This paper introduces an adaptive finite element method (AFEM) using the newest vertex bisection and marking exclusively according to the error estimator without special treatment of oscillation. By the combination ...This paper introduces an adaptive finite element method (AFEM) using the newest vertex bisection and marking exclusively according to the error estimator without special treatment of oscillation. By the combination of the global lower bound and the localized upper bound of the posteriori error estimator, perturbation of oscillation, and cardinality of the marked element set, it is proved that the AFEM is quasi-optimal for linear elasticity problems in two dimensions, and this conclusion is verified by the numerical examples.展开更多
The Burgers' equation with uncertain initial and boundary conditions is approximated using a Polynomial Chaos Expansion (PCE) approach where the solution is represented as a series of stochastic, orthogonal polynom...The Burgers' equation with uncertain initial and boundary conditions is approximated using a Polynomial Chaos Expansion (PCE) approach where the solution is represented as a series of stochastic, orthogonal polynomials. The resulting truncated PCE system is solved using a novel numerical discretization method based on spatial derivative operators satisfying the summation by parts property and weak boundary conditions to ensure stability. The resulting PCE solution yields an accurate quantitative description of the stochastic evolution of the system, provided that appropriate boundary conditions are available. The specification of the boundary data is shown to influence the solution; we will discuss the problematic implications of the lack of precisely characterized boundary data and possible ways of imposing stable and accurate boundary conditions.展开更多
The tangent polynomials Tn(z)are generalization of tangent numbers or the Euler zigzag numbers Tn.In particular,Tn(0)=Tn.These polynomials are closely related to Bernoulli,Euler and Genocchi polynomials.One of the ext...The tangent polynomials Tn(z)are generalization of tangent numbers or the Euler zigzag numbers Tn.In particular,Tn(0)=Tn.These polynomials are closely related to Bernoulli,Euler and Genocchi polynomials.One of the extensions and analogues of special polynomials that attract the attention of several mathematicians is the Apostol-type polynomials.One of these Apostol-type polynomials is the Apostol-tangent polynomials Tn(z,λ).Whenλ=1,Tn(z,1)=Tn(z).The use of hyperbolic functions to derive asymptotic approximations of polynomials together with saddle point method was applied to the Bernoulli and Euler polynomials by Lopez and Temme.The same method was applied to the Genocchi polynomials by Corcino et al.The essential steps in applying the method are(1)to obtain the integral representation of the polynomials under study using their exponential generating functions and the Cauchy integral formula,and(2)to apply the saddle point method.It is found out that the method is applicable to Apostol-tangent polynomials.As a result,asymptotic approximation of Apostol-tangent polynomials in terms of hyperbolic functions are derived for large values of the parameter n and uniform approximation with enlarged region of validity are also obtained.Moreover,higher-order Apostol-tangent polynomials are introduced.Using the same method,asymptotic approximation of higherorder Apostol-tangent polynomials in terms of hyperbolic functions are derived and uniform approximation with enlarged region of validity are also obtained.It is important to note that the consideration of Apostol-type polynomials and higher order Apostol-type polynomials were not done by Lopez and Temme.This part is first done in this paper.The accuracy of the approximations are illustrated by plotting the graphs of the exact values of the Apostol-tangent and higher-order Apostol-tangent polynomials and their corresponding approximate values for specific values of the parameters n,λand m.展开更多
In this paper, the electromagnetic scattering from a rectangular large open cavity embedded in an infinite ground plane is studied. By introducing a nonlocal artificial boundary condition, the scattering problem from ...In this paper, the electromagnetic scattering from a rectangular large open cavity embedded in an infinite ground plane is studied. By introducing a nonlocal artificial boundary condition, the scattering problem from the open cavity is reduced to a bounded domain problem. A compact fourth order finite difference scheme is then proposed to discrete the cavity scattering model in the rectangular domain, and a special treatment is enforced to approximate the boundary condition, which makes truncation errors reach (.9(h4) in the whole computational domain. A fast algorithm, exploiting the discrete Fourier transfor- mation in the horizontal and a Gaussian elimination in the vertical direction, is employed, which reduces the discrete system to a much smaller interface system. An effective pre- conditioner is presented for the BICGstab iterative solver to solve this interface system. Numerical results demonstrate the remarkable accuracy and efficiency of the proposed method. In particular, it can be used to solve the cavity model for the large wave number up to 600π.展开更多
In this paper,we investigate a priori and a posteriori error estimates of fully discrete H^(1)-Galerkin mixed finite element methods for parabolic optimal control prob-lems.The state variables and co-state variables a...In this paper,we investigate a priori and a posteriori error estimates of fully discrete H^(1)-Galerkin mixed finite element methods for parabolic optimal control prob-lems.The state variables and co-state variables are approximated by the lowest order Raviart-Thomas mixed finite element and linear finite element,and the control vari-able is approximated by piecewise constant functions.The time discretization of the state and co-state are based on finite difference methods.First,we derive a priori error estimates for the control variable,the state variables and the adjoint state variables.Second,by use of energy approach,we derive a posteriori error estimates for optimal control problems,assuming that only the underlying mesh is static.A numerical example is presented to verify the theoretical results on a priori error estimates.展开更多
We present an algebraic version of an iterative multigrid method for obstacle problems,called projected algebraic multigrid(PAMG)here.We show that classical algebraic multigrid algorithms can easily be extended to dea...We present an algebraic version of an iterative multigrid method for obstacle problems,called projected algebraic multigrid(PAMG)here.We show that classical algebraic multigrid algorithms can easily be extended to deal with this kind of problem.This paves the way for efficient multigrid solution of obstacle problems with partial differential equations arising,for example,in financial engineering.展开更多
Population spatialization is widely used for spatially downscaling census population data to finer-scale.The core idea of modern population spatialization is to establish the association between ancillary data and pop...Population spatialization is widely used for spatially downscaling census population data to finer-scale.The core idea of modern population spatialization is to establish the association between ancillary data and population at the administrative-unit-level(AUlevel)and transfer it to generate the gridded population.However,the statistical characteristic of attributes at the pixel-level differs from that at the AU-level,thus leading to prediction bias via the cross-scale modeling(i.e.scale mismatch problem).In addition,integrating multi-source data simply as covariates may underutilize spatial semantics,and lead to incorrect population disaggregation;while neglecting the spatial autocorrelation of population generates excessively heterogeneous population distribution that contradicts to real-world situation.To address the scale mismatch in downscaling,this paper proposes a Cross-Scale Feature Construction(CSFC)method.More specifically,by grading pixel-level attributes,we construct the feature vector of pixel grade proportions to narrow the scale differences in feature representation between AU-level and pixel-level.Meanwhile,fine-grained building patch and mobile positioning data are utilized to adjust the population weighting layer generated from POI-density-based regression modeling.Spatial filtering is furtherly adopted to model the spatial autocorrelation effect of population and reduce the heterogeneity in population caused by pixel-level attribute discretization.Through the comparison with traditional feature construction method and the ablation experiments,the results demonstrate significant accuracy improvements in population spatialization and verify the effectiveness of weight correction steps.Furthermore,accuracy comparisons with WorldPop and GPW datasets quantitatively illustrate the advantages of the proposed method in fine-scale population spatialization.展开更多
基金Project supported by the National Natural Science Foundation of China(Nos.1120115911426102+4 种基金and 11571293)the Natural Science Foundation of Hunan Province(No.11JJ3135)the Foundation for Outstanding Young Teachers in Higher Education of Guangdong Province(No.Yq2013054)the Pearl River S&T Nova Program of Guangzhou(No.2013J2200063)the Construct Program of the Key Discipline in Hunan University of Science and Engineering
文摘This paper introduces an adaptive finite element method (AFEM) using the newest vertex bisection and marking exclusively according to the error estimator without special treatment of oscillation. By the combination of the global lower bound and the localized upper bound of the posteriori error estimator, perturbation of oscillation, and cardinality of the marked element set, it is proved that the AFEM is quasi-optimal for linear elasticity problems in two dimensions, and this conclusion is verified by the numerical examples.
基金Supported by the US Department of Energy under the PSAAP Program
文摘The Burgers' equation with uncertain initial and boundary conditions is approximated using a Polynomial Chaos Expansion (PCE) approach where the solution is represented as a series of stochastic, orthogonal polynomials. The resulting truncated PCE system is solved using a novel numerical discretization method based on spatial derivative operators satisfying the summation by parts property and weak boundary conditions to ensure stability. The resulting PCE solution yields an accurate quantitative description of the stochastic evolution of the system, provided that appropriate boundary conditions are available. The specification of the boundary data is shown to influence the solution; we will discuss the problematic implications of the lack of precisely characterized boundary data and possible ways of imposing stable and accurate boundary conditions.
基金funded by Cebu Normal University through its Research Institute for Computational Mathematics and Physics(RICMP).
文摘The tangent polynomials Tn(z)are generalization of tangent numbers or the Euler zigzag numbers Tn.In particular,Tn(0)=Tn.These polynomials are closely related to Bernoulli,Euler and Genocchi polynomials.One of the extensions and analogues of special polynomials that attract the attention of several mathematicians is the Apostol-type polynomials.One of these Apostol-type polynomials is the Apostol-tangent polynomials Tn(z,λ).Whenλ=1,Tn(z,1)=Tn(z).The use of hyperbolic functions to derive asymptotic approximations of polynomials together with saddle point method was applied to the Bernoulli and Euler polynomials by Lopez and Temme.The same method was applied to the Genocchi polynomials by Corcino et al.The essential steps in applying the method are(1)to obtain the integral representation of the polynomials under study using their exponential generating functions and the Cauchy integral formula,and(2)to apply the saddle point method.It is found out that the method is applicable to Apostol-tangent polynomials.As a result,asymptotic approximation of Apostol-tangent polynomials in terms of hyperbolic functions are derived for large values of the parameter n and uniform approximation with enlarged region of validity are also obtained.Moreover,higher-order Apostol-tangent polynomials are introduced.Using the same method,asymptotic approximation of higherorder Apostol-tangent polynomials in terms of hyperbolic functions are derived and uniform approximation with enlarged region of validity are also obtained.It is important to note that the consideration of Apostol-type polynomials and higher order Apostol-type polynomials were not done by Lopez and Temme.This part is first done in this paper.The accuracy of the approximations are illustrated by plotting the graphs of the exact values of the Apostol-tangent and higher-order Apostol-tangent polynomials and their corresponding approximate values for specific values of the parameters n,λand m.
基金Acknowledgments. The research of the second author was supported by the FRG grant of the Hong Kong Baptist University (No. FRG/08-09/II-35). The research of the third author was supported by the FRG grant of the Hong Kong Baptist University, the GIF Grants of Hong Kong Research Grants Council, and the Collaborative Research Fund of National Science Foundation of China (NSFC) under Grant No. G10729101,
文摘In this paper, the electromagnetic scattering from a rectangular large open cavity embedded in an infinite ground plane is studied. By introducing a nonlocal artificial boundary condition, the scattering problem from the open cavity is reduced to a bounded domain problem. A compact fourth order finite difference scheme is then proposed to discrete the cavity scattering model in the rectangular domain, and a special treatment is enforced to approximate the boundary condition, which makes truncation errors reach (.9(h4) in the whole computational domain. A fast algorithm, exploiting the discrete Fourier transfor- mation in the horizontal and a Gaussian elimination in the vertical direction, is employed, which reduces the discrete system to a much smaller interface system. An effective pre- conditioner is presented for the BICGstab iterative solver to solve this interface system. Numerical results demonstrate the remarkable accuracy and efficiency of the proposed method. In particular, it can be used to solve the cavity model for the large wave number up to 600π.
基金This work was supported by National Natural Science Foundation of China(11601014,11626037,11526036)China Postdoctoral Science Foundation(2016M 601359)+4 种基金Scientific and Technological Developing Scheme of Jilin Province(20160520108 JH,20170101037JC)Science and Technology Research Project of Jilin Provincial Depart-ment of Education(201646)Special Funding for Promotion of Young Teachers of Beihua University,Natural Science Foundation of Hunan Province(14JJ3135)the Youth Project of Hunan Provincial Education Department(15B096)the construct program of the key discipline in Hunan University of Science and Engineering.
文摘In this paper,we investigate a priori and a posteriori error estimates of fully discrete H^(1)-Galerkin mixed finite element methods for parabolic optimal control prob-lems.The state variables and co-state variables are approximated by the lowest order Raviart-Thomas mixed finite element and linear finite element,and the control vari-able is approximated by piecewise constant functions.The time discretization of the state and co-state are based on finite difference methods.First,we derive a priori error estimates for the control variable,the state variables and the adjoint state variables.Second,by use of energy approach,we derive a posteriori error estimates for optimal control problems,assuming that only the underlying mesh is static.A numerical example is presented to verify the theoretical results on a priori error estimates.
文摘We present an algebraic version of an iterative multigrid method for obstacle problems,called projected algebraic multigrid(PAMG)here.We show that classical algebraic multigrid algorithms can easily be extended to deal with this kind of problem.This paves the way for efficient multigrid solution of obstacle problems with partial differential equations arising,for example,in financial engineering.
基金National Natural Science Foundation of China[Grant Nos.42090010,U20A2091,41971349,and 41930107]National Key R&D Program of China[Grant Nos.2018YFC0809800 and 2017YFB0503704].
文摘Population spatialization is widely used for spatially downscaling census population data to finer-scale.The core idea of modern population spatialization is to establish the association between ancillary data and population at the administrative-unit-level(AUlevel)and transfer it to generate the gridded population.However,the statistical characteristic of attributes at the pixel-level differs from that at the AU-level,thus leading to prediction bias via the cross-scale modeling(i.e.scale mismatch problem).In addition,integrating multi-source data simply as covariates may underutilize spatial semantics,and lead to incorrect population disaggregation;while neglecting the spatial autocorrelation of population generates excessively heterogeneous population distribution that contradicts to real-world situation.To address the scale mismatch in downscaling,this paper proposes a Cross-Scale Feature Construction(CSFC)method.More specifically,by grading pixel-level attributes,we construct the feature vector of pixel grade proportions to narrow the scale differences in feature representation between AU-level and pixel-level.Meanwhile,fine-grained building patch and mobile positioning data are utilized to adjust the population weighting layer generated from POI-density-based regression modeling.Spatial filtering is furtherly adopted to model the spatial autocorrelation effect of population and reduce the heterogeneity in population caused by pixel-level attribute discretization.Through the comparison with traditional feature construction method and the ablation experiments,the results demonstrate significant accuracy improvements in population spatialization and verify the effectiveness of weight correction steps.Furthermore,accuracy comparisons with WorldPop and GPW datasets quantitatively illustrate the advantages of the proposed method in fine-scale population spatialization.