Accurately approximating higher order derivatives is an inherently difficult problem. It is shown that a random variable shape parameter strategy can improve the accuracy of approximating higher order derivatives with...Accurately approximating higher order derivatives is an inherently difficult problem. It is shown that a random variable shape parameter strategy can improve the accuracy of approximating higher order derivatives with Radial Basis Function methods. The method is used to solve fourth order boundary value problems. The use and location of ghost points are examined in order to enforce the extra boundary conditions that are necessary to make a fourth-order problem well posed. The use of ghost points versus solving an overdetermined linear system via least squares is studied. For a general fourth-order boundary value problem, the recommended approach is to either use one of two novel sets of ghost centers introduced here or else to use a least squares approach. When using either ghost centers or least squares, the random variable shape parameter strategy results in significantly better accuracy than when a constant shape parameter is used.展开更多
In this paper,hierarchical basis method for second order nonsymmetric andindefinite elliptic problem on a polygonal domain(possibly nonconvex)discreted by avertex-centered covolume method is constructed.
The reduced basis methods (RBM) have been demonstrated as a promising numerical technique for statics problems and are extended to structural dynamic problems in this paper. Direct step-by-step integration and mode su...The reduced basis methods (RBM) have been demonstrated as a promising numerical technique for statics problems and are extended to structural dynamic problems in this paper. Direct step-by-step integration and mode superposition are the most widely used methods in the field of the finite element analysis of structural dynamic response and solid mechanics. Herein these two methods are both transformed into reduced forms according to the proposed reduced basis methods. To generate a reduced surrogate model with small size, a greedy algorithm is suggested to construct sample set and reduced basis space adaptively in a prescribed training parameter space. For mode superposition method, the reduced basis space comprises the truncated eigenvectors from generalized eigenvalue problem associated with selected sample parameters. The reduced generalized eigenvalue problem is obtained by the projection of original generalized eigenvalue problem onto the reduced basis space. In the situation of direct integration, the solutions of the original increment formulation corresponding to the sample set are extracted to construct the reduced basis space. The reduced increment formulation is formed by the same method as mode superposition method. Numerical example is given in Section 5 to validate the efficiency of the presented reduced basis methods for structural dynamic problems.展开更多
In this article,a meshless method using the spacetime collocation for solving the two-dimensional backward heat conduction problem(BHCP)is proposed.The spacetime collocation meshless method(SCMM)is to derive the gener...In this article,a meshless method using the spacetime collocation for solving the two-dimensional backward heat conduction problem(BHCP)is proposed.The spacetime collocation meshless method(SCMM)is to derive the general solutions as the basis functions for the two-dimensional transient heat equation using the separation of variables.Numerical solutions of the heat conduction problem are expressed as a series using the addition theorem.Because the basis functions are the general solutions of the governing equation,the boundary points may be collocated on the spacetime boundary of the domain.The proposed method is verified by conducting several heat conduction problems.We also carry out numerical applications to compare the SCMM with other meshless methods.The results show that the SCMM is accurate and efficient.Furthermore,it is found that the recovered boundary data on inaccessible boundary can be obtained with high accuracy even though the over specified data are provided only at a 1/6 portion of the spacetime boundary.展开更多
Two new regularization algorithms for solving the first-kind Volterra integral equation, which describes the pressure-rate deconvolution problem in well test data interpretation, are developed in this paper. The main ...Two new regularization algorithms for solving the first-kind Volterra integral equation, which describes the pressure-rate deconvolution problem in well test data interpretation, are developed in this paper. The main features of the problem are the strong nonuniform scale of the solution and large errors (up to 15%) in the input data. In both algorithms, the solution is represented as decomposition on special basic functions, which satisfy given a priori information on solution, and this idea allow us significantly to improve the quality of approximate solution and simplify solving the minimization problem. The theoretical details of the algorithms, as well as the results of numerical experiments for proving robustness of the algorithms, are presented.展开更多
A local meshless method is applied to find the numerical solutions of two classes of inverse problems in parabolic equations. The problem is reconstructing the source term using a solution specified at some internal p...A local meshless method is applied to find the numerical solutions of two classes of inverse problems in parabolic equations. The problem is reconstructing the source term using a solution specified at some internal points;one class is that the source term is time dependent, and the other class is that the source term is time and space dependent. Some numerical experiments are presented and discussed.展开更多
In this paper,we investigate a stochastic meshfree finite volume element method for an optimal control problem governed by the convection diffusion equations with random coefficients.There are two contributions of thi...In this paper,we investigate a stochastic meshfree finite volume element method for an optimal control problem governed by the convection diffusion equations with random coefficients.There are two contributions of this paper.Firstly,we establish a scheme to approximate the optimality system by using the finite volume element method in the physical space and the meshfree method in the probability space,which is competitive for high-dimensional random inputs.Secondly,the a priori error estimates are derived for the state,the co-state and the control variables.Some numerical tests are carried out to confirm the theoretical results and demonstrate the efficiency of the proposed method.展开更多
Generalized Jacobi polynomials with indexes α,β∈ R are introduced and some basic properties are established. As examples of applications,the second- and fourth-order elliptic boundary value problems with Dirichlet ...Generalized Jacobi polynomials with indexes α,β∈ R are introduced and some basic properties are established. As examples of applications,the second- and fourth-order elliptic boundary value problems with Dirichlet or Robin boundary conditions are considered,and the generalized Jacobi spectral schemes are proposed. For the diagonalization of discrete systems,the Jacobi-Sobolev orthogonal basis functions are constructed,which allow the exact solutions and the approximate solutions to be represented in the forms of infinite and truncated Jacobi series. Error estimates are obtained and numerical results are provided to illustrate the effectiveness and the spectral accuracy.展开更多
To optimize the algorithms for the dihedral hidden subgroup problem,we present a new algorithm based on lattice basis reduction algorithm.For n\120,we reduce the dihedral hidden subgroup problem to shortest vector pro...To optimize the algorithms for the dihedral hidden subgroup problem,we present a new algorithm based on lattice basis reduction algorithm.For n\120,we reduce the dihedral hidden subgroup problem to shortest vector problem.A subroutine is given to get a transition quantum state by constructing a phase filter function,and then the measurement basis are derived based on the lattice basis reduction algorithm for solving low density subset sum problem.Finally,the parity of slope s is revealed by the measurement.This algorithm needs preparing mn quantum states,m qubits to store and O(n2)classical space,which is superior to existing algorithms.展开更多
Let X＊ be a free monoid over an alphabet X and W be a finite language over X. Let S（W） be the Rees quotient X＊/I（W）, where I（W） is the ideal of X＊ consisting of all elements of X＊ that are not subwords of W....Let X＊ be a free monoid over an alphabet X and W be a finite language over X. Let S（W） be the Rees quotient X＊/I（W）, where I（W） is the ideal of X＊ consisting of all elements of X＊ that are not subwords of W. Then S（W） is a finite monoid with zero and is called the discrete syntactic monoid of W. W is called finitely based if the monoid S（W） is finitely based. In this paper, we give some sufficient conditions for a monoid to be non-finitely based. Using these conditions and other results, we describe all finitely based 2-limited words over a three-element alphabet. Furthermore, an explicit algorithm is given to decide that whether or not a 2-limited word in which there are exactly two non-linear letters is finitely based.展开更多
文摘Accurately approximating higher order derivatives is an inherently difficult problem. It is shown that a random variable shape parameter strategy can improve the accuracy of approximating higher order derivatives with Radial Basis Function methods. The method is used to solve fourth order boundary value problems. The use and location of ghost points are examined in order to enforce the extra boundary conditions that are necessary to make a fourth-order problem well posed. The use of ghost points versus solving an overdetermined linear system via least squares is studied. For a general fourth-order boundary value problem, the recommended approach is to either use one of two novel sets of ghost centers introduced here or else to use a least squares approach. When using either ghost centers or least squares, the random variable shape parameter strategy results in significantly better accuracy than when a constant shape parameter is used.
基金This work was supported by the National Natural Science Foundation of China under grant 10071015
文摘In this paper,hierarchical basis method for second order nonsymmetric andindefinite elliptic problem on a polygonal domain(possibly nonconvex)discreted by avertex-centered covolume method is constructed.
文摘The reduced basis methods (RBM) have been demonstrated as a promising numerical technique for statics problems and are extended to structural dynamic problems in this paper. Direct step-by-step integration and mode superposition are the most widely used methods in the field of the finite element analysis of structural dynamic response and solid mechanics. Herein these two methods are both transformed into reduced forms according to the proposed reduced basis methods. To generate a reduced surrogate model with small size, a greedy algorithm is suggested to construct sample set and reduced basis space adaptively in a prescribed training parameter space. For mode superposition method, the reduced basis space comprises the truncated eigenvectors from generalized eigenvalue problem associated with selected sample parameters. The reduced generalized eigenvalue problem is obtained by the projection of original generalized eigenvalue problem onto the reduced basis space. In the situation of direct integration, the solutions of the original increment formulation corresponding to the sample set are extracted to construct the reduced basis space. The reduced increment formulation is formed by the same method as mode superposition method. Numerical example is given in Section 5 to validate the efficiency of the presented reduced basis methods for structural dynamic problems.
文摘In this article,a meshless method using the spacetime collocation for solving the two-dimensional backward heat conduction problem(BHCP)is proposed.The spacetime collocation meshless method(SCMM)is to derive the general solutions as the basis functions for the two-dimensional transient heat equation using the separation of variables.Numerical solutions of the heat conduction problem are expressed as a series using the addition theorem.Because the basis functions are the general solutions of the governing equation,the boundary points may be collocated on the spacetime boundary of the domain.The proposed method is verified by conducting several heat conduction problems.We also carry out numerical applications to compare the SCMM with other meshless methods.The results show that the SCMM is accurate and efficient.Furthermore,it is found that the recovered boundary data on inaccessible boundary can be obtained with high accuracy even though the over specified data are provided only at a 1/6 portion of the spacetime boundary.
文摘Two new regularization algorithms for solving the first-kind Volterra integral equation, which describes the pressure-rate deconvolution problem in well test data interpretation, are developed in this paper. The main features of the problem are the strong nonuniform scale of the solution and large errors (up to 15%) in the input data. In both algorithms, the solution is represented as decomposition on special basic functions, which satisfy given a priori information on solution, and this idea allow us significantly to improve the quality of approximate solution and simplify solving the minimization problem. The theoretical details of the algorithms, as well as the results of numerical experiments for proving robustness of the algorithms, are presented.
文摘A local meshless method is applied to find the numerical solutions of two classes of inverse problems in parabolic equations. The problem is reconstructing the source term using a solution specified at some internal points;one class is that the source term is time dependent, and the other class is that the source term is time and space dependent. Some numerical experiments are presented and discussed.
基金supported by the National Natural Science Foundation of China(Nos.11701253,11971259,11801216)Natural Science Foundation of Shandong Province(No.ZR2017BA010)。
文摘In this paper,we investigate a stochastic meshfree finite volume element method for an optimal control problem governed by the convection diffusion equations with random coefficients.There are two contributions of this paper.Firstly,we establish a scheme to approximate the optimality system by using the finite volume element method in the physical space and the meshfree method in the probability space,which is competitive for high-dimensional random inputs.Secondly,the a priori error estimates are derived for the state,the co-state and the control variables.Some numerical tests are carried out to confirm the theoretical results and demonstrate the efficiency of the proposed method.
基金the National Natural Science Foundation of China (Nos.11571238,11601332,91130014,11471312 and 91430216).
文摘Generalized Jacobi polynomials with indexes α,β∈ R are introduced and some basic properties are established. As examples of applications,the second- and fourth-order elliptic boundary value problems with Dirichlet or Robin boundary conditions are considered,and the generalized Jacobi spectral schemes are proposed. For the diagonalization of discrete systems,the Jacobi-Sobolev orthogonal basis functions are constructed,which allow the exact solutions and the approximate solutions to be represented in the forms of infinite and truncated Jacobi series. Error estimates are obtained and numerical results are provided to illustrate the effectiveness and the spectral accuracy.
基金supported by a grant from the Major State Basic Research Development Program of China (973 Program) (2013CB338002)
文摘To optimize the algorithms for the dihedral hidden subgroup problem,we present a new algorithm based on lattice basis reduction algorithm.For n\120,we reduce the dihedral hidden subgroup problem to shortest vector problem.A subroutine is given to get a transition quantum state by constructing a phase filter function,and then the measurement basis are derived based on the lattice basis reduction algorithm for solving low density subset sum problem.Finally,the parity of slope s is revealed by the measurement.This algorithm needs preparing mn quantum states,m qubits to store and O(n2)classical space,which is superior to existing algorithms.
基金Supported by National Natural Science Foundation of China (Grant No. 10971086)Mathematical Tianyuan Foundation of China (Grant No. 11126186)+1 种基金Natural Science Foundation of Gansu Province (Grant No.1107RJZA218)Fundamental Research Funds for Central Universities (Grant No. lzujbky-2012-12)
文摘Let X＊ be a free monoid over an alphabet X and W be a finite language over X. Let S（W） be the Rees quotient X＊/I（W）, where I（W） is the ideal of X＊ consisting of all elements of X＊ that are not subwords of W. Then S（W） is a finite monoid with zero and is called the discrete syntactic monoid of W. W is called finitely based if the monoid S（W） is finitely based. In this paper, we give some sufficient conditions for a monoid to be non-finitely based. Using these conditions and other results, we describe all finitely based 2-limited words over a three-element alphabet. Furthermore, an explicit algorithm is given to decide that whether or not a 2-limited word in which there are exactly two non-linear letters is finitely based.
基金supported by the National Science and Technology Major Projects(No.2011ZX05020-008)Well Logging Advanced Technique and Application Basis Research Project of Petrochina Company(No.2011A-3901)
