Gabriele Santin

A new stable basis for radial basis function interpolation

1 Radial Basis Functions 5 1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.2 Native Space . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.2.1 Embeddings . . . . . . . . . . . . . . . . . . . . . . . . 9 1.2.2 An integral operator and a “natural” basis . . . . . . . 10 1.2.3 Other inner products . . . . . . . . . . . . . . . . . . . 11 1.3 Error bounds and stability estimates . . . . . . . . . . . . . . 12

Partition of unity interpolation using stable kernel-based techniques

Abstract In this paper we propose a new stable and accurate approximation technique which is extremely effective for interpolating large scattered data sets. The Partition of Unity (PU) method is performed considering Radial Basis Functions (RBFs) as local approximants and using locally supported weights. In particular, the approach consists in computing, for each PU subdomain, a stable basis. Such technique, taking advantage of the local scheme, leads to a significant benefit in terms of stability, especially for flat kernels. Furthermore, an optimized searching procedure is applied to build the local stable bases, thus rendering the method more efficient.

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 ℋ

An algebraic cubature formula on curvilinear polygons

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Greedy Kernel Approximation for Sparse Surrogate Modeling

in which they are reproducing. Continuous kernels on compact domains have an expansion into eigenfunctions that are both L2-orthonormal and orthogonal in ℋ

A rescaled method for RBF approximation

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Approximating basins of attraction for dynamical systems via stable radial bases

(Mercer expansion). This paper examines the corresponding eigenspaces and proves that they have optimality properties among all other subspaces of ℋ

Numerical modelling of a peripheral arterial stenosis using dimensionally reduced models and kernel methods: Numerical modelling of a peripheral arterial stenosis using dimensionally reduced models and kernel methods

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Fast computation of orthonormal basis for RBF spaces through Krylov space methods

. These results have strong connections to n-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 n-dimensional subspaces spanned by translates of the kernel with respect to n 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.

Convergence rate of the data-independent P-greedy algorithm in kernel-based approximation

Abstract We implement in Matlab a Gauss-like cubature formula on bivariate domains whose boundary is a piecewise smooth Jordan curve (curvilinear polygons). The key tools are Green’s integral formula, together with the recent software package Chebfun to approximate the boundary curve close to machine precision by piecewise Chebyshev interpolation. Several tests are presented, including some comparisons of this new routine ChebfunGauss with the recent SplineGauss that approximates the boundary by splines.