of some connectionistic models, commonly used to construct approximation schemes for optimization problems with multivariable functions as admissible solutions, are investigated. Such models are made up of linear combinations of computational units
with adjustable parameters. The relationship between model complexity (number of computational units) and approximation error is investigated using tools from Statistical Learning Theory, such as Talagrand's
inequality, fat-shattering dimension, and Rademacher's complexity. For some families of multivariable functions, estimates of the approximation accuracy of models with certain computational units are derived in dependence of the Rademacher's complexities of the families. The
estimates improve previously-available ones, which were expressed in terms of V C dimension and derived by exploiting union-bound techniques. The results are applied to approximation schemes with certain radial-basis-functions as computational units, for which it is shown that
the estimates do not exhibit the curse of dimensionality with respect to the number of variables
A High-Order Kernel Method for Diffusion and Reaction-Diffusion Equations on Surfaces
In this paper we present a high-order kernel method for numerically solving
diffusion and reaction-diffusion partial differential equations (PDEs) on
smooth, closed surfaces embedded in . For two-dimensional
surfaces embedded in , these types of problems have received
growing interest in biology, chemistry, and computer graphics to model such
things as diffusion of chemicals on biological cells or membranes, pattern
formations in biology, nonlinear chemical oscillators in excitable media, and
texture mappings. Our kernel method is based on radial basis functions (RBFs)
and uses a semi-discrete approach (or the method-of-lines) in which the surface
derivative operators that appear in the PDEs are approximated using
collocation. The method only requires nodes at "scattered" locations on the
surface and the corresponding normal vectors to the surface. Additionally, it
does not rely on any surface-based metrics and avoids any intrinsic coordinate
systems, and thus does not suffer from any coordinate distortions or
singularities. We provide error estimates for the kernel-based approximate
surface derivative operators and numerically study the accuracy and stability
of the method. Applications to different non-linear systems of PDEs that arise
in biology and chemistry are also presented
Sampling and Approximation of Bandlimited Volumetric Data
We present an approximation scheme for functions in three dimensions, that
requires only their samples on the Cartesian grid, under the assumption that
the functions are sufficiently concentrated in both space and frequency. The
scheme is based on expanding the given function in the basis of generalized
prolate spheroidal wavefunctions, with the expansion coefficients given by
weighted dot products between the samples of the function and the samples of
the basis functions. As numerical implementations require all expansions to be
finite, we present a truncation rule for the expansions. Finally, we derive a
bound on the overall approximation error in terms of the assumed
space/frequency concentration
Numerical solutions of a boundary value problem on the sphere using radial basis functions
Boundary value problems on the unit sphere arise naturally in geophysics and
oceanography when scientists model a physical quantity on large scales. Robust
numerical methods play an important role in solving these problems. In this
article, we construct numerical solutions to a boundary value problem defined
on a spherical sub-domain (with a sufficiently smooth boundary) using radial
basis functions (RBF). The error analysis between the exact solution and the
approximation is provided. Numerical experiments are presented to confirm
theoretical estimates
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