We present a mathematical and numerical study for a pointwise optimal control problem governed by a variable-coefficient Riesz-fractional diffusion equation.Due to the impact of the variable diffusivity coefficient,ex...We present a mathematical and numerical study for a pointwise optimal control problem governed by a variable-coefficient Riesz-fractional diffusion equation.Due to the impact of the variable diffusivity coefficient,existing regularity results for their constantcoefficient counterparts do not apply,while the bilinear forms of the state(adjoint)equation may lose the coercivity that is critical in error estimates of the finite element method.We reformulate the state equation as an equivalent constant-coefficient fractional diffusion equation with the addition of a variable-coefficient low-order fractional advection term.First order optimality conditions are accordingly derived and the smoothing properties of the solutions are analyzed by,e.g.,interpolation estimates.The weak coercivity of the resulting bilinear forms are proven via the Garding inequality,based on which we prove the optimal-order convergence estimates of the finite element method for the(adjoint)state variable and the control variable.Numerical experiments substantiate the theoretical predictions.展开更多
A framework for algebraic multilevel preconditioning methods is presented for solving largesparse systems of linear equations with symmetric positive definite coefficient matrices, whicharise in the discretization of ...A framework for algebraic multilevel preconditioning methods is presented for solving largesparse systems of linear equations with symmetric positive definite coefficient matrices, whicharise in the discretization of second order elliptic boundary value problems by the finite elementmethod. This framework covers not only all known algebraic multilevel preconditioning methods,but yields also new ones. It is shown that all preconditioners within this framework have optimalorders of complexities for problems in two-dimensional (2-D) and three-dimensional(3-D) problemdomains, and their relative condition numbers are bounded uniformly with respect to the numbersof both the levels and the nodes.展开更多
A class of hybrid algebraic multilevel preconditioning methods are presented for solving the systems of linear equations with symmetric positive definite matrices resulting from the discretization of many second orde...A class of hybrid algebraic multilevel preconditioning methods are presented for solving the systems of linear equations with symmetric positive definite matrices resulting from the discretization of many second order elliptic boundary value problems by the finite element method. The new preconditioners are shown to be of optimal orders of complexities for 2-D and 3-D problem domains, and their relative condition numbers are estimated to be bounded uniformly with respect to the numbers of both levels and nodes.1980 Mathematics Subject Classification (1985 Revision ). AMS (MOS ): 65F10,65N20, 65N301 CR:Gl. 3.展开更多
A framework for parallel algebraic multilevel preconditioning methods presented for solving large sparse systems of linear equstions with symmetric positive definite coefficient matrices,which arise in suitable finite...A framework for parallel algebraic multilevel preconditioning methods presented for solving large sparse systems of linear equstions with symmetric positive definite coefficient matrices,which arise in suitable finite element discretizations of many second-order self-adjoint elliptic boundary value problems. This framework not only covers all known parallel algebraic multilevel preconditioning methods, but also yields new ones. It is shown that all preconditioners within this framework have optimal orders of complexities for problems in two-dimensional(2-D) and three-dimensional (3-D) problem domains, and their relative condition numbers are bounded uniformly with respect to the numbers of both levels and nodes.展开更多
We develop a family of characteristic discontinuous Galerkin methods for transient advection-diffusion equations,including the characteristic NIPG,OBB,IIPG,and SIPG schemes.The derived schemes possess combined advanta...We develop a family of characteristic discontinuous Galerkin methods for transient advection-diffusion equations,including the characteristic NIPG,OBB,IIPG,and SIPG schemes.The derived schemes possess combined advantages of EulerianLagrangian methods and discontinuous Galerkin methods.An optimal-order error estimate in the L2 norm and a superconvergence estimate in a weighted energy norm are proved for the characteristic NIPG,IIPG,and SIPG scheme.Numerical experiments are presented to confirm the optimal-order spatial and temporal convergence rates of these schemes as proved in the theorems and to show that these schemes compare favorably to the standard NIPG,OBB,IIPG,and SIPG schemes in the context of advection-diffusion equations.展开更多
针对多输入多输出系统中常用的非线性检测算法,如排序QR分解(Sorted QR Decomposition,SQRD)、球型译码(Sphere Decoding,SD)、K-Best或QRM(QR Decomposition and M algorithm)等,提出了一种具有最优检测顺序的QR快速分解方法,作为检测...针对多输入多输出系统中常用的非线性检测算法,如排序QR分解(Sorted QR Decomposition,SQRD)、球型译码(Sphere Decoding,SD)、K-Best或QRM(QR Decomposition and M algorithm)等,提出了一种具有最优检测顺序的QR快速分解方法,作为检测前的预处理操作。该算法首先对信道矩阵进行第一次QR分解,根据所得上三角矩阵R可确定最优的检测顺序,并按该顺序对R进行列重排。然后对R进行第二次QR分解,即得具有最优检测顺序的QR分解结果。与现有的基于R对角元素的模值排序的QR分解算法相比,本算法可保证检测顺序最优从而性能最优。仿真结果表明天线配置为4×4和6×6时,在误码率10^(-3)处可节约信噪比分别为:1dB和2dB;与现有的基于信干噪比排序的QR分解算法相比,本算法与其性能一致的基础上可节约25%的复乘法次数和33%的复加法次数。展开更多
Based on the newly-developed element energy projection （EEP） method with optimal super-convergence order for computation of super-convergent results, an improved self-adaptive strategy for one-dimensional finite ele...Based on the newly-developed element energy projection （EEP） method with optimal super-convergence order for computation of super-convergent results, an improved self-adaptive strategy for one-dimensional finite element method （FEM） is proposed. In the strategy, a posteriori errors are estimated by comparing FEM solutions to EEP super-convergent solutions with optimal order of super-convergence, meshes are refined by using the error-averaging method. Quasi-FEM solutions are used to replace the true FEM solutions in the adaptive process. This strategy has been found to be simple, clear, efficient and reliable. For most problems, only one adaptive step is needed to produce the required FEM solutions which pointwise satisfy the user specified error tolerances in the max-norm. Taking the elliptical ordinary differential equation of the second order as the model problem, this paper describes the fundamental idea, implementation strategy and computational algorithm and representative numerical examples are given to show the effectiveness and reliability of the proposed approach.展开更多
基金supported by the National Natural Science Foundation of China(11971276,12171287)Natural Science Foundation of Shandong Province(ZR2016JL004)+1 种基金supported by the China Postdoctoral Science Foundation(2021TQ0017,2021M700244)International Postdoctoral Exchange Fellowship Program(Talent-Introduction Program)(YJ20210019)。
文摘We present a mathematical and numerical study for a pointwise optimal control problem governed by a variable-coefficient Riesz-fractional diffusion equation.Due to the impact of the variable diffusivity coefficient,existing regularity results for their constantcoefficient counterparts do not apply,while the bilinear forms of the state(adjoint)equation may lose the coercivity that is critical in error estimates of the finite element method.We reformulate the state equation as an equivalent constant-coefficient fractional diffusion equation with the addition of a variable-coefficient low-order fractional advection term.First order optimality conditions are accordingly derived and the smoothing properties of the solutions are analyzed by,e.g.,interpolation estimates.The weak coercivity of the resulting bilinear forms are proven via the Garding inequality,based on which we prove the optimal-order convergence estimates of the finite element method for the(adjoint)state variable and the control variable.Numerical experiments substantiate the theoretical predictions.
文摘A framework for algebraic multilevel preconditioning methods is presented for solving largesparse systems of linear equations with symmetric positive definite coefficient matrices, whicharise in the discretization of second order elliptic boundary value problems by the finite elementmethod. This framework covers not only all known algebraic multilevel preconditioning methods,but yields also new ones. It is shown that all preconditioners within this framework have optimalorders of complexities for problems in two-dimensional (2-D) and three-dimensional(3-D) problemdomains, and their relative condition numbers are bounded uniformly with respect to the numbersof both the levels and the nodes.
文摘A class of hybrid algebraic multilevel preconditioning methods are presented for solving the systems of linear equations with symmetric positive definite matrices resulting from the discretization of many second order elliptic boundary value problems by the finite element method. The new preconditioners are shown to be of optimal orders of complexities for 2-D and 3-D problem domains, and their relative condition numbers are estimated to be bounded uniformly with respect to the numbers of both levels and nodes.1980 Mathematics Subject Classification (1985 Revision ). AMS (MOS ): 65F10,65N20, 65N301 CR:Gl. 3.
文摘A framework for parallel algebraic multilevel preconditioning methods presented for solving large sparse systems of linear equstions with symmetric positive definite coefficient matrices,which arise in suitable finite element discretizations of many second-order self-adjoint elliptic boundary value problems. This framework not only covers all known parallel algebraic multilevel preconditioning methods, but also yields new ones. It is shown that all preconditioners within this framework have optimal orders of complexities for problems in two-dimensional(2-D) and three-dimensional (3-D) problem domains, and their relative condition numbers are bounded uniformly with respect to the numbers of both levels and nodes.
文摘We develop a family of characteristic discontinuous Galerkin methods for transient advection-diffusion equations,including the characteristic NIPG,OBB,IIPG,and SIPG schemes.The derived schemes possess combined advantages of EulerianLagrangian methods and discontinuous Galerkin methods.An optimal-order error estimate in the L2 norm and a superconvergence estimate in a weighted energy norm are proved for the characteristic NIPG,IIPG,and SIPG scheme.Numerical experiments are presented to confirm the optimal-order spatial and temporal convergence rates of these schemes as proved in the theorems and to show that these schemes compare favorably to the standard NIPG,OBB,IIPG,and SIPG schemes in the context of advection-diffusion equations.
文摘针对多输入多输出系统中常用的非线性检测算法,如排序QR分解(Sorted QR Decomposition,SQRD)、球型译码(Sphere Decoding,SD)、K-Best或QRM(QR Decomposition and M algorithm)等,提出了一种具有最优检测顺序的QR快速分解方法,作为检测前的预处理操作。该算法首先对信道矩阵进行第一次QR分解,根据所得上三角矩阵R可确定最优的检测顺序,并按该顺序对R进行列重排。然后对R进行第二次QR分解,即得具有最优检测顺序的QR分解结果。与现有的基于R对角元素的模值排序的QR分解算法相比,本算法可保证检测顺序最优从而性能最优。仿真结果表明天线配置为4×4和6×6时,在误码率10^(-3)处可节约信噪比分别为:1dB和2dB;与现有的基于信干噪比排序的QR分解算法相比,本算法与其性能一致的基础上可节约25%的复乘法次数和33%的复加法次数。
基金the National Natural Science Foundation of China(No.50678093)Program for Changjiang Scholars and Innovative Research Team in University(No.IRT00736)
文摘Based on the newly-developed element energy projection （EEP） method with optimal super-convergence order for computation of super-convergent results, an improved self-adaptive strategy for one-dimensional finite element method （FEM） is proposed. In the strategy, a posteriori errors are estimated by comparing FEM solutions to EEP super-convergent solutions with optimal order of super-convergence, meshes are refined by using the error-averaging method. Quasi-FEM solutions are used to replace the true FEM solutions in the adaptive process. This strategy has been found to be simple, clear, efficient and reliable. For most problems, only one adaptive step is needed to produce the required FEM solutions which pointwise satisfy the user specified error tolerances in the max-norm. Taking the elliptical ordinary differential equation of the second order as the model problem, this paper describes the fundamental idea, implementation strategy and computational algorithm and representative numerical examples are given to show the effectiveness and reliability of the proposed approach.
