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

Correction to: Jumping with variably scaled discontinuous kernels (VSDKs)

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Recursive POD expansion for the advection-diffusion-reaction equation

This paper deals with the approximation of advection-diffusion-reaction equation solution by reduced order methods. We use the Recursive POD approximation for multivariate functions introduced in [M. AZAÏEZ, F. BEN BELGACEM, T. CHACÓN REBOLLO, Recursive POD expansion for reactiondiffusion equation, Adv.Model. and Simul. in Eng. Sci. (2016) 3:3. DOI 10.1186/s40323-016-0060-1] and applied to the low tensor representation of the solution of the reaction-diffusion partial differential equation. In this contribution we extend the Recursive POD approximation for multivariate functions with an arbitrary number of parameters, for which we prove general error estimates. The method is used to approximate the solutions of the advection-diffusion-reaction equation. We prove spectral error estimates, in which the spectral convergence rate depends only on the diffusion interval, while the error estimates are affected by a factor that grows exponentially with the advection velocity, and are independent of the reaction rate if this lives in a bounded set. These error estimates are based upon the analyticity of the solution of these equations as a function of the parameters (advection velocity, diffusion, reaction rate). We present several numerical tests, strongly consistent with the theoretical error estimates.Ministerio de Economía y CompetitividadAgence nationale de la rechercheGruppo Nazionale per il Calcolo ScientificoUE ERA-PLANE

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

Jumping with variably scaled discontinuous kernels (VSDKs)

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