Both the Newton interpolating polynomials and the Thiele-type interpolating continued fractions based on inverse differences are used to construct a kind of bivariate blending rational interpolants and an error estima...Both the Newton interpolating polynomials and the Thiele-type interpolating continued fractions based on inverse differences are used to construct a kind of bivariate blending rational interpolants and an error estimation is given.展开更多
Newton＇s polynomial interpolation may be the favorite linear interpolation,associated continued fractions interpolation is a new type nonlinear interpolation.We use those two interpolation to construct a new kind of ...Newton＇s polynomial interpolation may be the favorite linear interpolation,associated continued fractions interpolation is a new type nonlinear interpolation.We use those two interpolation to construct a new kind of bivariate blending rational interpolants.Characteristic theorem is discussed.We give some new blending interpolation formulae.展开更多
In this paper we apply a new method to choose suitable free parameters of the planar cubic G1 Hermite interpolant. The method provides the cubic G1 Hermite interpolant with minimal quadratic oscillation in average. We...In this paper we apply a new method to choose suitable free parameters of the planar cubic G1 Hermite interpolant. The method provides the cubic G1 Hermite interpolant with minimal quadratic oscillation in average. We can use the method to construct the optimal shape-preserving interpolant. Some numerical examples are presented to illustrate the effectiveness of the method.展开更多
By making use of Thiele-type bivariate branched continued fractions and Sumelson inverse,we construct a few kinds of bivariate vector valued rational interpolonts (BVRIs) over rectangular grids and find out certain re...By making use of Thiele-type bivariate branched continued fractions and Sumelson inverse,we construct a few kinds of bivariate vector valued rational interpolonts (BVRIs) over rectangular grids and find out certain relations among these BVRIs such as boundary identity and duality.展开更多
It is well-known that Hermite rational interpolation gives a better approximation than Hermite polynomial interpolation,especially for large sequences of interpolation points,but it is difficult to solve the problem o...It is well-known that Hermite rational interpolation gives a better approximation than Hermite polynomial interpolation,especially for large sequences of interpolation points,but it is difficult to solve the problem of convergence and control the occurrence of real poles.In this paper,we establish a family of linear Hermite barycentric rational interpolants r that has no real poles on any interval and in the case k=0,1,2,the function r^(k)(x)converges to f^(k)(x)at the rate of O(h^3d+3-k)as h→0 on any real interpolation interval,regardless of the distribution of the interpolation points.Also,the function r(x)is linear in data.展开更多
It is well-known that interpolation by rational functions results in a more accurate approximation than the polynomials interpolation.However,classical rational interpolation has some deficiencies such as uncontrollab...It is well-known that interpolation by rational functions results in a more accurate approximation than the polynomials interpolation.However,classical rational interpolation has some deficiencies such as uncontrollable poles and low convergence order.In contrast with the classical rational interpolants,the generalized barycentric rational interpolants which depend linearly on the interpolated values,yield infinite smooth approximation with no poles in real numbers.In this paper,a numerical collocation approach,based on the generalized barycentric rational interpolation and Gaussian quadrature formula,was introduced to approximate the solution of Volterra-Fredholm integral equations.Three types of points in the solution domain are used as interpolation nodes.The obtained numerical results confirm that the barycentric rational interpolants are efficient tools for solving Volterra-Fredholm integral equations.Moreover,integral equations with Runge’s function as an exact solution,no oscillation occurrs in the obtained approximate solutions so that the Runge’s phenomenon is avoided.展开更多
Efficient algorithms are established for the computation of bivariate lacunary vector valued rational interpolants based on the branched continued fractions and a numerical example is given to show how the algorithms ...Efficient algorithms are established for the computation of bivariate lacunary vector valued rational interpolants based on the branched continued fractions and a numerical example is given to show how the algorithms are implemented,展开更多
In this note we present a constructive proof of symmetrical determinantal formulas forthe numerator and denominator of an ordinary rational interpolant,consider the confluencecase and give new determinantal formulas o...In this note we present a constructive proof of symmetrical determinantal formulas forthe numerator and denominator of an ordinary rational interpolant,consider the confluencecase and give new determinantal formulas of the rational interpolant by means of Lagrange’sbasis functions.展开更多
In this paper the concept of first boundary condition （i）（i = 0, 1, 2,…, n） is proposed based on [1], the existence of two times spline interpolant under first boundary condition is proved using constructivity me...In this paper the concept of first boundary condition （i）（i = 0, 1, 2,…, n） is proposed based on [1], the existence of two times spline interpolant under first boundary condition is proved using constructivity method and the uniqueness of the two times spline interpolant under first boundary condition（n） is proved too.展开更多
The aim of this paper is to lay a algebraic geometry foundation for constructing smoothing interpolants on curved side element. Some interpolation theorems in polynomial space are given. The main results effectively f...The aim of this paper is to lay a algebraic geometry foundation for constructing smoothing interpolants on curved side element. Some interpolation theorems in polynomial space are given. The main results effectively for CAGD are presented.展开更多
In this paper,a necessary and sufficient condition for the existence of a kind of bivariate vector valued rational interpolants over rectangular grids is given.This criterion is an algebraic method,i.e.,by solving a s...In this paper,a necessary and sufficient condition for the existence of a kind of bivariate vector valued rational interpolants over rectangular grids is given.This criterion is an algebraic method,i.e.,by solving a system of equations based on the given data,we can directly test whether the relevant interpolant exists or not.By coming up with our method, the problem of how to deal with scalar equations and vector equations in the same system of equations is solved.After testing existence,an expression of the corresponding bivariate vector-valued rational interpolant can be constructed consequently.In addition,the way to get the expression is different from the one by making use of Thiele-type bivariate branched vector-valued continued fractions and Samelson inverse which are commonly used to construct the bivariate vector-valued rational interpolants.Compared with the Thiele-type method,the one given in this paper is more direct.Finally,some numerical examples are given to illustrate the result.展开更多
This paper is dedicated to the expansion of the framework of general interpolant observables introduced by Azouani,Olson,and Titi for continuous data assimilation of nonlinear partial differential equations.The main f...This paper is dedicated to the expansion of the framework of general interpolant observables introduced by Azouani,Olson,and Titi for continuous data assimilation of nonlinear partial differential equations.The main feature of this expanded framework is its mesh-free aspect,which allows the observational data itself to dictate the subdivision of the domain via partition of unity in the spirit of the so-called Partition of Unity Method by Babuska and Melenk.As an application of this framework,we consider a nudging-based scheme for data assimilation applied to the context of the two-dimensional Navier-Stokes equations as a paradigmatic example and establish convergence to the reference solution in all higher-order Sobolev topologies in a periodic,mean-free setting.The convergence analysis also makes use of absorbing ball bounds in higherorder Sobolev norms,for which explicit bounds appear to be available in the literature only up to H^(2);such bounds are additionally proved for all integer levels of Sobolev regularity above H^(2).展开更多
The post-processing procedure is given by a interpolant postprocessing of the finit element solution by appropriately-define finite dimensional subspaces, The corresponding superconvergence are established on general ...The post-processing procedure is given by a interpolant postprocessing of the finit element solution by appropriately-define finite dimensional subspaces, The corresponding superconvergence are established on general quasi-regular finite element partitions.展开更多
A new kind of matrix-valued rational interpolants is recursively established by means of generalized Samelson inverse for matrices, with scalar numerator and matrix-valued denominator. In this respect, it is essential...A new kind of matrix-valued rational interpolants is recursively established by means of generalized Samelson inverse for matrices, with scalar numerator and matrix-valued denominator. In this respect, it is essentially different from that of the previous works [7, 9], where the matrix-valued rational interpolants is in Thiele-type continued fraction form with matrix-valued numerator and scalar denominator. For both univariate and bivariate cases, sufficient conditions for existence, characterisation and uniqueness in some sense are proved respectively, and an error formula for the univariate interpolating function is also given. The results obtained in this paper are illustrated with some numerical examples.展开更多
Human motion modeling is a core technology in computer animation,game development,and humancomputer interaction.In particular,generating natural and coherent in-between motion using only the initial and terminal frame...Human motion modeling is a core technology in computer animation,game development,and humancomputer interaction.In particular,generating natural and coherent in-between motion using only the initial and terminal frames remains a fundamental yet unresolved challenge.Existing methods typically rely on dense keyframe inputs or complex prior structures,making it difficult to balance motion quality and plausibility under conditions such as sparse constraints,long-term dependencies,and diverse motion styles.To address this,we propose a motion generation framework based on a frequency-domain diffusion model,which aims to better model complex motion distributions and enhance generation stability under sparse conditions.Our method maps motion sequences to the frequency domain via the Discrete Cosine Transform(DCT),enabling more effective modeling of low-frequency motion structures while suppressing high-frequency noise.A denoising network based on self-attention is introduced to capture long-range temporal dependencies and improve global structural awareness.Additionally,a multi-objective loss function is employed to jointly optimize motion smoothness,pose diversity,and anatomical consistency,enhancing the realism and physical plausibility of the generated sequences.Comparative experiments on the Human3.6M and LaFAN1 datasets demonstrate that our method outperforms state-of-the-art approaches across multiple performance metrics,showing stronger capabilities in generating intermediate motion frames.This research offers a new perspective and methodology for human motion generation and holds promise for applications in character animation,game development,and virtual interaction.展开更多
Smooth interpolants defined over tetrahedra are currently being developed for they have many applications in geography, solid modeling, finite element analysis, etc. In this paper, we will characterize a certain class...Smooth interpolants defined over tetrahedra are currently being developed for they have many applications in geography, solid modeling, finite element analysis, etc. In this paper, we will characterize a certain class of C-1 discrete tetrahedral interpolants with only C-1 data required. As special cases of the class characterized, we give two C-1 discrete tetrahedral interpolants which have concise expressions.展开更多
Algorithms to generate a triangular or a quadrilateral interpolant with G^1-continuity are given in this paper for arbitrary scattered data with associated normal vectors over a prescribed triangular or quadrilateral ...Algorithms to generate a triangular or a quadrilateral interpolant with G^1-continuity are given in this paper for arbitrary scattered data with associated normal vectors over a prescribed triangular or quadrilateral decomposition. The interpolants are constructed with a general method to generate surfaces from moving Bezier curves under geometric constraints. With the algorithm, we may obtain interpolants in complete symbolic parametric forms, leading to a fast computation of the interpolant. A dynamic interpolation solid modelling software package DISM is implemented based on the algorithm which can be used to generate and manipulate solid objects in an interactive way.展开更多
Focuses on a study that presented an axiomatic definition to bivariate vector valued rational interpolation on distinct plane interpolation points. Definition; Existence and uniqueness; Connection.
In order to solve the problem of the variable coefficient ordinary differen-tial equation on the bounded domain,the Lagrange interpolation method is used to approximate the exact solution of the equation,and the error...In order to solve the problem of the variable coefficient ordinary differen-tial equation on the bounded domain,the Lagrange interpolation method is used to approximate the exact solution of the equation,and the error between the numerical solution and the exact solution is obtained,and then compared with the error formed by the difference method,it is concluded that the Lagrange interpolation method is more effective in solving the variable coefficient ordinary differential equation.展开更多
Substantial advancements have been achieved in Tunnel Boring Machine(TBM)technology and monitoring systems,yet the presence of missing data impedes accurate analysis and interpretation of TBM monitoring results.This s...Substantial advancements have been achieved in Tunnel Boring Machine(TBM)technology and monitoring systems,yet the presence of missing data impedes accurate analysis and interpretation of TBM monitoring results.This study aims to investigate the issue of missing data in extensive TBM datasets.Through a comprehensive literature review,we analyze the mechanism of missing TBM data and compare different imputation methods,including statistical analysis and machine learning algorithms.We also examine the impact of various missing patterns and rates on the efficacy of these methods.Finally,we propose a dynamic interpolation strategy tailored for TBM engineering sites.The research results show that K-Nearest Neighbors(KNN)and Random Forest(RF)algorithms can achieve good interpolation results;As the missing rate increases,the interpolation effect of different methods will decrease;The interpolation effect of block missing is poor,followed by mixed missing,and the interpolation effect of sporadic missing is the best.On-site application results validate the proposed interpolation strategy's capability to achieve robust missing value interpolation effects,applicable in ML scenarios such as parameter optimization,attitude warning,and pressure prediction.These findings contribute to enhancing the efficiency of TBM missing data processing,offering more effective support for large-scale TBM monitoring datasets.展开更多
文摘Both the Newton interpolating polynomials and the Thiele-type interpolating continued fractions based on inverse differences are used to construct a kind of bivariate blending rational interpolants and an error estimation is given.
基金Supported by the Project Foundation of the Department of Education of Anhui Province(KJ2008A027,KJ2010B182,KJ2011B152,KJ2011B137)Supported by the Grant of Scientific Research Foundation for Talents of Hefei University(11RC05)Supported by the Grant of Scientific Research Foundation Hefei University(11KY06ZR)
文摘Newton＇s polynomial interpolation may be the favorite linear interpolation,associated continued fractions interpolation is a new type nonlinear interpolation.We use those two interpolation to construct a new kind of bivariate blending rational interpolants.Characteristic theorem is discussed.We give some new blending interpolation formulae.
基金The Scientific Research Fund (18A415) of Hunan Provincial Education Department
文摘In this paper we apply a new method to choose suitable free parameters of the planar cubic G1 Hermite interpolant. The method provides the cubic G1 Hermite interpolant with minimal quadratic oscillation in average. We can use the method to construct the optimal shape-preserving interpolant. Some numerical examples are presented to illustrate the effectiveness of the method.
基金Supported by the National Natural Science Foundation of China.
文摘By making use of Thiele-type bivariate branched continued fractions and Sumelson inverse,we construct a few kinds of bivariate vector valued rational interpolonts (BVRIs) over rectangular grids and find out certain relations among these BVRIs such as boundary identity and duality.
基金Supported by the National Natural Science Foundation of China(Grant No.11601224)the Science Foundation of Ministry of Education of China(Grant No.18YJC790069)+1 种基金the Natural Science Foundation of Jiangsu Higher Education Institutions of China(Grant No.18KJD110007)the National Statistical Science Research Project of China(Grant No.2018LY28)。
文摘It is well-known that Hermite rational interpolation gives a better approximation than Hermite polynomial interpolation,especially for large sequences of interpolation points,but it is difficult to solve the problem of convergence and control the occurrence of real poles.In this paper,we establish a family of linear Hermite barycentric rational interpolants r that has no real poles on any interval and in the case k=0,1,2,the function r^(k)(x)converges to f^(k)(x)at the rate of O(h^3d+3-k)as h→0 on any real interpolation interval,regardless of the distribution of the interpolation points.Also,the function r(x)is linear in data.
文摘It is well-known that interpolation by rational functions results in a more accurate approximation than the polynomials interpolation.However,classical rational interpolation has some deficiencies such as uncontrollable poles and low convergence order.In contrast with the classical rational interpolants,the generalized barycentric rational interpolants which depend linearly on the interpolated values,yield infinite smooth approximation with no poles in real numbers.In this paper,a numerical collocation approach,based on the generalized barycentric rational interpolation and Gaussian quadrature formula,was introduced to approximate the solution of Volterra-Fredholm integral equations.Three types of points in the solution domain are used as interpolation nodes.The obtained numerical results confirm that the barycentric rational interpolants are efficient tools for solving Volterra-Fredholm integral equations.Moreover,integral equations with Runge’s function as an exact solution,no oscillation occurrs in the obtained approximate solutions so that the Runge’s phenomenon is avoided.
基金Supported by-the National Natural Science Foundation of China
文摘Efficient algorithms are established for the computation of bivariate lacunary vector valued rational interpolants based on the branched continued fractions and a numerical example is given to show how the algorithms are implemented,
文摘In this note we present a constructive proof of symmetrical determinantal formulas forthe numerator and denominator of an ordinary rational interpolant,consider the confluencecase and give new determinantal formulas of the rational interpolant by means of Lagrange’sbasis functions.
文摘In this paper the concept of first boundary condition （i）（i = 0, 1, 2,…, n） is proposed based on [1], the existence of two times spline interpolant under first boundary condition is proved using constructivity method and the uniqueness of the two times spline interpolant under first boundary condition（n） is proved too.
文摘The aim of this paper is to lay a algebraic geometry foundation for constructing smoothing interpolants on curved side element. Some interpolation theorems in polynomial space are given. The main results effectively for CAGD are presented.
基金the National Natural Science Foundation of China （No. 60473114） the Natural Science Foundation of Auhui Province （No. 070416227）+2 种基金 the Natural Science Research Scheme of Education Department of Anhui Province （No. KJ2008B246） Colleges and Universities in Anhui Province Young Teachers Subsidy Scheme （No. 2008jq1110） the Science Research Foundation of Chaohu College （No. XLY-200705）.
文摘In this paper,a necessary and sufficient condition for the existence of a kind of bivariate vector valued rational interpolants over rectangular grids is given.This criterion is an algebraic method,i.e.,by solving a system of equations based on the given data,we can directly test whether the relevant interpolant exists or not.By coming up with our method, the problem of how to deal with scalar equations and vector equations in the same system of equations is solved.After testing existence,an expression of the corresponding bivariate vector-valued rational interpolant can be constructed consequently.In addition,the way to get the expression is different from the one by making use of Thiele-type bivariate branched vector-valued continued fractions and Samelson inverse which are commonly used to construct the bivariate vector-valued rational interpolants.Compared with the Thiele-type method,the one given in this paper is more direct.Finally,some numerical examples are given to illustrate the result.
基金partially supported by the award PSC-CUNY64335-0052,jointly funded by The Professional Staff Congress and The City University of New York。
文摘This paper is dedicated to the expansion of the framework of general interpolant observables introduced by Azouani,Olson,and Titi for continuous data assimilation of nonlinear partial differential equations.The main feature of this expanded framework is its mesh-free aspect,which allows the observational data itself to dictate the subdivision of the domain via partition of unity in the spirit of the so-called Partition of Unity Method by Babuska and Melenk.As an application of this framework,we consider a nudging-based scheme for data assimilation applied to the context of the two-dimensional Navier-Stokes equations as a paradigmatic example and establish convergence to the reference solution in all higher-order Sobolev topologies in a periodic,mean-free setting.The convergence analysis also makes use of absorbing ball bounds in higherorder Sobolev norms,for which explicit bounds appear to be available in the literature only up to H^(2);such bounds are additionally proved for all integer levels of Sobolev regularity above H^(2).
基金the Foundation of Natonal Natural Science of China and the Foundation of Aducation of Hunan Province.
文摘The post-processing procedure is given by a interpolant postprocessing of the finit element solution by appropriately-define finite dimensional subspaces, The corresponding superconvergence are established on general quasi-regular finite element partitions.
文摘A new kind of matrix-valued rational interpolants is recursively established by means of generalized Samelson inverse for matrices, with scalar numerator and matrix-valued denominator. In this respect, it is essentially different from that of the previous works [7, 9], where the matrix-valued rational interpolants is in Thiele-type continued fraction form with matrix-valued numerator and scalar denominator. For both univariate and bivariate cases, sufficient conditions for existence, characterisation and uniqueness in some sense are proved respectively, and an error formula for the univariate interpolating function is also given. The results obtained in this paper are illustrated with some numerical examples.
基金supported by the National Natural Science Foundation of China(Grant No.72161034).
文摘Human motion modeling is a core technology in computer animation,game development,and humancomputer interaction.In particular,generating natural and coherent in-between motion using only the initial and terminal frames remains a fundamental yet unresolved challenge.Existing methods typically rely on dense keyframe inputs or complex prior structures,making it difficult to balance motion quality and plausibility under conditions such as sparse constraints,long-term dependencies,and diverse motion styles.To address this,we propose a motion generation framework based on a frequency-domain diffusion model,which aims to better model complex motion distributions and enhance generation stability under sparse conditions.Our method maps motion sequences to the frequency domain via the Discrete Cosine Transform(DCT),enabling more effective modeling of low-frequency motion structures while suppressing high-frequency noise.A denoising network based on self-attention is introduced to capture long-range temporal dependencies and improve global structural awareness.Additionally,a multi-objective loss function is employed to jointly optimize motion smoothness,pose diversity,and anatomical consistency,enhancing the realism and physical plausibility of the generated sequences.Comparative experiments on the Human3.6M and LaFAN1 datasets demonstrate that our method outperforms state-of-the-art approaches across multiple performance metrics,showing stronger capabilities in generating intermediate motion frames.This research offers a new perspective and methodology for human motion generation and holds promise for applications in character animation,game development,and virtual interaction.
文摘Smooth interpolants defined over tetrahedra are currently being developed for they have many applications in geography, solid modeling, finite element analysis, etc. In this paper, we will characterize a certain class of C-1 discrete tetrahedral interpolants with only C-1 data required. As special cases of the class characterized, we give two C-1 discrete tetrahedral interpolants which have concise expressions.
文摘Algorithms to generate a triangular or a quadrilateral interpolant with G^1-continuity are given in this paper for arbitrary scattered data with associated normal vectors over a prescribed triangular or quadrilateral decomposition. The interpolants are constructed with a general method to generate surfaces from moving Bezier curves under geometric constraints. With the algorithm, we may obtain interpolants in complete symbolic parametric forms, leading to a fast computation of the interpolant. A dynamic interpolation solid modelling software package DISM is implemented based on the algorithm which can be used to generate and manipulate solid objects in an interactive way.
基金the National Natural Science Foundation of China (19871054).
文摘Focuses on a study that presented an axiomatic definition to bivariate vector valued rational interpolation on distinct plane interpolation points. Definition; Existence and uniqueness; Connection.
文摘In order to solve the problem of the variable coefficient ordinary differen-tial equation on the bounded domain,the Lagrange interpolation method is used to approximate the exact solution of the equation,and the error between the numerical solution and the exact solution is obtained,and then compared with the error formed by the difference method,it is concluded that the Lagrange interpolation method is more effective in solving the variable coefficient ordinary differential equation.
基金supported by the National Natural Science Foundation of China(Grant No.52409151)the Programme of Shenzhen Key Laboratory of Green,Efficient and Intelligent Construction of Underground Metro Station(Programme No.ZDSYS20200923105200001)the Science and Technology Major Project of Xizang Autonomous Region of China(XZ202201ZD0003G).
文摘Substantial advancements have been achieved in Tunnel Boring Machine(TBM)technology and monitoring systems,yet the presence of missing data impedes accurate analysis and interpretation of TBM monitoring results.This study aims to investigate the issue of missing data in extensive TBM datasets.Through a comprehensive literature review,we analyze the mechanism of missing TBM data and compare different imputation methods,including statistical analysis and machine learning algorithms.We also examine the impact of various missing patterns and rates on the efficacy of these methods.Finally,we propose a dynamic interpolation strategy tailored for TBM engineering sites.The research results show that K-Nearest Neighbors(KNN)and Random Forest(RF)algorithms can achieve good interpolation results;As the missing rate increases,the interpolation effect of different methods will decrease;The interpolation effect of block missing is poor,followed by mixed missing,and the interpolation effect of sporadic missing is the best.On-site application results validate the proposed interpolation strategy's capability to achieve robust missing value interpolation effects,applicable in ML scenarios such as parameter optimization,attitude warning,and pressure prediction.These findings contribute to enhancing the efficiency of TBM missing data processing,offering more effective support for large-scale TBM monitoring datasets.
