In this paper, the authors give a different and more precise analysis of the stability of the classical Gauss-Laguerre quadrature rule for the Cauchy P.V. integrals on the half line. Moreover, in order to obtain this ...In this paper, the authors give a different and more precise analysis of the stability of the classical Gauss-Laguerre quadrature rule for the Cauchy P.V. integrals on the half line. Moreover, in order to obtain this result they give some new estimates for the distance of the zeros of the Laguerre polynomials that can be useful also in other contests.展开更多
Three recent breakthroughs due to AI in arts and science serve as motivation:An award winning digital image,protein folding,fast matrix multiplication.Many recent developments in artificial neural networks,particularl...Three recent breakthroughs due to AI in arts and science serve as motivation:An award winning digital image,protein folding,fast matrix multiplication.Many recent developments in artificial neural networks,particularly deep learning(DL),applied and relevant to computational mechanics(solid,fluids,finite-element technology)are reviewed in detail.Both hybrid and pure machine learning(ML)methods are discussed.Hybrid methods combine traditional PDE discretizations with ML methods either(1)to help model complex nonlinear constitutive relations,(2)to nonlinearly reduce the model order for efficient simulation(turbulence),or(3)to accelerate the simulation by predicting certain components in the traditional integration methods.Here,methods(1)and(2)relied on Long-Short-Term Memory(LSTM)architecture,with method(3)relying on convolutional neural networks.Pure ML methods to solve(nonlinear)PDEs are represented by Physics-Informed Neural network(PINN)methods,which could be combined with attention mechanism to address discontinuous solutions.Both LSTM and attention architectures,together with modern and generalized classic optimizers to include stochasticity for DL networks,are extensively reviewed.Kernel machines,including Gaussian processes,are provided to sufficient depth for more advanced works such as shallow networks with infinite width.Not only addressing experts,readers are assumed familiar with computational mechanics,but not with DL,whose concepts and applications are built up from the basics,aiming at bringing first-time learners quickly to the forefront of research.History and limitations of AI are recounted and discussed,with particular attention at pointing out misstatements or misconceptions of the classics,even in well-known references.Positioning and pointing control of a large-deformable beam is given as an example.展开更多
This paper is concerned with obtaining the approximate solution for Volterra- Hammerstein integral equation with a regular kernel. We choose the Gauss points associated with the Legendre weight function w（x） = 1 as ...This paper is concerned with obtaining the approximate solution for Volterra- Hammerstein integral equation with a regular kernel. We choose the Gauss points associated with the Legendre weight function w（x） = 1 as the collocation points. The Legendre collocation discretization is proposed for Volterra-Hammerstein integral equation. We provide an error analysis which justifies that the errors of approximate solution decay exponentially in L2 norm and L^∞ norm. We give two numerical examples in order to illustrate the validity of the proposed Legendre spectral collocation method.展开更多
In this article we use Chebyshev spectral collocation method to deal with the Volterra integral equation which has two kinds of delay items. We use linear transformation to make the interval into a fixed interval [-1,...In this article we use Chebyshev spectral collocation method to deal with the Volterra integral equation which has two kinds of delay items. We use linear transformation to make the interval into a fixed interval [-1, 1]. Then we use the Gauss quadrature formula to approximate the solution. With the help of lemmas, we get the result that the numerical error decay exponentially in the infinity norm and the Chebyshev weighted Hilbert space norm. Some numerical experiments are given to confirm our theoretical prediction.展开更多
A new analytical expression is presented for the instantaneous power Probability Density Function （PDF） of receiver signals over composite K-u/gamma fading channels. Moreover, the exact expression of channel capacit...A new analytical expression is presented for the instantaneous power Probability Density Function （PDF） of receiver signals over composite K-u/gamma fading channels. Moreover, the exact expression of channel capacity is derived in the form of an infinite series, while an accurate approximation expression is obtained in closed form. To reveal the implications of the model parameters on capacity, we provide an expression for the case of a high-SNR environment. The relationship of the presented results with previously reported results on generalised-K and K fading channels is also discussed. Finally, numerical and simulation results are presented to prove the correctness of our derived expressions.展开更多
Making an exact computation of added resistance in sea waves is of high interest due to the economic effects relating to ship design and operation. In this paper, a B-spline based method is developed for computation o...Making an exact computation of added resistance in sea waves is of high interest due to the economic effects relating to ship design and operation. In this paper, a B-spline based method is developed for computation of added resistance. Based on the potential flow assumption, the velocity potential is computed using Green＇s formula. The Kochin function is applied to compute added resistance using Maruo＇s far-field method, the body surface is described by a B-spline curve and potentials and normal derivation of potentials are also described by B-spline basis functions and B-spline derivations. A collocation approach is applied for numerical computation, and integral equations are then evaluated by applying Gauss–Legendre quadrature. Computations are performed for a spheroid and different hull forms; results are validated by a comparison with experimental results. All results obtained with the present method show good agreement with experimental results.展开更多
Chebfun is a Matlab-based software system that overloads Matlab's discrete operations for vectors and matrices to analogous continuous operations for functions and operators.We begin by describing Chebfun's fa...Chebfun is a Matlab-based software system that overloads Matlab's discrete operations for vectors and matrices to analogous continuous operations for functions and operators.We begin by describing Chebfun's fast capabilities for Clenshaw-Curtis and also Gauss-Legendre,-Jacobi,-Hermite,and-Laguerre quadrature,based on algorithms of Waldvogel and Glaser,Liu and Rokhlin.Then we consider how such methods can be applied to quadrature problems including 2D integrals over rectangles,fractional derivatives and integrals,functions defined on unbounded intervals,and the fast computation of weights for barycentric interpolation.展开更多
A class of numerical methods is developed for second order Volterra integrodifferential equations by using a Legendre spectral approach.We provide a rigorous error analysis for the proposed methods,which shows that t...A class of numerical methods is developed for second order Volterra integrodifferential equations by using a Legendre spectral approach.We provide a rigorous error analysis for the proposed methods,which shows that the numerical errors decay exponentially in the L∞-norm and L2-norm.Numerical examples illustrate the convergence and effectiveness of the numerical methods.展开更多
文摘In this paper, the authors give a different and more precise analysis of the stability of the classical Gauss-Laguerre quadrature rule for the Cauchy P.V. integrals on the half line. Moreover, in order to obtain this result they give some new estimates for the distance of the zeros of the Laguerre polynomials that can be useful also in other contests.
文摘Three recent breakthroughs due to AI in arts and science serve as motivation:An award winning digital image,protein folding,fast matrix multiplication.Many recent developments in artificial neural networks,particularly deep learning(DL),applied and relevant to computational mechanics(solid,fluids,finite-element technology)are reviewed in detail.Both hybrid and pure machine learning(ML)methods are discussed.Hybrid methods combine traditional PDE discretizations with ML methods either(1)to help model complex nonlinear constitutive relations,(2)to nonlinearly reduce the model order for efficient simulation(turbulence),or(3)to accelerate the simulation by predicting certain components in the traditional integration methods.Here,methods(1)and(2)relied on Long-Short-Term Memory(LSTM)architecture,with method(3)relying on convolutional neural networks.Pure ML methods to solve(nonlinear)PDEs are represented by Physics-Informed Neural network(PINN)methods,which could be combined with attention mechanism to address discontinuous solutions.Both LSTM and attention architectures,together with modern and generalized classic optimizers to include stochasticity for DL networks,are extensively reviewed.Kernel machines,including Gaussian processes,are provided to sufficient depth for more advanced works such as shallow networks with infinite width.Not only addressing experts,readers are assumed familiar with computational mechanics,but not with DL,whose concepts and applications are built up from the basics,aiming at bringing first-time learners quickly to the forefront of research.History and limitations of AI are recounted and discussed,with particular attention at pointing out misstatements or misconceptions of the classics,even in well-known references.Positioning and pointing control of a large-deformable beam is given as an example.
基金supported by National Natural Science Foundation of China(11401347,91430104,11671157,61401255,11426193)Shandong Province Natural Science Foundation(ZR2014AP003)
文摘This paper is concerned with obtaining the approximate solution for Volterra- Hammerstein integral equation with a regular kernel. We choose the Gauss points associated with the Legendre weight function w（x） = 1 as the collocation points. The Legendre collocation discretization is proposed for Volterra-Hammerstein integral equation. We provide an error analysis which justifies that the errors of approximate solution decay exponentially in L2 norm and L^∞ norm. We give two numerical examples in order to illustrate the validity of the proposed Legendre spectral collocation method.
基金Supported by Guangdong Provincial Education Projects(2021KTSCX071,HSGDJG21356-372)Project of Hanshan Normal University(521036).
文摘In this article we use Chebyshev spectral collocation method to deal with the Volterra integral equation which has two kinds of delay items. We use linear transformation to make the interval into a fixed interval [-1, 1]. Then we use the Gauss quadrature formula to approximate the solution. With the help of lemmas, we get the result that the numerical error decay exponentially in the infinity norm and the Chebyshev weighted Hilbert space norm. Some numerical experiments are given to confirm our theoretical prediction.
基金supported by the National NatNatural Science Foundation of China under Grants No. 61132003,No. 61101237the Open Research Fund of National Mobile Communications Research Laboratory,Southeast University under Grant No. 2012D07
文摘A new analytical expression is presented for the instantaneous power Probability Density Function （PDF） of receiver signals over composite K-u/gamma fading channels. Moreover, the exact expression of channel capacity is derived in the form of an infinite series, while an accurate approximation expression is obtained in closed form. To reveal the implications of the model parameters on capacity, we provide an expression for the case of a high-SNR environment. The relationship of the presented results with previously reported results on generalised-K and K fading channels is also discussed. Finally, numerical and simulation results are presented to prove the correctness of our derived expressions.
文摘Making an exact computation of added resistance in sea waves is of high interest due to the economic effects relating to ship design and operation. In this paper, a B-spline based method is developed for computation of added resistance. Based on the potential flow assumption, the velocity potential is computed using Green＇s formula. The Kochin function is applied to compute added resistance using Maruo＇s far-field method, the body surface is described by a B-spline curve and potentials and normal derivation of potentials are also described by B-spline basis functions and B-spline derivations. A collocation approach is applied for numerical computation, and integral equations are then evaluated by applying Gauss–Legendre quadrature. Computations are performed for a spheroid and different hull forms; results are validated by a comparison with experimental results. All results obtained with the present method show good agreement with experimental results.
基金supported by the MathWorks,Inc.,King Abdullah University of Science and Technology (KAUST) (Award No. KUK-C1-013-04)the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013)/ERC (Grant Agreement No. 291068)
文摘Chebfun is a Matlab-based software system that overloads Matlab's discrete operations for vectors and matrices to analogous continuous operations for functions and operators.We begin by describing Chebfun's fast capabilities for Clenshaw-Curtis and also Gauss-Legendre,-Jacobi,-Hermite,and-Laguerre quadrature,based on algorithms of Waldvogel and Glaser,Liu and Rokhlin.Then we consider how such methods can be applied to quadrature problems including 2D integrals over rectangles,fractional derivatives and integrals,functions defined on unbounded intervals,and the fast computation of weights for barycentric interpolation.
基金the Foundation for Talent Introduction of Guangdong Provincial University,Guangdong Province Universities and Colleges Pearl River Scholar Funded Scheme(2008),National Science Foundation of China(10971074).
文摘A class of numerical methods is developed for second order Volterra integrodifferential equations by using a Legendre spectral approach.We provide a rigorous error analysis for the proposed methods,which shows that the numerical errors decay exponentially in the L∞-norm and L2-norm.Numerical examples illustrate the convergence and effectiveness of the numerical methods.
