The energy norm convergence rate of the finite element solution of the heat equation is reduced by the time-regularity of the exact solution. This paper presents an adaptive finite element treatment of time-dependent ...The energy norm convergence rate of the finite element solution of the heat equation is reduced by the time-regularity of the exact solution. This paper presents an adaptive finite element treatment of time-dependent singularities on the one-dimensional heat equation. The method is based on a Fourier decomposition of the solution and an extraction formula of the coefficients of the singularities coupled with a predictor-corrector algorithm. The method recovers the optimal convergence rate of the finite element method on a quasi-uniform mesh refinement. Numerical results are carried out to show the efficiency of the method.展开更多
A globally convergent infeasible-interior-point predictor-corrector algorithm is presented for the second-order cone programming (SOCP) by using the Alizadeh- Haeberly-Overton (AHO) search direction. This algorith...A globally convergent infeasible-interior-point predictor-corrector algorithm is presented for the second-order cone programming (SOCP) by using the Alizadeh- Haeberly-Overton (AHO) search direction. This algorithm does not require the feasibility of the initial points and iteration points. Under suitable assumptions, it is shown that the algorithm can find an -approximate solution of an SOCP in at most O(√n ln(ε0/ε)) iterations. The iteration-complexity bound of our algorithm is almost the same as the best known bound of feasible interior point algorithms for the SOCP.展开更多
This article presents a polynomial predictor-corrector interior-point algorithm for convex quadratic programming based on a modified predictor-corrector interior-point algorithm. In this algorithm, there is only one c...This article presents a polynomial predictor-corrector interior-point algorithm for convex quadratic programming based on a modified predictor-corrector interior-point algorithm. In this algorithm, there is only one corrector step after each predictor step, where Step 2 is a predictor step and Step 4 is a corrector step in the algorithm. In the algorithm, the predictor step decreases the dual gap as much as possible in a wider neighborhood of the central path and the corrector step draws iteration points back to a narrower neighborhood and make a reduction for the dual gap. It is shown that the algorithm has O(√nL) iteration complexity which is the best result for convex quadratic programming so far.展开更多
A conformal optical system refers to the one whose first optical surface conforms to both aerodynamic and imaging requirements. Appropriate correction is required because a conformal dome induces significant aberratio...A conformal optical system refers to the one whose first optical surface conforms to both aerodynamic and imaging requirements. Appropriate correction is required because a conformal dome induces significant aberrations. This paper intends to explain that an effective solution to the easy-fabrication conic surface corrector compensates aberrations induced by a coaxial aspheric dome. A conformal optical system with an ellipsoid MgF2 conformal dome, which has a fineness ratio of 2.0, is designed as an example. The field of regard angle is -4-30 degrees with a +2 degree instantaneous field of view. The system's ultimate value of modulation transfer function is close to the diffraction limit, which indicates that the performance of the conic conformal optical system with a fixed conic corrector meets the imaging requirements.展开更多
A three-dimensional (3D) predictor-corrector finite difference method for standing wave is developed. It is applied to solve the 3D nonlinear potential flow equa- tions with a free surface. The 3D irregular tank is ...A three-dimensional (3D) predictor-corrector finite difference method for standing wave is developed. It is applied to solve the 3D nonlinear potential flow equa- tions with a free surface. The 3D irregular tank is mapped onto a fixed cubic tank through the proper coordinate transform schemes. The cubic tank is distributed by the staggered meshgrid, and the staggered meshgrid is used to denote the variables of the flow field. The predictor-corrector finite difference method is given to develop the difference equa- tions of the dynamic boundary equation and kinematic boundary equation. Experimental results show that, using the finite difference method of the predictor-corrector scheme, the numerical solutions agree well with the published results. The wave profiles of the standing wave with different amplitudes and wave lengths are studied. The numerical solutions are also analyzed and presented graphically.展开更多
Multi-conjugation adaptive optics(MCAOs) have been investigated and used in the large aperture optical telescopes for high-resolution imaging with large field of view(FOV).The atmospheric tomographic phase reconst...Multi-conjugation adaptive optics(MCAOs) have been investigated and used in the large aperture optical telescopes for high-resolution imaging with large field of view(FOV).The atmospheric tomographic phase reconstruction and projection of three-dimensional turbulence volume onto wavefront correctors,such as deformable mirrors(DMs) or liquid crystal wavefront correctors(LCWCs),is a very important step in the data processing of an MCAO's controller.In this paper,a method according to the wavefront reconstruction performance of MCAO is presented to evaluate the optimized configuration of multi laser guide stars(LGSs) and the reasonable conjugation heights of LCWCs.Analytical formulations are derived for the different configurations and are used to generate optimized parameters for MCAO.Several examples are given to demonstrate our LGSs configuration optimization method.Compared with traditional methods,our method has minimum wavefront tomographic error,which will be helpful to get higher imaging resolution at large FOV in MCAO.展开更多
Based on the ideas of infeasible interior-point methods and predictor-corrector algorithms, two interior-point predictor-corrector algorithms for the second-order cone programming (SOCP) are presented. The two algor...Based on the ideas of infeasible interior-point methods and predictor-corrector algorithms, two interior-point predictor-corrector algorithms for the second-order cone programming (SOCP) are presented. The two algorithms use the Newton direction and the Euler direction as the predictor directions, respectively. The corrector directions belong to the category of the Alizadeh-Haeberly-Overton (AHO) directions. These algorithms are suitable to the cases of feasible and infeasible interior iterative points. A simpler neighborhood of the central path for the SOCP is proposed, which is the pivotal difference from other interior-point predictor-corrector algorithms. Under some assumptions, the algorithms possess the global, linear, and quadratic convergence. The complexity bound O(rln(εo/ε)) is obtained, where r denotes the number of the second-order cones in the SOCP problem. The numerical results show that the proposed algorithms are effective.展开更多
A finite volume element predictor-corrector method for a class of nonlinear parabolic system of equations is presented and analyzed. Suboptimal L^2 error estimate for the finite volume element predictor-corrector meth...A finite volume element predictor-corrector method for a class of nonlinear parabolic system of equations is presented and analyzed. Suboptimal L^2 error estimate for the finite volume element predictor-corrector method is derived. A numerical experiment shows that the numerical results are consistent with theoretical analysis.展开更多
Theory has it that increasing the step length improves the accuracy of a method. In order to affirm this we increased the step length of the concept in [1] by one to get k = 5. The technique of collocation and interpo...Theory has it that increasing the step length improves the accuracy of a method. In order to affirm this we increased the step length of the concept in [1] by one to get k = 5. The technique of collocation and interpolation of the power series approximate solution at some selected grid points is considered so as to generate continuous linear multistep methods with constant step sizes. Two, three and four interpolation points are considered to generate the continuous predictor-corrector methods which are implemented in block method respectively. The proposed methods when tested on some numerical examples performed more efficiently than those of [1]. Interestingly the concept of self starting [2] and that of constant order are reaffirmed in our new methods.展开更多
Direct dynamics simulations are a useful and general approach for studying the atomistic properties of complex chemical systems because they do not require fitting an analytic potential energy function.Hessian-based p...Direct dynamics simulations are a useful and general approach for studying the atomistic properties of complex chemical systems because they do not require fitting an analytic potential energy function.Hessian-based predictor-corrector integrators are a widely used approach for calculating the trajectories of moving atoms in direct dynamics simulations.We employ a monodromy matrix to propose a tool for evaluating the accuracy of integrators in the trajectory calculation.We choose a general velocity Verlet as a different object.We also simulate molecular with hydrogen(CO_2) and molecular with hydrogen(H_2O) motions.Comparing the eigenvalues of monodromy matrix,many simulations show that Hessian-based predictor-corrector integrators perform well for Hessian updates and non-Hessian updates.Hessian-based predictor-corrector integrator with Hessian update has a strong performance in the H_2O simulations.Hessian-based predictor-corrector integrator with Hessian update has a strong performance when the integrating step of the velocity Verlet approach is tripled for the predicting step.In the CO_2 simulations,a strong performance occurs when the integrating step is a multiple of five.展开更多
The simplified Newton method, at the expense of fast convergence, reduces the work required by Newton method by reusing the initial Jacobian matrix. The composite Newton method attempts to balance the trade-off betwee...The simplified Newton method, at the expense of fast convergence, reduces the work required by Newton method by reusing the initial Jacobian matrix. The composite Newton method attempts to balance the trade-off between expense and fast convergence by composing one Newton step with one simplified Newton step. Recently, Mehrotra suggested a predictor-corrector variant of primal-dual interior point method for linear programming. It is currently the interiorpoint method of the choice for linear programming. In this work we propose a predictor-corrector interior-point algorithm for convex quadratic programming. It is proved that the algorithm is equivalent to a level-1 perturbed composite Newton method. Computations in the algorithm do not require that the initial primal and dual points be feasible. Numerical experiments are made.展开更多
A new class of generalized mixed implicit quasi-equilibrium problems (GMIQEP) with four-functions is introduced and studied. The new class of equilibrium problems includes many known generalized equilibrium problems...A new class of generalized mixed implicit quasi-equilibrium problems (GMIQEP) with four-functions is introduced and studied. The new class of equilibrium problems includes many known generalized equilibrium problems and generalized mixed implicit quasi-variational inequality problems as many special cases. By employing the auxiliary principle technique, some predictor-corrector iterative algorithms for solving the GMIQEP are suggested and analyzed. The convergence of the suggested algorithm only requires the continuity and the partially relaxed implicit strong monotonicity of the mappings展开更多
In this paper, we present and analyze modified families of predictor-corrector iterative methods for finding simple zeros of univariate nonlinear equations, permitting near the root. The main advantage of our methods ...In this paper, we present and analyze modified families of predictor-corrector iterative methods for finding simple zeros of univariate nonlinear equations, permitting near the root. The main advantage of our methods is that they perform better and moreover, have the same efficiency indices as that of existing multipoint iterative methods. Furthermore, the convergence analysis of the new methods is discussed and several examples are given to illustrate their efficiency.展开更多
Based on wide-neighbourhood , we in this paper produce aprodictor-corrector algorithm for linear programming, which possesse onepredictor step and two corrector steps. Its iteration complixity bound isO(n2/3 L).Our al...Based on wide-neighbourhood , we in this paper produce aprodictor-corrector algorithm for linear programming, which possesse onepredictor step and two corrector steps. Its iteration complixity bound isO(n2/3 L).Our algorithm improve on the iteration complexity of O(n3/4 L) ofthe predictor-corrector algorithm proposed by Y.Zhang and D.Zhang.展开更多
It has been shown in various papers that most interior-point algorithms for linear optimization and their analysis can be generalized to P_*(κ) linear complementarity problems.This paper presents an extension of t...It has been shown in various papers that most interior-point algorithms for linear optimization and their analysis can be generalized to P_*(κ) linear complementarity problems.This paper presents an extension of the recent variant of Mehrotra's second order algorithm for linear optimijation.It is shown that the iteration-complexity bound of the algorithm is O(4κ + 3)√14κ + 5 nlog(x0)Ts0/ε,which is similar to that of the corresponding algorithm for linear optimization.展开更多
Mehrotra' s recent suggestion of a predictor-eorrector variant of primal-dual interior-point method for linear programming is currently the interior-point method of choice for linear programming.In this work the a...Mehrotra' s recent suggestion of a predictor-eorrector variant of primal-dual interior-point method for linear programming is currently the interior-point method of choice for linear programming.In this work the authors give a predictor-corrector interior-point algorithm for monotone variational inequality problems. The algorithm was proved to be equivalent to a level-1 perturbed composite Newton method. Computations in the algorithm do not require the initial iteration to be feasible. Numerical results of experiments are presented.展开更多
We present the new predictor-corrector methods for systems of nonlinear differential equations, based on themethod of esponential time differencing. We compare the present schemes with the explicit multistep exponenti...We present the new predictor-corrector methods for systems of nonlinear differential equations, based on themethod of esponential time differencing. We compare the present schemes with the explicit multistep exponential time differecing and Adams Bashforth-Moulton method. The numerical results show that the schemes are more accurate and more efficient than Adams predictor-corrector method. The exponential time differencing method has been developed and perfected by the present studies.展开更多
The numerical solution of large scale multi-dimensional convection diffusion equations often requires efficient parallel algorithms.In this work,we consider the extension of a recently proposed non-overlapping domain ...The numerical solution of large scale multi-dimensional convection diffusion equations often requires efficient parallel algorithms.In this work,we consider the extension of a recently proposed non-overlapping domain decomposition method for two dimensional time dependent convection diffusion equations with variable coefficients. By combining predictor-corrector technique,modified upwind differences with explicitimplicit coupling,the method under consideration provides intrinsic parallelism while maintaining good stability and accuracy.Moreover,for multi-dimensional problems, the method can be readily implemented on a multi-processor system and does not have the limitation on the choice of subdomains required by some other similar predictor-corrector or stabilized schemes.These properties of the method are demonstrated in this work through both rigorous mathematical analysis and numerical experiments.展开更多
文摘The energy norm convergence rate of the finite element solution of the heat equation is reduced by the time-regularity of the exact solution. This paper presents an adaptive finite element treatment of time-dependent singularities on the one-dimensional heat equation. The method is based on a Fourier decomposition of the solution and an extraction formula of the coefficients of the singularities coupled with a predictor-corrector algorithm. The method recovers the optimal convergence rate of the finite element method on a quasi-uniform mesh refinement. Numerical results are carried out to show the efficiency of the method.
基金the National Science Foundation(60574075, 60674108)
文摘A globally convergent infeasible-interior-point predictor-corrector algorithm is presented for the second-order cone programming (SOCP) by using the Alizadeh- Haeberly-Overton (AHO) search direction. This algorithm does not require the feasibility of the initial points and iteration points. Under suitable assumptions, it is shown that the algorithm can find an -approximate solution of an SOCP in at most O(√n ln(ε0/ε)) iterations. The iteration-complexity bound of our algorithm is almost the same as the best known bound of feasible interior point algorithms for the SOCP.
基金Project supported by the National Science Foundation of China (60574071) the Foundation for University Key Teacher by the Ministry of Education.
文摘This article presents a polynomial predictor-corrector interior-point algorithm for convex quadratic programming based on a modified predictor-corrector interior-point algorithm. In this algorithm, there is only one corrector step after each predictor step, where Step 2 is a predictor step and Step 4 is a corrector step in the algorithm. In the algorithm, the predictor step decreases the dual gap as much as possible in a wider neighborhood of the central path and the corrector step draws iteration points back to a narrower neighborhood and make a reduction for the dual gap. It is shown that the algorithm has O(√nL) iteration complexity which is the best result for convex quadratic programming so far.
基金Project supported by the Aeronautical Science Foundation of China (Grant No.20060112102)
文摘A conformal optical system refers to the one whose first optical surface conforms to both aerodynamic and imaging requirements. Appropriate correction is required because a conformal dome induces significant aberrations. This paper intends to explain that an effective solution to the easy-fabrication conic surface corrector compensates aberrations induced by a coaxial aspheric dome. A conformal optical system with an ellipsoid MgF2 conformal dome, which has a fineness ratio of 2.0, is designed as an example. The field of regard angle is -4-30 degrees with a +2 degree instantaneous field of view. The system's ultimate value of modulation transfer function is close to the diffraction limit, which indicates that the performance of the conic conformal optical system with a fixed conic corrector meets the imaging requirements.
基金supported by the Yunnan Provincial Applied Basic Research Program of China(No. KKSY201207019)
文摘A three-dimensional (3D) predictor-corrector finite difference method for standing wave is developed. It is applied to solve the 3D nonlinear potential flow equa- tions with a free surface. The 3D irregular tank is mapped onto a fixed cubic tank through the proper coordinate transform schemes. The cubic tank is distributed by the staggered meshgrid, and the staggered meshgrid is used to denote the variables of the flow field. The predictor-corrector finite difference method is given to develop the difference equa- tions of the dynamic boundary equation and kinematic boundary equation. Experimental results show that, using the finite difference method of the predictor-corrector scheme, the numerical solutions agree well with the published results. The wave profiles of the standing wave with different amplitudes and wave lengths are studied. The numerical solutions are also analyzed and presented graphically.
基金Project supported by the National Natural Science Foundation of China(Grant Nos.11174274,11174279,61205021,11204299,61475152,and 61405194)the State Key Laboratory of Applied Optics,Changchun Institute of Optics,Fine Mechanics and Physics,Chinese Academy of Sciences
文摘Multi-conjugation adaptive optics(MCAOs) have been investigated and used in the large aperture optical telescopes for high-resolution imaging with large field of view(FOV).The atmospheric tomographic phase reconstruction and projection of three-dimensional turbulence volume onto wavefront correctors,such as deformable mirrors(DMs) or liquid crystal wavefront correctors(LCWCs),is a very important step in the data processing of an MCAO's controller.In this paper,a method according to the wavefront reconstruction performance of MCAO is presented to evaluate the optimized configuration of multi laser guide stars(LGSs) and the reasonable conjugation heights of LCWCs.Analytical formulations are derived for the different configurations and are used to generate optimized parameters for MCAO.Several examples are given to demonstrate our LGSs configuration optimization method.Compared with traditional methods,our method has minimum wavefront tomographic error,which will be helpful to get higher imaging resolution at large FOV in MCAO.
基金supported by the National Natural Science Foundation of China (Nos. 71061002 and 11071158)the Natural Science Foundation of Guangxi Province of China (Nos. 0832052 and 2010GXNSFB013047)
文摘Based on the ideas of infeasible interior-point methods and predictor-corrector algorithms, two interior-point predictor-corrector algorithms for the second-order cone programming (SOCP) are presented. The two algorithms use the Newton direction and the Euler direction as the predictor directions, respectively. The corrector directions belong to the category of the Alizadeh-Haeberly-Overton (AHO) directions. These algorithms are suitable to the cases of feasible and infeasible interior iterative points. A simpler neighborhood of the central path for the SOCP is proposed, which is the pivotal difference from other interior-point predictor-corrector algorithms. Under some assumptions, the algorithms possess the global, linear, and quadratic convergence. The complexity bound O(rln(εo/ε)) is obtained, where r denotes the number of the second-order cones in the SOCP problem. The numerical results show that the proposed algorithms are effective.
基金The Major State Research Program (G1999030803) of China and the NNSF (G10271066, 19972023) of China.
文摘A finite volume element predictor-corrector method for a class of nonlinear parabolic system of equations is presented and analyzed. Suboptimal L^2 error estimate for the finite volume element predictor-corrector method is derived. A numerical experiment shows that the numerical results are consistent with theoretical analysis.
文摘Theory has it that increasing the step length improves the accuracy of a method. In order to affirm this we increased the step length of the concept in [1] by one to get k = 5. The technique of collocation and interpolation of the power series approximate solution at some selected grid points is considered so as to generate continuous linear multistep methods with constant step sizes. Two, three and four interpolation points are considered to generate the continuous predictor-corrector methods which are implemented in block method respectively. The proposed methods when tested on some numerical examples performed more efficiently than those of [1]. Interestingly the concept of self starting [2] and that of constant order are reaffirmed in our new methods.
基金Project(2016JJ2029)supported by Hunan Provincial Natural Science Foundation of ChinaProject(2016WLZC014)supported by the Open Research Fund of Hunan Provincial Key Laboratory of Network Investigational TechnologyProject(2015HNWLFZ059)supported by the Open Research Fund of Key Laboratory of Network Crime Investigation of Hunan Provincial Colleges,China
文摘Direct dynamics simulations are a useful and general approach for studying the atomistic properties of complex chemical systems because they do not require fitting an analytic potential energy function.Hessian-based predictor-corrector integrators are a widely used approach for calculating the trajectories of moving atoms in direct dynamics simulations.We employ a monodromy matrix to propose a tool for evaluating the accuracy of integrators in the trajectory calculation.We choose a general velocity Verlet as a different object.We also simulate molecular with hydrogen(CO_2) and molecular with hydrogen(H_2O) motions.Comparing the eigenvalues of monodromy matrix,many simulations show that Hessian-based predictor-corrector integrators perform well for Hessian updates and non-Hessian updates.Hessian-based predictor-corrector integrator with Hessian update has a strong performance in the H_2O simulations.Hessian-based predictor-corrector integrator with Hessian update has a strong performance when the integrating step of the velocity Verlet approach is tripled for the predicting step.In the CO_2 simulations,a strong performance occurs when the integrating step is a multiple of five.
文摘The simplified Newton method, at the expense of fast convergence, reduces the work required by Newton method by reusing the initial Jacobian matrix. The composite Newton method attempts to balance the trade-off between expense and fast convergence by composing one Newton step with one simplified Newton step. Recently, Mehrotra suggested a predictor-corrector variant of primal-dual interior point method for linear programming. It is currently the interiorpoint method of the choice for linear programming. In this work we propose a predictor-corrector interior-point algorithm for convex quadratic programming. It is proved that the algorithm is equivalent to a level-1 perturbed composite Newton method. Computations in the algorithm do not require that the initial primal and dual points be feasible. Numerical experiments are made.
基金Project supported by the Natural Science Foundation of Sichuan Educational Commission (No.2003A081)
文摘A new class of generalized mixed implicit quasi-equilibrium problems (GMIQEP) with four-functions is introduced and studied. The new class of equilibrium problems includes many known generalized equilibrium problems and generalized mixed implicit quasi-variational inequality problems as many special cases. By employing the auxiliary principle technique, some predictor-corrector iterative algorithms for solving the GMIQEP are suggested and analyzed. The convergence of the suggested algorithm only requires the continuity and the partially relaxed implicit strong monotonicity of the mappings
文摘In this paper, we present and analyze modified families of predictor-corrector iterative methods for finding simple zeros of univariate nonlinear equations, permitting near the root. The main advantage of our methods is that they perform better and moreover, have the same efficiency indices as that of existing multipoint iterative methods. Furthermore, the convergence analysis of the new methods is discussed and several examples are given to illustrate their efficiency.
文摘Based on wide-neighbourhood , we in this paper produce aprodictor-corrector algorithm for linear programming, which possesse onepredictor step and two corrector steps. Its iteration complixity bound isO(n2/3 L).Our algorithm improve on the iteration complexity of O(n3/4 L) ofthe predictor-corrector algorithm proposed by Y.Zhang and D.Zhang.
基金supported by the Natural Science Foundation of Hubei Province of China(2008CDZ047)
文摘It has been shown in various papers that most interior-point algorithms for linear optimization and their analysis can be generalized to P_*(κ) linear complementarity problems.This paper presents an extension of the recent variant of Mehrotra's second order algorithm for linear optimijation.It is shown that the iteration-complexity bound of the algorithm is O(4κ + 3)√14κ + 5 nlog(x0)Ts0/ε,which is similar to that of the corresponding algorithm for linear optimization.
文摘Mehrotra' s recent suggestion of a predictor-eorrector variant of primal-dual interior-point method for linear programming is currently the interior-point method of choice for linear programming.In this work the authors give a predictor-corrector interior-point algorithm for monotone variational inequality problems. The algorithm was proved to be equivalent to a level-1 perturbed composite Newton method. Computations in the algorithm do not require the initial iteration to be feasible. Numerical results of experiments are presented.
基金The project supported by National Natural Science Foundation of China under Grant No.19902002
文摘We present the new predictor-corrector methods for systems of nonlinear differential equations, based on themethod of esponential time differencing. We compare the present schemes with the explicit multistep exponential time differecing and Adams Bashforth-Moulton method. The numerical results show that the schemes are more accurate and more efficient than Adams predictor-corrector method. The exponential time differencing method has been developed and perfected by the present studies.
基金the National Natural Science Foundation of China(No.10571017)supported in part by the National Natural Science Foundation of China(No.60533020)supported in part by NSF DMS 0712744
文摘The numerical solution of large scale multi-dimensional convection diffusion equations often requires efficient parallel algorithms.In this work,we consider the extension of a recently proposed non-overlapping domain decomposition method for two dimensional time dependent convection diffusion equations with variable coefficients. By combining predictor-corrector technique,modified upwind differences with explicitimplicit coupling,the method under consideration provides intrinsic parallelism while maintaining good stability and accuracy.Moreover,for multi-dimensional problems, the method can be readily implemented on a multi-processor system and does not have the limitation on the choice of subdomains required by some other similar predictor-corrector or stabilized schemes.These properties of the method are demonstrated in this work through both rigorous mathematical analysis and numerical experiments.