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13 pages, 2332 KiB

Open AccessArticle

Automatic Sleep Staging Based on Single-Channel EEG Signal Using Null Space Pursuit Decomposition Algorithm

by Weiwei Xiao, Rongqian Linghu, Huan Li and Fengzhen Hou

Axioms 2023, 12(1), 30; https://doi.org/10.3390/axioms12010030 - 27 Dec 2022

Cited by 2 | Viewed by 2128

Sleep quality is related to people’s physical and mental health, so an accurate assessment of sleep quality is key to recognizing sleep disorders and taking effective interventions. To address the shortcomings of traditional manual and automatic staging methods, such as being time-consuming and [...] Read more.

Sleep quality is related to people’s physical and mental health, so an accurate assessment of sleep quality is key to recognizing sleep disorders and taking effective interventions. To address the shortcomings of traditional manual and automatic staging methods, such as being time-consuming and having low classification accuracy, an automatic sleep staging method based on the null space pursuit (NSP) decomposition algorithm of single-channel electroencephalographic (EEG) signals is proposed, which provides a new way for EEG signal decomposition and automatic identification of sleep stages. First, the single-channel EEG signal data from the Sleep-EDF database, DREAMS Subject database, and Sleep Heart Health Study database (SHHS), available on PhysioNet, were preprocessed, respectively. Second, the preprocessed single-channel EEG signals were decomposed by the NSP algorithm. Third, we extracted nine features in the time domain of the nonlinear dynamics and statistics from the original EEG signal and the six simple signals that were decomposed. Finally, the extreme gradient boosting (XGBOOST) algorithm was used to construct a classification model to classify and identify the 63 extracted EEG signal features for automatic sleep staging. The experimental results showed that, on the Sleep-EDF database, the accuracy of four and five categories were 93.59% and 92.89%, respectively; on the DREAMS Subject database, the accuracy rates of four and five categories were 91.32% and 90.01%, respectively; on the SHHS database, the accuracy rates of four and five categories were 90.25% and 88.37%, respectively. The experimental results show that the automatic sleep staging model proposed in this work has high classification accuracy and efficiency, as well as strong applicability and robustness. Full article

(This article belongs to the Special Issue Numerical Computation, Approximation of Functions and Applied Mathematics)

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17 pages, 382 KiB

Open AccessArticle

Approximation Properties of the Blending-Type Bernstein–Durrmeyer Operators

by Yu-Jie Liu, Wen-Tao Cheng, Wen-Hui Zhang and Pei-Xin Ye

Axioms 2023, 12(1), 5; https://doi.org/10.3390/axioms12010005 - 21 Dec 2022

Cited by 2 | Viewed by 1358

Abstract

We construct the blending-type modified Bernstein–Durrmeyer operators and investigate their approximation properties. First, we derive the Voronovskaya-type asymptotic theorem for this type of operator. Then, the local and global approximation theorems are obtained by using the classical modulus of continuity and K-functional. [...] Read more.

We construct the blending-type modified Bernstein–Durrmeyer operators and investigate their approximation properties. First, we derive the Voronovskaya-type asymptotic theorem for this type of operator. Then, the local and global approximation theorems are obtained by using the classical modulus of continuity and K-functional. Finally, we derive the rate of convergence for functions with a derivative of bounded variation. The results show that the new operators have good approximation properties. Full article

(This article belongs to the Special Issue Numerical Computation, Approximation of Functions and Applied Mathematics)

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22 pages, 380 KiB

Open AccessArticle

Construction of Eigenfunctions to One Nonlocal Second-Order Differential Operator with Double Involution

by Batirkhan Turmetov and Valery Karachik

Axioms 2022, 11(10), 543; https://doi.org/10.3390/axioms11100543 - 11 Oct 2022

Cited by 1 | Viewed by 1147

Abstract

In this paper, we study the eigenfunctions to one nonlocal second-order differential operator with double involution. We give an explicit form of the eigenfunctions to the boundary value problem in the unit ball with Dirichlet conditions on the boundary. For the problem under [...] Read more.

In this paper, we study the eigenfunctions to one nonlocal second-order differential operator with double involution. We give an explicit form of the eigenfunctions to the boundary value problem in the unit ball with Dirichlet conditions on the boundary. For the problem under consideration, the completeness of the system of eigenfunctions is established. Full article

(This article belongs to the Special Issue Numerical Computation, Approximation of Functions and Applied Mathematics)

29 pages, 502 KiB

Open AccessArticle

Optimality of the Approximation and Learning by the Rescaled Pure Super Greedy Algorithms

by Wenhui Zhang, Peixin Ye, Shuo Xing and Xu Xu

Axioms 2022, 11(9), 437; https://doi.org/10.3390/axioms11090437 - 28 Aug 2022

Cited by 3 | Viewed by 1653

Abstract

We propose the Weak Rescaled Pure Super Greedy Algorithm (WRPSGA) for approximation with respect to a dictionary

D

in Hilbert space. The WRPSGA is simpler than some popular greedy algorithms. We show that the convergence rate of the RPSGA on the closure of [...] Read more.

We propose the Weak Rescaled Pure Super Greedy Algorithm (WRPSGA) for approximation with respect to a dictionary

D

in Hilbert space. The WRPSGA is simpler than some popular greedy algorithms. We show that the convergence rate of the RPSGA on the closure of the convex hull of the

μ

-coherent dictionary

D

is optimal. Then, we design the Rescaled Pure Super Greedy Learning Algorithm (RPSGLA) for kernel-based supervised learning. We prove that the convergence rate of the RPSGLA can be arbitrarily close to the best rate

O (m^{- 1})

under some mild assumptions. Full article

(This article belongs to the Special Issue Numerical Computation, Approximation of Functions and Applied Mathematics)

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19 pages, 1133 KiB

Open AccessArticle

A Stable Generalized Finite Element Method Coupled with Deep Neural Network for Interface Problems with Discontinuities

by Ying Jiang, Minghui Nian and Qinghui Zhang

Axioms 2022, 11(8), 384; https://doi.org/10.3390/axioms11080384 - 5 Aug 2022

Cited by 1 | Viewed by 1894

Abstract

The stable generalized finite element method (SGFEM) is an improved version of generalized or extended FEM (GFEM/XFEM), which (i) uses simple and unfitted meshes, (ii) reaches optimal convergence orders, and (iii) is stable and robust in the sense that conditioning is of the [...] Read more.

The stable generalized finite element method (SGFEM) is an improved version of generalized or extended FEM (GFEM/XFEM), which (i) uses simple and unfitted meshes, (ii) reaches optimal convergence orders, and (iii) is stable and robust in the sense that conditioning is of the same order as that of FEM and does not get bad as interfaces approach boundaries of elements. This paper designs the SGFEM for the discontinuous interface problem (DIP) by coupling a deep neural network (DNN). The main idea is to construct a function using the DNN, which captures the discontinuous interface condition, and transform the DIP to an (approximately) equivalent continuous interface problem (CIP) based on the DNN function such that the SGFEM for CIPs can be applied. The SGFEM for the DIP is a conforming method that maintains the features (i)–(iii) of SGFEM and is free from penalty terms. The approximation error of the proposed SGFEM is analyzed mathematically, which is split into an error of SGFEM of the CIP and a learning error of the DNN. The learning dimension of DNN is one dimension less than that of the domain and can be implemented efficiently. It is known that the DNN enjoys advantages in nonlinear approximations and high-dimensional problems. Therefore, the proposed SGFEM coupled with the DNN has great potential in the high-dimensional interface problem with interfaces of complex geometries. Numerical experiments verify the efficiency and optimal convergence of the proposed method. Full article

(This article belongs to the Special Issue Numerical Computation, Approximation of Functions and Applied Mathematics)

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20 pages, 323 KiB

Open AccessArticle

Some Upper Bounds for RKHS Approximation by Bessel Functions

by Mingdang Tian, Baohuai Sheng and Shuhua Wang

Axioms 2022, 11(5), 233; https://doi.org/10.3390/axioms11050233 - 17 May 2022

Cited by 5 | Viewed by 2001

Abstract

A reproducing kernel Hilbert space (RKHS) approximation problem arising from learning theory is investigated. Some K-functionals and moduli of smoothness with respect to RKHSs are defined with Fourier–Bessel series and Fourier–Bessel transforms, respectively. Their equivalent relation is shown, with which the upper [...] Read more.

A reproducing kernel Hilbert space (RKHS) approximation problem arising from learning theory is investigated. Some K-functionals and moduli of smoothness with respect to RKHSs are defined with Fourier–Bessel series and Fourier–Bessel transforms, respectively. Their equivalent relation is shown, with which the upper bound estimate for the best RKHS approximation is provided. The convergence rate is bounded with the defined modulus of smoothness, which shows that the RKHS approximation can attain the same approximation ability as that of the Fourier–Bessel series and Fourier–Bessel transform. In particular, it is shown that for a RKHS produced by the Bessel operator, the convergence rate sums up to the bound of a corresponding convolution operator approximation. The investigations show some new applications of Bessel functions. The results obtained can be used to bound the approximation error in learning theory. Full article

(This article belongs to the Special Issue Numerical Computation, Approximation of Functions and Applied Mathematics)

27 pages, 7638 KiB

Open AccessEditor’s ChoiceArticle

Improvement of the WENO-NIP Scheme for Hyperbolic Conservation Laws

by Ruo Li and Wei Zhong

Axioms 2022, 11(5), 190; https://doi.org/10.3390/axioms11050190 - 20 Apr 2022

Cited by 4 | Viewed by 2956

Abstract

The WENO-NIP scheme was obtained by developing a class of

L_{1}

-norm smoothness indicators based on Newton interpolation polynomial. It recovers the optimal convergence order in smooth regions regardless of critical points and achieves better resolution than the classical WENO-JS scheme. However, [...] Read more.

The WENO-NIP scheme was obtained by developing a class of

L_{1}

-norm smoothness indicators based on Newton interpolation polynomial. It recovers the optimal convergence order in smooth regions regardless of critical points and achieves better resolution than the classical WENO-JS scheme. However, the WENO-NIP scheme produces severe spurious oscillations when solving 1D linear advection problems with discontinuities at long output times, and it is also very oscillatory near discontinuities for 1D Riemann problems. In this paper, we find that the spectral property of WENO-NIP exhibits the negative dissipation characteristic, and this is the reason why WENO-NIP is unstable near discontinuities. Using this knowledge, we develop a way of improving the WENO-NIP scheme by introducing an additional term to eliminate the negative dissipation interval. The proposed scheme, denoted as WENO-NIP+, maintains the same convergence property, as well as the same low-dissipation property, as the corresponding WENO-NIP scheme. Numerical examples confirm that the proposed scheme is much more stable near discontinuities for 1D linear advection problems with large output times and 1D Riemann problems than the WENO-NIP scheme. Furthermore, the new scheme is far less dissipative in the region with high-frequency waves. In addition, the improved WENO-NIP+ scheme can remove or at least greatly decrease the post-shock oscillations that are commonly produced by the WENO-NIP scheme when simulating 2D Euler equations with strong shocks. Full article

(This article belongs to the Special Issue Numerical Computation, Approximation of Functions and Applied Mathematics)

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17 pages, 591 KiB

Open AccessArticle

Almost Optimality of the Orthogonal Super Greedy Algorithm for μ-Coherent Dictionaries

by Chunfang Shao, Jincai Chang, Peixin Ye, Wenhui Zhang and Shuo Xing

Axioms 2022, 11(5), 186; https://doi.org/10.3390/axioms11050186 - 20 Apr 2022

Cited by 3 | Viewed by 1859

Abstract

We study the approximation capability of the orthogonal super greedy algorithm (OSGA) with respect to

μ

-coherent dictionaries in Hilbert spaces. We establish the Lebesgue-type inequalities for OSGA, which show that the OSGA provides an almost optimal approximation on the first [...] Read more.

We study the approximation capability of the orthogonal super greedy algorithm (OSGA) with respect to

μ

-coherent dictionaries in Hilbert spaces. We establish the Lebesgue-type inequalities for OSGA, which show that the OSGA provides an almost optimal approximation on the first

[1 / (18 μ s)]

steps. Moreover, we improve the asymptotic constant in the Lebesgue-type inequality of OGA obtained by Livshitz E D. Full article

(This article belongs to the Special Issue Numerical Computation, Approximation of Functions and Applied Mathematics)

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17 pages, 358 KiB

Open AccessArticle

Exponential Multistep Methods for Stiff Delay Differential Equations

by Rui Zhan, Weihong Chen, Xinji Chen and Runjie Zhang

Axioms 2022, 11(5), 185; https://doi.org/10.3390/axioms11050185 - 19 Apr 2022

Cited by 3 | Viewed by 2289

Abstract

Stiff delay differential equations are frequently utilized in practice, but their numerical simulations are difficult due to the complicated interaction between the stiff and delay terms. At the moment, only a few low-order algorithms offer acceptable convergent and stable features. Exponential integrators are [...] Read more.

Stiff delay differential equations are frequently utilized in practice, but their numerical simulations are difficult due to the complicated interaction between the stiff and delay terms. At the moment, only a few low-order algorithms offer acceptable convergent and stable features. Exponential integrators are a type of efficient numerical approach for stiff problems that can eliminate the influence of stiffness on the scheme by precisely dealing with the stiff term. This study is concerned with two exponential multistep methods of Adams type for stiff delay differential equations. For semilinear delay differential equations, applying the linear multistep method directly to the integral form of the equation yields the exponential multistep method. It is shown that the proposed k-step method is stiffly convergent of order k. On the other hand, we can follow the strategy of the Rosenbrock method to linearize the equation along the numerical solution in each step. The so-called exponential Rosenbrock multistep method is constructed by applying the exponential multistep method to the transformed form of the semilinear delay differential equation. This method can be easily extended to nonlinear delay differential equations. The main contribution of this study is that we show that the k-step exponential Rosenbrock multistep method is stiffly convergent of order

k + 1

within the framework of a strongly continuous semigroup on Banach space. As a result, the methods developed in this study may be utilized to solve abstract stiff delay differential equations and can be served as time matching methods for delay partial differential equations. Numerical experiments are presented to demonstrate the theoretical results. Full article

(This article belongs to the Special Issue Numerical Computation, Approximation of Functions and Applied Mathematics)

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