In this paper, a modified version of the Classical Lagrange Multiplier method is developed for convex quadratic optimization problems. The method, which is evolved from the first order derivative test for optimality o...In this paper, a modified version of the Classical Lagrange Multiplier method is developed for convex quadratic optimization problems. The method, which is evolved from the first order derivative test for optimality of the Lagrangian function with respect to the primary variables of the problem, decomposes the solution process into two independent ones, in which the primary variables are solved for independently, and then the secondary variables, which are the Lagrange multipliers, are solved for, afterward. This is an innovation that leads to solving independently two simpler systems of equations involving the primary variables only, on one hand, and the secondary ones on the other. Solutions obtained for small sized problems (as preliminary test of the method) demonstrate that the new method is generally effective in producing the required solutions.展开更多
The distributed Lagrange multiplier/fictitious domain(DLM/FD)-mixed finite element method is developed and analyzed in this paper for a transient Stokes interface problem with jump coefficients.The semi-and fully disc...The distributed Lagrange multiplier/fictitious domain(DLM/FD)-mixed finite element method is developed and analyzed in this paper for a transient Stokes interface problem with jump coefficients.The semi-and fully discrete DLM/FD-mixed finite element scheme are developed for the first time for this problem with a moving interface,where the arbitrary Lagrangian-Eulerian(ALE)technique is employed to deal with the moving and immersed subdomain.Stability and optimal convergence properties are obtained for both schemes.Numerical experiments are carried out for different scenarios of jump coefficients,and all theoretical results are validated.展开更多
The Lagrange multiplier method plays an important role in establishing generalized variational principles notonly in tluid mechallics. but also in elasticity. Sometimes, however, one may come across variational crisi...The Lagrange multiplier method plays an important role in establishing generalized variational principles notonly in tluid mechallics. but also in elasticity. Sometimes, however, one may come across variational crisis(somemultipliers vanish identically). failing to achieve his aim. The crisis is caused by the fact that the Inultipliers are treatedas independent variables in the process of variatioll. but after identification they become functions of the originalindependent variables. To overcome it, a Inodified Lagrange multiplier method or semi-inverse method has beenproposed to deduce generalized varistional principles. Some e-camples are given to illustrate its convenience andeffectiveness of the novel method.展开更多
A parameter-free approach is proposed to determine the Lagrange multiplier for the constraint of material volume in the level set method.It is inspired by the procedure of determining the threshold of sensitivity numb...A parameter-free approach is proposed to determine the Lagrange multiplier for the constraint of material volume in the level set method.It is inspired by the procedure of determining the threshold of sensitivity number in the BESO method.It first computes the difference between the volume of current design and the upper bound of volume.Then,the Lagrange multiplier is regarded as the threshold of sensitivity number to remove the redundant material.Numerical examples proved that this approach is effective to constrain the volume.More importantly,there is no parameter in the proposed approach,which makes it convenient to use.In addition,the convergence is stable,and there is no big oscillation.展开更多
Chemical process optimization can be described as large-scale nonlinear constrained minimization. The modified augmented Lagrange multiplier methods (MALMM) for large-scale nonlinear constrained minimization are studi...Chemical process optimization can be described as large-scale nonlinear constrained minimization. The modified augmented Lagrange multiplier methods (MALMM) for large-scale nonlinear constrained minimization are studied in this paper. The Lagrange function contains the penalty terms on equality and inequality constraints and the methods can be applied to solve a series of bound constrained sub-problems instead of a series of unconstrained sub-problems. The steps of the methods are examined in full detail. Numerical experiments are made for a variety of problems, from small to very large-scale, which show the stability and effectiveness of the methods in large-scale
problems.展开更多
In this paper, a unified matrix recovery model was proposed for diverse corrupted matrices. Resulting from the separable structure of the proposed model, the convex optimization problem can be solved efficiently by ad...In this paper, a unified matrix recovery model was proposed for diverse corrupted matrices. Resulting from the separable structure of the proposed model, the convex optimization problem can be solved efficiently by adopting an inexact augmented Lagrange multiplier (IALM) method. Additionally, a random projection accelerated technique (IALM+RP) was adopted to improve the success rate. From the preliminary numerical comparisons, it was indicated that for the standard robust principal component analysis (PCA) problem, IALM+RP was at least two to six times faster than IALM with an insignificant reduction in accuracy; and for the outlier pursuit (OP) problem, IALM+RP was at least 6.9 times faster, even up to 8.3 times faster when the size of matrix was 2 000×2 000.展开更多
This paper, with a finite element method, studies the interaction of a coupled incompressible fluid-rigid structure system with a free surface subjected to external wave excitations. With this fully coupled model, the...This paper, with a finite element method, studies the interaction of a coupled incompressible fluid-rigid structure system with a free surface subjected to external wave excitations. With this fully coupled model, the rigid structure is taken as ＂fictitious＂ fluid with zero strain rate. Both fluid and structure are described by velocity and pressure. The whole domain, including fluid region and structure region, is modeled by the incompressible Navier-Stokes equations which are discretized with fixed Eulerian mesh. However, to keep the structure＇ s rigid body shape and behavior, a rigid body constraint is enforced on the ＂fictitious＂ fluid domain by use of the Distributed Lagrange Multipher/Fictitious Domain （DLM/ FD） method which is originally introduced to solve particulate flow problems by Glowinski et al. For the verification of the model presented herein, a 2D numerical wave tank is established to simulate small amplitude wave propagations, and then numerical results are compared with analytical solutions. Finally, a 2D example of fluid-structure interaction under wave dynamic forces provides convincing evidences for the method excellent solution quality and fidelity.展开更多
Recently many research works have been conducted and published regarding fractional order differential equations. There are several approaches available for numerical approximations of the solution of fractional order...Recently many research works have been conducted and published regarding fractional order differential equations. There are several approaches available for numerical approximations of the solution of fractional order diffusion equations. Spectral collocation method based on Lagrange’s basis polynomials to approximate numerical solutions of one-dimensional (1D) space fractional diffusion equations are introduced in this research paper. The proposed form of approximate solution satisfies non-zero Dirichlet’s boundary conditions on both boundaries. Collocation scheme produce a system of first order Ordinary Differential Equations (ODE) from the fractional diffusion equation. We applied this method with four different sets of collocation points to compare their performance.展开更多
Due to the ability to model various complex phenomena where classical calculus failed, fractional calculus is getting enormous attention recently. There are several approaches available for numerical approximations of...Due to the ability to model various complex phenomena where classical calculus failed, fractional calculus is getting enormous attention recently. There are several approaches available for numerical approximations of various types of fractional differential equations. For fractional diffusion equations spectral collocation is one of the efficient and most popular ap-proximation techniques. In this research, we introduce spectral collocation method based on Lagrange’s basis polynomials for numerical approximations of two-dimensional (2D) space fractional diffusion equations where spatial fractional derivative is described in Riemann-Liouville sense. We consider four different types of nodes to generate Lagrange’s basis polynomials and as collocation points in the proposed spectral collocation technique. Spectral collocation method converts the diffusion equation into a system of ordinary differential equations (ODE) for time variable and we use 4th order Runge-Kutta method to solve the resulting system of ODE. Two examples are considered to verify the efficiency of different types of nodes in the proposed method. We compare approximated solution with exact solution and find that Lagrange’s spectral collocation method gives very high accuracy approximation. Among the four types of nodes, nodes from Jacobi polynomial give highest accuracy and nodes from Chebyshev polynomials of 1st kind give lowest accuracy in the proposed method.展开更多
This paper introduces the Lagrangian relaxation method to solve multiobjective optimization problems. It is often required to use the appropriate technique to determine the Lagrangian multipliers in the relaxation met...This paper introduces the Lagrangian relaxation method to solve multiobjective optimization problems. It is often required to use the appropriate technique to determine the Lagrangian multipliers in the relaxation method that leads to finding the optimal solution to the problem. Our analysis aims to find a suitable technique to generate Lagrangian multipliers, and later these multipliers are used in the relaxation method to solve Multiobjective optimization problems. We propose a search-based technique to generate Lagrange multipliers. In our paper, we choose a suitable and well-known scalarization method that transforms the original multiobjective into a scalar objective optimization problem. Later, we solve this scalar objective problem using Lagrangian relaxation techniques. We use Brute force techniques to sort optimum solutions. Finally, we analyze the results, and efficient methods are recommended.展开更多
文摘In this paper, a modified version of the Classical Lagrange Multiplier method is developed for convex quadratic optimization problems. The method, which is evolved from the first order derivative test for optimality of the Lagrangian function with respect to the primary variables of the problem, decomposes the solution process into two independent ones, in which the primary variables are solved for independently, and then the secondary variables, which are the Lagrange multipliers, are solved for, afterward. This is an innovation that leads to solving independently two simpler systems of equations involving the primary variables only, on one hand, and the secondary ones on the other. Solutions obtained for small sized problems (as preliminary test of the method) demonstrate that the new method is generally effective in producing the required solutions.
基金P.Sun was supported by NSF Grant DMS-1418806C.S.Zhang was partially supported by the National Key Research and Development Program of China(Grant No.2016YFB0201304)+1 种基金the Major Research Plan of National Natural Science Foundation of China(Grant Nos.91430215,91530323)the Key Research Program of Frontier Sciences of CAS.
文摘The distributed Lagrange multiplier/fictitious domain(DLM/FD)-mixed finite element method is developed and analyzed in this paper for a transient Stokes interface problem with jump coefficients.The semi-and fully discrete DLM/FD-mixed finite element scheme are developed for the first time for this problem with a moving interface,where the arbitrary Lagrangian-Eulerian(ALE)technique is employed to deal with the moving and immersed subdomain.Stability and optimal convergence properties are obtained for both schemes.Numerical experiments are carried out for different scenarios of jump coefficients,and all theoretical results are validated.
文摘The Lagrange multiplier method plays an important role in establishing generalized variational principles notonly in tluid mechallics. but also in elasticity. Sometimes, however, one may come across variational crisis(somemultipliers vanish identically). failing to achieve his aim. The crisis is caused by the fact that the Inultipliers are treatedas independent variables in the process of variatioll. but after identification they become functions of the originalindependent variables. To overcome it, a Inodified Lagrange multiplier method or semi-inverse method has beenproposed to deduce generalized varistional principles. Some e-camples are given to illustrate its convenience andeffectiveness of the novel method.
基金This research work is supported by the National Natural Science Foundation of China(Grant No.51975227).
文摘A parameter-free approach is proposed to determine the Lagrange multiplier for the constraint of material volume in the level set method.It is inspired by the procedure of determining the threshold of sensitivity number in the BESO method.It first computes the difference between the volume of current design and the upper bound of volume.Then,the Lagrange multiplier is regarded as the threshold of sensitivity number to remove the redundant material.Numerical examples proved that this approach is effective to constrain the volume.More importantly,there is no parameter in the proposed approach,which makes it convenient to use.In addition,the convergence is stable,and there is no big oscillation.
文摘Chemical process optimization can be described as large-scale nonlinear constrained minimization. The modified augmented Lagrange multiplier methods (MALMM) for large-scale nonlinear constrained minimization are studied in this paper. The Lagrange function contains the penalty terms on equality and inequality constraints and the methods can be applied to solve a series of bound constrained sub-problems instead of a series of unconstrained sub-problems. The steps of the methods are examined in full detail. Numerical experiments are made for a variety of problems, from small to very large-scale, which show the stability and effectiveness of the methods in large-scale
problems.
基金Supported by National Natural Science Foundation of China (No.51275348)College Students Innovation and Entrepreneurship Training Program of Tianjin University (No.201210056339)
文摘In this paper, a unified matrix recovery model was proposed for diverse corrupted matrices. Resulting from the separable structure of the proposed model, the convex optimization problem can be solved efficiently by adopting an inexact augmented Lagrange multiplier (IALM) method. Additionally, a random projection accelerated technique (IALM+RP) was adopted to improve the success rate. From the preliminary numerical comparisons, it was indicated that for the standard robust principal component analysis (PCA) problem, IALM+RP was at least two to six times faster than IALM with an insignificant reduction in accuracy; and for the outlier pursuit (OP) problem, IALM+RP was at least 6.9 times faster, even up to 8.3 times faster when the size of matrix was 2 000×2 000.
基金This study is supported by the National Natural Science Foundation of China (Grant No50579046) the Science Foundation of Tianjin Municipal Commission of Science and Technology (Grant No043114711)
文摘This paper, with a finite element method, studies the interaction of a coupled incompressible fluid-rigid structure system with a free surface subjected to external wave excitations. With this fully coupled model, the rigid structure is taken as ＂fictitious＂ fluid with zero strain rate. Both fluid and structure are described by velocity and pressure. The whole domain, including fluid region and structure region, is modeled by the incompressible Navier-Stokes equations which are discretized with fixed Eulerian mesh. However, to keep the structure＇ s rigid body shape and behavior, a rigid body constraint is enforced on the ＂fictitious＂ fluid domain by use of the Distributed Lagrange Multipher/Fictitious Domain （DLM/ FD） method which is originally introduced to solve particulate flow problems by Glowinski et al. For the verification of the model presented herein, a 2D numerical wave tank is established to simulate small amplitude wave propagations, and then numerical results are compared with analytical solutions. Finally, a 2D example of fluid-structure interaction under wave dynamic forces provides convincing evidences for the method excellent solution quality and fidelity.
文摘Recently many research works have been conducted and published regarding fractional order differential equations. There are several approaches available for numerical approximations of the solution of fractional order diffusion equations. Spectral collocation method based on Lagrange’s basis polynomials to approximate numerical solutions of one-dimensional (1D) space fractional diffusion equations are introduced in this research paper. The proposed form of approximate solution satisfies non-zero Dirichlet’s boundary conditions on both boundaries. Collocation scheme produce a system of first order Ordinary Differential Equations (ODE) from the fractional diffusion equation. We applied this method with four different sets of collocation points to compare their performance.
文摘Due to the ability to model various complex phenomena where classical calculus failed, fractional calculus is getting enormous attention recently. There are several approaches available for numerical approximations of various types of fractional differential equations. For fractional diffusion equations spectral collocation is one of the efficient and most popular ap-proximation techniques. In this research, we introduce spectral collocation method based on Lagrange’s basis polynomials for numerical approximations of two-dimensional (2D) space fractional diffusion equations where spatial fractional derivative is described in Riemann-Liouville sense. We consider four different types of nodes to generate Lagrange’s basis polynomials and as collocation points in the proposed spectral collocation technique. Spectral collocation method converts the diffusion equation into a system of ordinary differential equations (ODE) for time variable and we use 4th order Runge-Kutta method to solve the resulting system of ODE. Two examples are considered to verify the efficiency of different types of nodes in the proposed method. We compare approximated solution with exact solution and find that Lagrange’s spectral collocation method gives very high accuracy approximation. Among the four types of nodes, nodes from Jacobi polynomial give highest accuracy and nodes from Chebyshev polynomials of 1st kind give lowest accuracy in the proposed method.
文摘This paper introduces the Lagrangian relaxation method to solve multiobjective optimization problems. It is often required to use the appropriate technique to determine the Lagrangian multipliers in the relaxation method that leads to finding the optimal solution to the problem. Our analysis aims to find a suitable technique to generate Lagrangian multipliers, and later these multipliers are used in the relaxation method to solve Multiobjective optimization problems. We propose a search-based technique to generate Lagrange multipliers. In our paper, we choose a suitable and well-known scalarization method that transforms the original multiobjective into a scalar objective optimization problem. Later, we solve this scalar objective problem using Lagrangian relaxation techniques. We use Brute force techniques to sort optimum solutions. Finally, we analyze the results, and efficient methods are recommended.
