1,721,009 research outputs found
Data-Driven Kernel Designs for Optimized Greedy Schemes: A Machine Learning Perspective
Full text linkThanks to their easy implementation via radial basis functions (RBFs), meshfree
kernel methods have proved to be an effective tool for, e.g., scattered data interpolation, PDE collocation, and classification and regression tasks. Their accuracy might depend on a length scale
hyperparameter, which is often tuned via cross-validation schemes. Here we leverage approaches
and tools from the machine learning community to introduce two-layered kernel machines, which
generalize the classical RBF approaches that rely on a single hyperparameter. Indeed, the proposed
learning strategy returns a kernel that is optimized not only in the Euclidean directions, but that
further incorporates kernel rotations. The kernel optimization is shown to be robust by using recently improved calculations of cross-validation scores. Finally, the use of greedy approaches, and
specifically of the vectorial kernel orthogonal greedy algorithm (VKOGA), allows us to construct
an optimized basis that adapts to the data. Beyond a rigorous analysis on the convergence of the
so-constructed two-layered (2L)-KOGA, its benefits are highlighted on both synthesized and real
benchmark dataset
Fast and stable rational RBF-based partition of unity interpolation
Full text linkWe perform a local computation via the Partition of Unity (PU) method of rational Radial Basis Function (RBF) interpolants. We investigate the well-posedness of the problem and we provide error bounds. The resulting scheme, efficiently implemented by means of the Deflation Accelerated Conjugate Gradient (DACG), enables us to deal with huge data sets and, thanks to the use of Variably Scaled Kernels (VSKs), it turns out to be stable. For functions with steep gradients or discontinuities, which are truly common in applications, the results show that the new proposed method outperforms the classical and rescaled PU schemes
RBF-Based Partition of Unity Methods for Elliptic PDEs: Adaptivity and Stability Issues Via Variably Scaled Kernels
No full textAnalysis of a new class of rational RBF expansions
Full text linkWe propose a new method, namely an eigen-rational kernel-based scheme, for multivariate interpolation
via mesh-free methods. It consists of a fractional radial basis function (RBF) expansion, with the
denominator depending on the eigenvector associated to the largest eigenvalue of the kernel matrix.
Classical bounds in terms of Lebesgue constants and convergence rates with respect to the mesh size of the
eigen-rational interpolant are indeed comparable with those of classical kernel-based methods. However,
the proposed approach takes advantage of rescaling the classical RBF expansion providing more robust
approximations. Theoretical analysis, numerical experiments and applications support our findings
Classifier-dependent feature selection via greedy methods
Full text linkThe purpose of this study is to introduce a new approach to feature ranking for classification tasks, called in what follows greedy feature selection. In statistical learning, feature selection is usually realized by means of methods that are independent of the classifier applied to perform the prediction using that reduced number of features. Instead, the greedy feature selection identifies the most important feature at each step and according to the selected classifier. The benefits of such scheme are investigated in terms of model capacity indicators, such as the Vapnik-Chervonenkis dimension or the kernel alignment. This theoretical study proves that the iterative greedy algorithm is able to construct classifiers whose complexity capacity grows at each step. The proposed method is then tested numerically on various datasets and compared to the state-of-the-art techniques. The results show that our iterative scheme is able to truly capture only a few relevant features, and may improve, especially for real and noisy data, the accuracy scores of other techniques. The greedy scheme is also applied to the challenging application of predicting geo-effective manifestations of the active Sun
A greedy non-intrusive reduced order model for shallow water equations
Full text linkIn this work, we develop Non-Intrusive Reduced Order Models (NIROMs) that combine Proper Orthogonal Decomposition (POD) with a Radial Basis Function (RBF) interpolation method to construct efficient reduced order models for time-dependent problems arising in large scale environmental flow applications. The performance of the POD-RBF NIROM is compared with a traditional nonlinear POD (NPOD) model by evaluating the accuracy and robustness for test problems representative of riverine flows. Different greedy algorithms are studied in order to determine a near-optimal distribution of interpolation points for the RBF approximation. A new power-scaled residual greedy (psr-greedy) algorithm is proposed to address some of the primary drawbacks of the existing greedy approaches. The relative performances of these greedy algorithms are studied with numerical experiments using realistic two-dimensional (2D) shallow water flow applications involving coastal and riverine dynamics. (C) 2021 Elsevier Inc. All rights reserved
Visibility Interpolation in Solar Hard X-Ray Imaging: Application to RHESSI and STIX
Full text linkSpace telescopes for solar hard X-ray imaging provide observations made of sampled Fourier components of the incoming photon flux. The aim of this study is to design an image reconstruction method relying on enhanced visibility interpolation in the Fourier domain. The interpolation-based method is applied to synthetic visibilities generated by means of the simulation software implemented within the framework of the Spectrometer/Telescope for Imaging X-rays (STIX) mission on board Solar Orbiter. An application to experimental visibilities observed by the Reuven Ramaty High Energy Solar Spectroscopic Imager (RHESSI) is also considered. In order to interpolate these visibility data, we have utilized an approach based on Variably Scaled Kernels (VSKs), which are able to realize feature augmentation by exploiting prior information on the flaring source and which are used here, for the first time, in the context of inverse problems. When compared to an interpolation-based reconstruction algorithm previously introduced for RHESSI, VSKs offer significantly better performance, particularly in the case of STIX imaging, which is characterized by a notably sparse sampling of the Fourier domain. In the case of RHESSI data, this novel approach is particularly reliable when the flaring sources are either characterized by narrow, ribbon-like shapes or high-resolution detectors are utilized for observations. The use of VSKs for interpolating hard X-ray visibilities allows remarkable image reconstruction accuracy when the information on the flaring source is encoded by a small set of scattered Fourier data and when the visibility surface is affected by significant oscillations in the frequency domain
Jumping with variably scaled discontinuous kernels (VSDKs)
Full text linkIn this paper we address the problem of approximating functions with discontinuities via kernel-based methods. The main result is the construction of discontinuous kernel-based basis functions. The linear spaces spanned by these discontinuous kernels lead to a very flexible tool which sensibly or completely reduces the well-known Gibbs phenomenon in reconstructing functions with jumps. For the new basis we provide error bounds and numerical results that support our claims. The method is also effectively tested for approximating satellite images
Rational RBF-based partition of unity method for efficiently and accurately approximating 3D objects
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