Article
Mathematics, Applied
Qi Ye
Summary: In this article, the concept of positive definite multi-kernels is developed using the knowledge of positive definite tensors, to construct kernel-based interpolants for scattered data. The techniques of reproducing kernel Banach spaces are applied to show the optimal recoveries and error analysis for a specific class of strictly positive definite multi-kernels.
APPLIED AND COMPUTATIONAL HARMONIC ANALYSIS
(2023)
Article
Computer Science, Artificial Intelligence
Fanghui Liu, Lei Shi, Xiaolin Huang, Jie Yang, Johan A. K. Suykens
Summary: This paper examines the asymptotic properties of regularized least squares with indefinite kernels in RKKS, showing a globally optimal solution on a sphere and deriving learning rates. It is the first work on the approximation analysis of regularized learning algorithms in RKKS.
Article
Mathematics, Applied
Baasansuren Jadamba, Akhtar A. Khan, Fabio Raciti, Miguel Sama
Summary: This paper develops a stochastic approximation approach for estimating the flexural rigidity within the framework of variational inequalities. The nonlinear inverse problem is analyzed as a stochastic optimization problem using an energy least-squares formulation. A stochastic variational inequality is solved by a stochastic auxiliary problem principle-based iterative scheme, which satisfies the necessary and sufficient optimality condition for the optimization problem. The convergence analysis for the proposed iterative scheme is given under general conditions on the random noise. Detailed computational results demonstrate the feasibility and efficacy of the proposed methodology.
COMMUNICATIONS IN NONLINEAR SCIENCE AND NUMERICAL SIMULATION
(2022)
Article
Mathematics, Applied
Li He
Summary: This paper examines the properties of the Hardy-Sobolev space H82 and its relation to the Dirichlet space. By proving the equivalence between the density of the range of Cw in H82 and the density of polynomials in a specific Dirichlet space, several conclusions are drawn.
Article
Computer Science, Information Systems
Hao Xu
Summary: This paper presents an unsupervised manifold learning algorithm on the SPD matrix manifold for data dimensional reduction. By constructing polynomial kernel matrix, weight matrix, and sparsity preserving matrix, and utilizing polynomial mapping with geodesic distance, the proposed approach achieves dimensional reduction of SPD matrix data.
INFORMATION SCIENCES
(2022)
Article
Computer Science, Theory & Methods
Anat Amir, David Levin, Francis J. Narcowich, Joseph D. Ward
Summary: The paper discusses extrapolating data of a Cm function sampled at scattered sites on a Lipschitz region O in Rd to points outside of O in a computationally efficient way. It introduces a novel two-step moving least squares procedure and highlights improved meshfree approximation error estimates when using certain local Lagrange kernels.
FOUNDATIONS OF COMPUTATIONAL MATHEMATICS
(2022)
Article
Mathematics
Weiqi Zhou
Summary: We demonstrate that it is possible to have a kernel function that realizes the Riesz representation map even in the absence of a reproducing kernel in a Hilbert space. Constructions are presented for spaces which are the Fourier transform of weighted L-2 spaces. Under a mild assumption on the weight function, we are able to reproduce Riesz representatives of all functionals by taking a limit procedure from computable integrals over compact sets, despite the fact that the kernel is not necessarily in the underlying Hilbert space. Distributional kernels are also examined.
Article
Neurosciences
Kisung You, Hae-Jeong Park
Summary: Functional networks are commonly represented using symmetric positive definite matrices on the Riemannian manifold, considering all pairwise interactions. Despite the geometric properties of the SPD manifold, there are limited studies focusing on connectivity analysis.
Article
Mathematics, Applied
W. Franca, V. A. Menegatto
Summary: This paper investigates positive definite functions given by general integral transforms on a product of metric spaces. When one of the metric spaces includes R-d, the paper provides constructions of continuous and strictly positive definite functions using hypergeometric functions and conditionally negative definite functions. These constructions complement and generalize the original contribution by Gneiting.
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS
(2022)
Article
Geochemistry & Geophysics
Zheng Yang, Yongqiang Cheng, Hao Wu, Xiang Li, Hongqiang Wang
Summary: This article presents a novel detection method based on matrix information geometry for detecting radar targets submerged in a strong sea-clutter background. The method incorporates filtering and manifold projection to enhance the discriminative power between the target and clutter, resulting in improved detection performance.
IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING
(2023)
Article
Mathematics, Applied
Fleurianne Bertrand, Daniele Boffi
Summary: The study focuses on the approximation of the spectrum of least-squares operators in linear elasticity problems. By considering two different formulations and conducting numerical experiments, the theoretical results are confirmed.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2021)
Article
Mathematics, Applied
Juan C. Garcia-Ardila, Misael E. Marriaga
Summary: This article studies moment functionals associated with Hermite, Laguerre, or Jacobi classical orthogonal polynomials. We investigate the approximation of polynomials in H-r (u), a Sobolev space consisting of functions with consecutive derivatives up to order r belonging to the L-2 space associated with u. This requires the simultaneous approximation of a function f and its consecutive derivatives up to order N ≤ r. We construct orthogonal polynomials that achieve such simultaneous approximation and provide error estimates in terms of E-n(f((r))), the error of best approximation of f((r)) in L-2(u).
NUMERICAL ALGORITHMS
(2023)
Article
Automation & Control Systems
Fei Lu, Mauro Maggioni, Sui Tang
Summary: This paper discusses the inverse problem of learning interaction laws from observation data, and proposes an estimator based on minimizing a regularized least squares functional. Numerical simulations show that the learnability condition is satisfied in practical models, and the estimators are robust to noise in the observations. The estimators are also capable of producing accurate predictions of trajectories in large time intervals.
JOURNAL OF MACHINE LEARNING RESEARCH
(2021)
Article
Mathematics, Applied
Limei Zhang, Hong Zheng, Feng Liu
Summary: In this study, the MLS-NMM is used for the first time to discretize 3D steady heat conduction problems of FGMs. The influence domains of nodes in the MLS are employed as mathematical patches to construct the MC, while the shape functions of MLS-nodes serve as weight functions subordinate to the MC. Compared with the traditional NMM, the proposed MLS-NMM eliminates cutting operations in generating the physical cover and significantly reduces computation complexity. Numerical experiments demonstrate that the MLS-NMM offers advantages of both MLS and NMM in solving steady heat conduction.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2023)
Article
Mathematics, Applied
Fleurianne Bertrand, Daniele Boffi, Lucia Gastaldi
Summary: We discuss the approximation of the eigensolutions associated with the Maxwell eigenvalue problem using least-squares finite elements. Our novel least-squares formulation is designed to attain the minimum at the solution of the two first-order equations obtained from the Maxwell curl curl equation. The convergence of the finite element approximation is studied and numerical tests show that the method provides optimally convergent results with edge elements, and even in the presence of singular solutions, nodal elements can be successfully employed for the approximation of our problem.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2023)