arXiv subjects

Roberto Cavoretto

Publications and source records attributed to Roberto Cavoretto.

At least 19 recordsLinked to original sources

arXiv2026

Adaptive Non-Linear Partition of Unity Methods for Scattered Data Interpolation with Discontinuities

Scattered data approximation with discontinuities is challenging due to the Gibbs phenomenon, which significantly reduces accuracy near interfaces. The recently introduced Non-Linear Partition of Unity Method (NL-PUM) addresses this by combining Radial Basis Function (RBF) interpolation with a non-linear Weighted Essentially Non-Oscillatory (WENO) strategy. While effective, NL-PUM's performance relies heavily on two fixed hyperparameters: the RBF shape parameter and the patch radius. This work extends NL-PUM by adapting both hyperparameters locally using Leave-One-Out Cross-Validation (LOOCV) minimized via Global Optimization with Optimistic Improvement (GOOI). Our main innovation is a smoothness indicator linking a discontinuity-aware shrinkage process to LOOCV-based radius selection: patches in smooth regions remain unchanged, while those near discontinuities automatically shrink to avoid crossing the interface. The resulting method, LOOCV-NL-PUM-GOOI, requires no prior knowledge of interface geometry and introduces no extra cost beyond standard adaptive shape parameter selection. Numerical experiments on synthetic test functions with jump discontinuities and a real-data application to Norwegian Fjords elevation data confirm that this approach substantially reduces approximation error near discontinuities while preserving full accuracy in smooth regions.

math.NA↗

arXiv2026

Adaptive RBF-KAN: A Comparative Evaluation of Dynamic Shape Parameters in Kolmogorov-Arnold Networks

Kolmogorov-Arnold Networks (KANs) approximate multivariate functions using learnable univariate edge functions, typically parameterized by B-spline bases. Although effective, spline-based implementations can be computationally expensive. A modified version of KANs, called FastKAN, improves efficiency by replacing splines with Gaussian radial basis functions (RBFs), but it relies on a fixed kernel and shape parameter. In this work, we extend the RBF-based KAN framework by introducing a broader family of radial basis kernels and by initializing the kernel shape parameter using leave-one-out cross-validation (LOOCV). To the best of our knowledge, this is the first study that integrates LOOCV-based kernel scale estimation with deep KAN training. We also introduce Mat\'ern and Wendland kernels into the KAN framework for the first time, enabling more flexible basis representations beyond the Gaussian kernel used in FastKAN. The LOOCV estimate provides a data-driven initialization of the kernel scale, which is subsequently refined during network training. The proposed adaptive RBF-KAN is evaluated on several two-dimensional benchmark functions. The results highlight the importance of kernel selection and adaptive shape parameters, with different kernels showing advantages for smooth functions, discontinuities, and oscillatory patterns. Overall, combining LOOCV-based initialization with adaptive kernel learning provides a practical strategy for improving RBF-based KAN models.

stat.ML↗

arXiv2026

Graph Neural Networks for Community Detection in Graph Signal Analysis

Community detection is a central problem in graph analysis, with applications ranging from network science to graph signal processing. In recent years, Graph Neural Networks (GNNs) have emerged as effective tools for learning low-dimensional representations of graph-structured data and have shown strong performance in clustering tasks, particularly on large and high-dimensional graphs. This paper investigates the use of GNN-based community detection within a graph signal interpolation framework. After reviewing the main classes of GNN architectures for community detection according to a standard taxonomy, we integrate the resulting graph communities into a Partition of Unity Method (PUM) for interpolation with Graph Basis Functions (GBFs). In this approach, GNN-derived communities are used to construct local subdomains on which GBF interpolants are computed and subsequently combined into a global approximation. Numerical experiments on benchmark %graph datasets, including geometric and urban network examples demonstrate that the proposed combination of GNN-based clustering and GBF-PUM interpolation yields accurate signal reconstructions. The results indicate that deep learning-based community detection can provide effective graph partitions for localized interpolation schemes, supporting its use in scalable graph signal analysis.

math.NA↗

arXiv2023

A Bayesian Approach for Simultaneously Radial Kernel Parameter Tuning in the Partition of Unity Method

In this paper, Bayesian optimisation is used to simultaneously search the optimal values of the shape parameter and the radius in radial basis function partition of unity interpolation problem. It is a probabilistic iterative approach that models the error function with a step-by-step self-updated Gaussian process, whereas partition of unity leverages a mesh-free method that allows us to reduce cost-intensive computations when the number of scattered data is very large, as the entire domain is decomposed into several smaller subdomains of variable radius. Numerical experiments on the scattered data interpolation problem show that the combination of these two tools sharply reduces the search time with respect to other techniques such as the leave one out cross validation.

math.NA↗

arXiv2023

Parameter Tuning in the Radial Kernel-Based Partition of Unity Method by Bayesian Optimization

In this paper, we employ Bayesian optimization to concurrently explore the optimal values for both the shape parameter and the radius in the partition of unity interpolation using radial basis functions. Bayesian optimization is a probabilistic, iterative approach that models the error function through a progressively self-updated Gaussian process. Meanwhile, the partition of unity approach harnesses a meshfree method, allowing us to significantly reduce computational expenses, particularly when considering a substantial number of scattered data points. This reduction in computational cost is achieved by decomposing the entire domain into several smaller subdomains, each of them with a variable radius. We provide an estimation of the complexity of our algorithm and carry out numerical experiments to illustrate the effectiveness of our approach, dealing with test and real-world datasets.

math.NA↗

arXiv2023

Node-Binded Communities for Interpolation on Graphs

Partition of unity methods (PUMs) on graphs represent straightforward and remarkably adaptable auxiliary techniques for graph signal processing. By relying solely on the intrinsic graph structure, we propose the generation of a partition of unity through centrality measures and modularity. Subsequently, we integrate PUMs with a local graph basis function (GBF) approximation approach to achieve low-cost global interpolation schemes.

math.NA↗

arXiv2023

Bayesian Approach for Radial Kernel Parameter Tuning

In this paper we present a new fast and accurate method for Radial Basis Function (RBF) approximation, including interpolation as a special case, which enables us to effectively find the optimal value of the RBF shape parameter. In particular, we propose a statistical technique, called Bayesian optimization, that consists in modelling the error function with a Gaussian process, by which, through an iterative process, the optimal shape parameter is selected. The process is step by step self-updated resulting in a relevant decrease in search time with respect to the classical leave one out cross validation technique. Numerical results deriving from some test examples and an application to real data show the performance of the proposed method.

math.NA↗

arXiv2023

Node-Bound Communities for Partition of Unity Interpolation on Graphs

Graph signal processing benefits significantly from the direct and highly adaptable supplementary techniques offered by partition of unity methods (PUMs) on graphs. In our approach, we demonstrate the generation of a partition of unity solely based on the underlying graph structure, employing an algorithm that relies exclusively on centrality measures and modularity, without requiring the input of the number of subdomains. Subsequently, we integrate PUMs with a local graph basis function (GBF) approximation method to develop cost-effective global interpolation schemes. We also discuss numerical experiments conducted on both synthetic and real datasets to assess the performance of this presented technique.

math.NA↗

arXiv2022

Adaptive LOOCV-based kernel methods for solving time-dependent BVPs

In this paper we propose an adaptive scheme for the solution of time-dependent boundary value problems (BVPs). To solve numerically these problems, we consider the kernel-based method of lines that allows us to split the spatial and time derivatives, dealing with each separately. This adaptive algorithm is based on a leave-one-out cross validation (LOOCV) technique, which is employed as an error indicator. By this scheme, we can first detect the domain areas where the error is estimated to be too large -- generally due to steep variations or quick changes in the solution -- and then accordingly enhance the numerical solution by applying a two-point refinement strategy. Numerical experiments show the efficacy and performance of our adaptive refinement method.

math.NA↗

arXiv2020

Partition of Unity Methods for Signal Processing on Graphs

Partition of unity methods (PUMs) on graphs are simple and highly adaptive auxiliary tools for graph signal processing. Based on a greedy-type metric clustering and augmentation scheme, we show how a partition of unity can be generated in an efficient way on graphs. We investigate how PUMs can be combined with a local graph basis function (GBF) approximation method in order to obtain low-cost global interpolation or classification schemes. From a theoretical point of view, we study necessary prerequisites for the partition of unity such that global error estimates of the PUM follow from corresponding local ones. Finally, properties of the PUM as cost-efficiency and approximation accuracy are investigated numerically.

eess.SP↗

arXiv2017

Numerical Methods for Pulmonary Image Registration

Due to complexity and invisibility of human organs, diagnosticians need to analyze medical images to determine where the lesion region is, and which kind of disease is, in order to make precise diagnoses. For satisfying clinical purposes through analyzing medical images, registration plays an essential role. For instance, in Image-Guided Interventions (IGI) and computer-aided surgeries, patient anatomy is registered to preoperative images to guide surgeons complete procedures. Medical image registration is also very useful in surgical planning, monitoring disease progression and for atlas construction. Due to the significance, the theories, methods, and implementation method of image registration constitute fundamental knowledge in educational training for medical specialists. In this chapter, we focus on image registration of a specific human organ, i.e. the lung, which is prone to be lesioned. For pulmonary image registration, the improvement of the accuracy and how to obtain it in order to achieve clinical purposes represents an important problem which should seriously be addressed. In this chapter, we provide a survey which focuses on the role of image registration in educational training together with the state-of-the-art of pulmonary image registration. In the first part, we describe clinical applications of image registration introducing artificial organs in Simulation-based Education. In the second part, we summarize the common methods used in pulmonary image registration and analyze popular papers to obtain a survey of pulmonary image registration.

q-bio.QM↗

arXiv2017

Hermite-Birkhoff interpolation on scattered data on the sphere and other manifolds

The Hermite-Birkhoff interpolation problem of a function given on arbitrarily distributed points on the sphere and other manifolds is considered. Each proposed interpolant is expressed as a linear combination of basis functions, the combination coefficients being incomplete Taylor expansions of the interpolated function at the interpolation points. The basis functions depend on the geodesic distance, are orthonormal with respect to the point-evaluation functionals, and have all derivatives equal zero up to a certain order at the interpolation points. A remarkable feature of such interpolants, which belong to the class of partition of unity methods, is that their construction does not require solving linear systems. Numerical tests are given to show the interpolation performance.

math.NA↗

arXiv2017

Optimal selection of local approximants in RBF-PU interpolation

The Partition of Unity (PU) method, performed with local Radial Basis Function (RBF) approximants, has been proved to be an effective tool for solving large scattered data interpolation problems. However, in order to achieve a good accuracy, the question about how many points we have to consider on each local subdomain, i.e. how large can be the local data sets, needs to be answered. Moreover, it is well-known that also the shape parameter affects the accuracy of the local RBF approximants and, as a consequence, of the PU interpolant. Thus here, both the shape parameter used to fit the local problems and the size of the associated linear systems are supposed to vary among the subdomains. They are selected by minimizing an a priori error estimate. As evident from extensive numerical experiments and applications provided in the paper, the proposed method turns out to be extremely accurate also when data with non-homogeneous density are considered.

math.NA↗

arXiv2016

On the topology preservation of Gneiting's functions in image registration

The purpose of image registration is to determine a transformation such that the transformed version of the source image is similar to the target one. In this paper we focus on landmark-based image registration using radial basis functions (RBFs) transformations, in particular on the topology preservation of compactly supported radial basis functions (CSRBFs) transformations. In [1] the performances of Gneiting's and Wu's functions are compared with the ones of other well known schemes in image registration, as thin plate spline and Wendland's functions. Several numerical experiments and real-life cases with medical images show differences in accuracy and smoothness of the considered interpolation methods, which can be explained taking into account their topology preservation properties. Here we analyze analytically and experimentally the topology preservation performances of Gneiting's functions, comparing results with the ones obtained in [2], where Wendland's and Wu's functions are considered.

math.NA↗

arXiv2016

RBF-PU Interpolation with Variable Subdomain Sizes and Shape Parameters

In this paper, we deal with the challenging computational issue of interpolating large data sets, with eventually non-homogeneous densities. To such scope, the Radial Basis Function Partition of Unity (RBF-PU) method has been proved to be a reliable numerical tool. However, there are not available techniques enabling us to efficiently select the sizes of the local PU subdomains which, together with the value of the RBF shape parameter, greatly influence the accuracy of the final fit. Thus here, by minimizing an \emph{a priori} error estimate, we propose a RBF-PU method by suitably selecting variable shape parameters and subdomain sizes. Numerical results and applications show performaces of the interpolation technique.

math.NA↗

arXiv2016

Hermite-Birkhoff Interpolation on Arbitrarily Distributed Data on the Sphere and Other Manifolds

We consider the problem of interpolating a function given on scattered points using Hermite-Birkhoff formulas on the sphere and other manifolds. We express each proposed interpolant as a linear combination of basis functions, the combination coefficients being incomplete Taylor expansions of the interpolated function at the interpolation points. The basis functions have the following features: (i) depend on the geodesic distance; (ii) are orthonormal with respect to the point-evaluation functionals; and (iii) have all derivatives equal zero up to a certain order at the interpolation points. Moreover, the construction of such interpolants, which belong to the class of partition of unity methods, takes advantage of not requiring any solution of linear systems.

math.NA↗

arXiv2015

Fast and flexible interpolation via PUM with applications in population dynamics

In this paper the Partition of Unity Method (PUM) is efficiently performed using Radial Basis Functions (RBFs) as local approximants. In particular, we present a new space-partitioning data structure extremely useful in applications because of its independence from the problem geometry. Moreover, we study, in the context of wild herbivores in forests, an application of such algorithm. This investigation shows that the ecosystem of the considered natural park is in a very delicate situation, for which the animal population could become extinguished. The determination of the so-called sensitivity surfaces, obtained with the new fast and flexible interpolation tool, indicates some possible preventive measures to the park administrators.

math.NA↗

arXiv2015

Approximating basins of attraction for dynamical systems via stable radial bases

In applied sciences, such as physics and biology, it is often required to model the evolution of populations via dynamical systems. In this paper, we focus on the problem of approximating the basins of attraction of such models in case of multi-stability. We propose to reconstruct the domains of attraction via an implicit interpolant using stable radial bases, obtaining the surfaces by partitioning the phase space into disjoint regions. An application to a competition model presenting jointly three stable equilibria is considered.

math.NA↗