58 research outputs found
Efficient computation of partition of unity interpolants through a block-based searching technique
In this paper we propose a new efficient interpolation tool, extremely
suitable for large scattered data sets. The partition of unity method is used
and performed by blending Radial Basis Functions (RBFs) as local approximants
and using locally supported weight functions. In particular we present a new
space-partitioning data structure based on a partition of the underlying
generic domain in blocks. This approach allows us to examine only a reduced
number of blocks in the search process of the nearest neighbour points, leading
to an optimized searching routine. Complexity analysis and numerical
experiments in two- and three-dimensional interpolation support our findings.
Some applications to geometric modelling are also considered. Moreover, the
associated software package written in \textsc{Matlab} is here discussed and
made available to the scientific community
Data-Driven Extrapolation Via Feature Augmentation Based on Variably Scaled Thin Plate Splines
The data driven extrapolation requires the definition of a functional model depending on the available data and has the application scope of providing reliable predictions on the unknown dynamics. Since data might be scattered, we drive our attention towards kernel models that have the advantage of being meshfree. Precisely, the proposed numerical method makes use of the so-called Variably Scaled Kernels (VSKs), which are introduced to implement a feature augmentation-like strategy based on discrete data. Due to the possible uncertainty on the data and since we are interested in modelling the behaviour of the target functions, we seek for a regularized solution by ridge regression. Focusing on polyharmonic splines, we investigate their implementation in the VSK setting and we provide error bounds in Beppo–Levi spaces. The performances of the method are then tested on functions showing exponential or rational decay. Comparisons with Support Vector Regression (SVR) are also carried out and highlight that the proposed approach is effective, particularly since it does not require to train complex architecture constructions
Efficient Reduced Basis Algorithm (ERBA) for Kernel-Based Approximation
The main purpose of this work is to provide an efficient scheme for constructing kernel-based reduced interpolation models. In the existing literature such problems are mainly addressed via the well-established knot insertion or knot removal schemes. Such iterative strategies are usually quite demanding from a computational point of view and our goal is to study an efficient implementation for data removal approaches, namely efficient reduced basis algorithm (ERBA). Focusing on kernel-based interpolation, the algorithm makes use of two iterative rules for removing data. The former, called ERBA-r, is based on classical residual evaluations. The latter, namely ERBA-p, is independent of the function values and relies on error bounds defined by the power function. In both cases, inspired by the so-called extended Rippa’s algorithm, our ERBA takes advantage of a fast implementation
RBF approximation of large datasets by partition of unity and local stabilization
We present an algorithm to approximate large dataset by Radial Basis Function
(RBF) techniques. The method couples a fast domain decomposition procedure with a
localized stabilization method. The resulting algorithm can efficiently deal with large
problems and it is robust with respect to the typical instability of kernel methods
Partition of unity interpolation using stable kernel-based techniques
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
Feature augmentation for the inversion of the Fourier transform with limited data
We investigate an interpolation/extrapolation method that, given scattered observations of the Fourier transform, approximates its inverse. The interpolation algorithm takes advantage of modeling the available data via a shape-driven interpolation based on variably scaled Kernels (VSKs), whose implementation is here tailored for inverse problems. The so-constructed interpolants are used as inputs for a standard iterative inversion scheme. After providing theoretical results concerning the spectrum of the VSK collocation matrix, we test the method on astrophysical imaging benchmarks
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