Article
Automation & Control Systems
Alain Celisse, Martin Wahl
Summary: The study focuses on constructing early stopping rules in nonparametric regression with unknown optimal iteration number using iterative learning algorithms. Oracle inequalities are established for empirical estimation error and prediction error, showing that classic discrepancy principle is adaptive for slow rates in hard learning scenario, while smoothed discrepancy principles are adaptive for faster rates. The approach relies on deviation inequalities for stopping rules in fixed design setting and change-of-norm arguments for random design setting.
JOURNAL OF MACHINE LEARNING RESEARCH
(2021)
Article
Mathematics
Guan-Tie Deng, Yun Huang, Tao Qian
Summary: In this paper, the theory of Bergman kernel is extended to the weighted case, obtaining a general form of weighted Bergman reproducing kernel which can be used to calculate concrete Bergman kernel functions for specific weights and domains.
JOURNAL OF GEOMETRIC ANALYSIS
(2021)
Article
Computer Science, Artificial Intelligence
Jiamin Liu, Wangli Xu, Fode Zhang, Heng Lian
Summary: Kernel Fisher discriminant (KFD) is a popular nonlinear extension of Fisher's linear discriminant, but its asymptotic properties have been rarely studied. In this study, we propose an operator-theoretical formulation of KFD and establish the convergence of the KFD solution to its population target. We also introduce a sketched estimation approach based on a m x n sketching matrix, which retains the asymptotic properties even when m is much smaller than n. Numerical results demonstrate the effectiveness of the sketched estimator.
IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE
(2023)
Article
Statistics & Probability
N. Makigusa
Summary: In this article, a new discrepancy measure called maximum variance discrepancy is introduced to measure the difference between two distributions in Hilbert spaces that cannot be found via the maximum mean discrepancy. A two-sample goodness of fit test based on this discrepancy measure is proposed. The asymptotic null distribution of this test is obtained, providing an efficient approximation method for the null distribution.
COMMUNICATIONS IN STATISTICS-THEORY AND METHODS
(2023)
Article
Mathematics, Applied
Holger Heitsch, Rene Henrion
Summary: The paper presents a fully explicit enumerative formula for calculating the spherical cap discrepancy, which can serve as a useful calibrating tool for testing the efficiency of sampling schemes. However, its application is limited to spheres of small dimension and moderate sample sizes.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2021)
Article
Mathematics, Interdisciplinary Applications
Dah-Chin Luor, Liang-Yu Hsieh
Summary: This paper investigates the connections between fractal interpolation functions (FIFs) and reproducing kernel Hilbert spaces (RKHSs). By establishing a fractal-type positive semi-definite kernel, it is shown that the span of linearly independent smooth FIFs is the corresponding RKHS. Furthermore, the nth derivatives of these FIFs, properties of related positive semi-definite kernels, and the importance of subspaces in curve-fitting applications are studied.
FRACTAL AND FRACTIONAL
(2023)
Article
Mathematics, Applied
Toni Karvonen, Simo Sarkka, Ken'ichiro Tanaka
Summary: In this study, we construct approximate Fekete point sets for kernel-based interpolation by maximising the determinant of a kernel Gram matrix obtained via truncation of an orthonormal expansion of the kernel. Uniform error estimates are proved for kernel interpolants at the resulting points. For the Gaussian kernel, it is shown that the approximate Fekete points in one dimension are the solution to a convex optimisation problem and that the interpolants converge with a super-exponential rate. Numerical examples are provided for the Gaussian kernel.
NUMERICAL ALGORITHMS
(2021)
Article
Mathematics
Zeyuan Song, Zuoren Sun
Summary: The central problem of this study is to represent any holomorphic and square integrable function on the Kepler manifold in the series form based on Fourier analysis. Three different domains on the Kepler manifold are considered and the weak pre-orthogonal adaptive Fourier decomposition (POAFD) is proposed. The weak maximal selection principle is shown to select the coefficient of the series, and a convergence theorem is proved to demonstrate the accuracy of the method.
Article
Mathematics, Applied
S. Leweke, O. Hauk, V Michel
Summary: This article introduces a method based on vector-valued splines to reconstruct neuronal current in the brain using non-invasive measurements. The method is efficient, accurate, and capable of handling irregularly distributed data. It shows promising results in synthetic test cases and real data acquired during a visual stimulation task.
Article
Mathematics, Applied
Ernesto De Vito, Nicole Muecke, Lorenzo Rosasco
Summary: This study focuses on reproducing kernel Hilbert spaces (RKHS) on a Riemannian manifold, exploring the conditions under which Sobolev spaces are RKHS and characterizing their reproducing kernels. The introduction of smoother diffusion spaces is also discussed, with detailed examples illustrating the general results. The paper presents a self-contained study of connections between Sobolev spaces, differential operators, and RKHS on Riemannian manifolds, aiming to provide a useful resource for researchers interested in the topic.
ANALYSIS AND APPLICATIONS
(2021)
Article
Mathematics, Applied
Wei Qu, Tao Qian, Haichou Li, Kehe Zhu
Summary: This study explores the best kernel approximation problem for analytic functions on the unit disk D in the reproducing kernel Hilbert space H, proving the existence of the best kernel approximation for weighted Bergman spaces with standard weights.
APPLIED MATHEMATICS AND COMPUTATION
(2022)
Article
Computer Science, Interdisciplinary Applications
J-L Akian, L. Bonnet, H. Owhadi, E. Savin
Summary: This paper introduces algorithms for selecting/designing kernels in Gaussian process regression/kriging surrogate modeling techniques. It presents two classes of algorithms: kernel flow, which selects the best kernel by minimizing the loss of accuracy caused by removing a portion of the dataset, and spectral kernel ridge regression, which selects the best kernel by minimizing the norm of the function to be approximated in the associated RKHS. The effectiveness of both approaches is demonstrated through numerical examples.
JOURNAL OF COMPUTATIONAL PHYSICS
(2022)
Article
Mathematics, Applied
Andreas Oslandsbotn, Zeljko Kereta, Valeriya Naumova, Yoav Freund, Alexander Cloninger
Summary: In this paper, we propose StreaMRAK, a streaming version of Kernel Ridge Regression (KRR), which improves on existing KRR schemes by dividing the problem into multiple levels of resolution and continuously integrating new samples. The results show that the proposed algorithm is fast and accurate.
APPLIED MATHEMATICS AND COMPUTATION
(2022)
Article
Computer Science, Artificial Intelligence
Xing Wei, Yunfeng Qiu, Zhiheng Ma, Xiaopeng Hong, Yihong Gong
Summary: There has been a growing interest in using computer vision and machine learning techniques to count crowds. Most existing methods heavily rely on fully-supervised learning and require a lot of labeled data. To address this issue, the study focuses on the semi-supervised learning paradigm and proposes a multiple representation learning method to train several models.
IEEE TRANSACTIONS ON IMAGE PROCESSING
(2023)
Article
Automation & Control Systems
Xiao Fang, Malay Ghosh
Summary: This paper focuses on statistical modeling and inference problems with sample sizes substantially smaller than the number of available covariates. It revisits a previous analysis and introduces a new class of global-local priors, providing results on posterior consistency and posterior contraction rates.
JOURNAL OF MACHINE LEARNING RESEARCH
(2023)
Article
Statistics & Probability
Josef Dick, Daniel Rudolf, Houying Zhu
ANNALS OF APPLIED PROBABILITY
(2016)
Article
Computer Science, Theory & Methods
Houying Zhu, Josef Dick
STATISTICS AND COMPUTING
(2017)
Article
Mathematics
Josef Dick, Aicke Hinrichs, Lev Markhasin, Friedrich Pillichshammer
ISRAEL JOURNAL OF MATHEMATICS
(2017)
Article
Mathematics, Applied
Josef Dick, Robert N. Gantner, Quoc T. Le Gia, Christoph Schwab
MATHEMATICAL MODELS & METHODS IN APPLIED SCIENCES
(2017)
Article
Mathematics, Applied
Josef Dick, Takashi Goda, Kosuke Suzuki, Takehito Yoshiki
NUMERISCHE MATHEMATIK
(2017)
Article
Mathematics, Applied
Josef Dick, Domingo Gomez-Perez, Friedrich Pillichshammer, Arne Winterhof
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY
(2017)
Article
Mathematics, Applied
Josef Dick, Aicke Hinrichs, Lev Markhasin, Friedrich Pillichshammer
Article
Mathematics, Applied
Josef Dick, Friedrich Pillichshammer, Kosuke Suzuki, Mario Ullrich, Takehito Yoshiki
FINITE FIELDS AND THEIR APPLICATIONS
(2018)
Article
Mathematics, Applied
Josef Dick, Christian Irrgeher, Gunther Leobacher, Friedrich Pillichshammer
SIAM JOURNAL ON NUMERICAL ANALYSIS
(2018)
Article
Mathematics, Applied
Josef Dick, Robert N. Gantner, Quoc T. Le Gia, Christoph Schwab
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2019)
Article
Mathematics, Applied
Josef Dick, Takashi Goda, Takehito Yoshiki
SIAM JOURNAL ON NUMERICAL ANALYSIS
(2019)
Article
Computer Science, Theory & Methods
D. Gunawan, M. -N. Tran, K. Suzuki, J. Dick, R. Kohn
STATISTICS AND COMPUTING
(2019)
Article
Statistics & Probability
Josef Dick, Daniel Rudolf, Houying Zhu
STATISTICS & PROBABILITY LETTERS
(2019)
Article
Mathematics, Applied
Josef Dick, Michael Feischl, Christoph Schwab
SIAM JOURNAL ON NUMERICAL ANALYSIS
(2019)
Article
Mathematics, Applied
Josef Dick, Friedrich Pillichshammer, Kosuke Suzuki, Mario Ullrich, Takehito Yoshiki
ANNALI DI MATEMATICA PURA ED APPLICATA
(2018)
Article
Mathematics, Applied
M. S. Bruzon, T. M. Garrido, R. de la Rosa
Summary: We study a family of generalized Zakharov-Kuznetsov modified equal width equations in (2+1)-dimensions involving an arbitrary function and three parameters. By using the Lie group theory, we classify the Lie point symmetries of these equations and obtain exact solutions. We also show that this family of equations admits local low-order multipliers and derive all local low-order conservation laws through the multiplier approach.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2024)
Article
Mathematics, Applied
Dohee Jung, Changbum Chun
Summary: The paper presents a general approach to enhance the Pade iterations for computing the matrix sign function by selecting an arbitrary three-point family of methods based on weight functions. The approach leads to a multi-parameter family of iterations and allows for the discovery of new methods. Convergence and stability analysis as well as numerical experiments confirm the improved performance of the new methods.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2024)
Article
Mathematics, Applied
Abhishek Yadav, Amit Setia, M. Thamban Nair
Summary: In this paper, we propose a Galerkin's residual-based numerical scheme for solving a system of Cauchy-type singular integral equations using Chebyshev polynomials. We prove the well-posedness of the system and derive a theoretical error bound and convergence order. The numerical examples validate the theoretical results.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2024)
Article
Mathematics, Applied
Fernando Chacon-Gomez, M. Eugenia Cornejo, Jesus Medina, Eloisa Ramirez-Poussa
Summary: The use of decision rules allows for reliable extraction of information and inference of conclusions from relational databases, but the concepts of decision algorithms need to be extended in fuzzy environments.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2024)
Article
Mathematics, Applied
Ilhame Amirali, Gabil M. Amiraliyev
Summary: This paper considers the one-dimensional initial-boundary problem for a pseudoparabolic equation with a time delay. To solve this problem numerically, a higher-order difference method is constructed and the error estimate for its solution is obtained. Based on the method of energy estimates, the fully discrete scheme is shown to be convergent of order four in space and of order two in time. The given numerical results illustrate the convergence and effectiveness of the numerical method.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2024)
Article
Mathematics, Applied
Tong-tong Shang, Guo-ji Tang, Wen-sheng Jia
Summary: The goal of this paper is to investigate a class of linear complementarity problems over tensor-spaces, denoted by TLCP, which is an extension of the classical linear complementarity problem. First, two classes of structured tensors over tensor-spaces (i.e., T-R tensor and T-RO tensor) are introduced and some equivalent characterizations are discussed. Then, the lower bound and upper bound of the solutions in the sense of the infinity norm of the TLCP are obtained when the problem has a solution.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2024)
Article
Mathematics, Applied
Fabio Difonzo, Pawel Przybylowicz, Yue Wu
Summary: This paper focuses on the existence, uniqueness, and approximation of solutions of delay differential equations (DDEs) with Caratheodory type right-hand side functions. It presents the construction of the randomized Euler scheme for DDEs and investigates its error. Furthermore, the paper reports the results of numerical experiments.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2024)
Article
Mathematics, Applied
Priyanka Roy, Geetanjali Panda, Dong Qiu
Summary: In this article, a gradient based descent line search scheme is proposed for solving interval optimization problems under generalized Hukuhara differentiability. The innovation and importance of these concepts are presented from practical and computational perspectives. The necessary condition for existence of critical point is presented in inclusion form of interval-valued gradient. Suitable efficient descent direction is chosen based on the monotonic property of the interval-valued function and specific interval ordering. Mathematical convergence of the scheme is proved under the assumption of Inexact line search. The theoretical developments are implemented with a set of interval test problems in different dimensions. A possible application in finance is provided and solved by the proposed scheme.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2024)
Article
Mathematics, Applied
Zhongqian Wang, Changqing Ye, Eric T. Chung
Summary: In this paper, the constrained energy minimizing generalized multiscale finite element method (CEM-GMsFEM) with mixed boundary conditions for elasticity equations in high contrast media is developed. The method offers advantages such as independence of target region's contrast from precision and significant impact of oversampling domain sizes on numerical accuracy. Furthermore, this is the first proof of convergence of CEM-GMsFEM with mixed boundary conditions for elasticity equations. Numerical experiments demonstrate the method's performance.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2024)
Article
Mathematics, Applied
Samaneh Soradi-Zeid, Maryam Alipour
Summary: The Laguerre polynomials are a new set of basic functions used to solve a specific class of optimal control problems specified by integro-differential equations, namely IOCP. The corresponding operational matrices of derivatives are calculated to extend the solution of the problem in terms of Laguerre polynomials. By considering the basis functions and using the collocation method, the IOCP is simplified into solving a system of nonlinear algebraic equations. The proposed method has been proven to have an error bound and convergence analysis for the approximate optimal value of the performance index. Finally, examples are provided to demonstrate the validity and applicability of this technique.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2024)
Article
Mathematics, Applied
Almudena P. Marquez, Maria L. Gandarias, Stephen C. Anco
Summary: A generalization of the KP equation involving higher-order dispersion is studied. The Lie point symmetries and conservation laws of the equation are obtained using Noether's theorem and the introduction of a potential. Sech-type line wave solutions are found and their features, including dark solitary waves on varying backgrounds, are discussed.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2024)
Article
Mathematics, Applied
Susanne Saminger-Platz, Anna Kolesarova, Adam Seliga, Radko Mesiar, Erich Peter Klement
Summary: In this article, we study real functions defined on the unit square satisfying basic properties and explore the conditions for generating bivariate copulas using parameterized transformations and other constructions.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2024)
Article
Mathematics, Applied
Lulu Tian, Nattaporn Chuenjarern, Hui Guo, Yang Yang
Summary: In this paper, a new local discontinuous Galerkin (LDG) algorithm is proposed to solve the incompressible Euler equation in two dimensions on overlapping meshes. The algorithm solves the vorticity, velocity field, and potential function on different meshes. The method employs overlapping meshes to ensure continuity of velocity along the interfaces of the primitive meshes, allowing for the application of upwind fluxes. The article introduces two sufficient conditions to maintain the maximum principle of vorticity.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2024)
Article
Mathematics, Applied
Cheng Wang, Jilu Wang, Steven M. Wise, Zeyu Xia, Liwei Xu
Summary: In this paper, a temporally second-order accurate numerical scheme for the Cahn-Hilliard-Magnetohydrodynamics system of equations is proposed and analyzed. The scheme utilizes a modified Crank-Nicolson-type approximation for time discretization and a mixed finite element method for spatial discretization. The modified Crank-Nicolson approximation allows for mass conservation and energy stability analysis. Error estimates are derived for the phase field, velocity, and magnetic fields, and numerical examples are presented to validate the proposed scheme's theoretical results.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2024)
Article
Mathematics, Applied
Mingyu He, Wenyuan Liao
Summary: This paper presents a numerical method for solving reaction-diffusion equations in spatially heterogeneous domains, which are commonly used to model biological applications. The method utilizes a fourth-order compact alternative directional implicit scheme based on Pade approximation-based operator splitting techniques. Stability analysis shows that the method is unconditionally stable, and numerical examples demonstrate its high efficiency and high order accuracy in both space and time.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2024)