In this paper, an improved splitting method, based on the completely square-conservative explicit difference schemes, is established. Not only can the time-direction precision of this method be higher than that of the...In this paper, an improved splitting method, based on the completely square-conservative explicit difference schemes, is established. Not only can the time-direction precision of this method be higher than that of the traditional splitting methods but also can the physical feature of mutual dependence of the fast and the slow stages that are calculated separately and splittingly be kept as well. Moreover, the method owns an universality, it can be generalized to other square-conservative difference schemes, such as the implicit and complete ones and the explicit and instantaneous ones. Good time benefits can be acquired when it is applied in the numerical simulations of the monthly mean currents of the South China Sea.展开更多
In this paper, equations of atmospheric and oceanic dynamics are reduced to a kind of evolutionary equation in operator form, based on which a conclusion that the separability of motion stages is relative is made and ...In this paper, equations of atmospheric and oceanic dynamics are reduced to a kind of evolutionary equation in operator form, based on which a conclusion that the separability of motion stages is relative is made and an issue that the tractional splitting methods established on the physical separability of the fast stage and the slow stage neglect the interaction between the two stages to some extent is shown. Also, three splitting patterns are summed up from the splitting methods in common use so that a comparison between them is carried out. The comparison shows that only the improved splitting pattern (ISP) can be in second order and keep the interaction well. Finally, the applications of some splitting methods on numerical simulations of typhoon tracks made clear that ISP owns the best effect and can save more than 80% CPU time.展开更多
In this work, a new numerical technique is proposed for the resolution of a fluid model based on three Boltzmann moments. The main purpose of this technique is to calculate electric and physical properties in the non-...In this work, a new numerical technique is proposed for the resolution of a fluid model based on three Boltzmann moments. The main purpose of this technique is to calculate electric and physical properties in the non-equilibrium electric discharge at low pressure. The transport and Poisson's equations form a self-consistent model. This equation system is written in cylindrical coordinates following the geometric shape of a plasma reactor. Our transport equation system is discretized using the finite volume approach and resolved by the N-BEE explicit scheme coupled to the time splitting method. This programming structure reduces computation time considerably. The 2D code is carried out and tested by comparing our results with those found in literature.展开更多
We propose a multi-symplectic wavelet splitting equations. Based on its mu]ti-symplectic formulation, method to solve the strongly coupled nonlinear SchrSdinger the strongly coupled nonlinear SchrSdinger equations can...We propose a multi-symplectic wavelet splitting equations. Based on its mu]ti-symplectic formulation, method to solve the strongly coupled nonlinear SchrSdinger the strongly coupled nonlinear SchrSdinger equations can be split into one linear multi-symplectic subsystem and one nonlinear infinite-dimensional Hamiltonian subsystem. For the linear subsystem, the multi-symplectic wavelet collocation method and the symplectic Euler method are employed in spatial and temporal discretization, respectively. For the nonlinear subsystem, the mid-point symplectic scheme is used. Numerical simulations show the effectiveness of the proposed method during long-time numerical calculation.展开更多
The Operator Splitting method is applied to differential equations occurring as mathematical models in financial models. This paper provides various operator splitting methods to obtain an effective and accurate solut...The Operator Splitting method is applied to differential equations occurring as mathematical models in financial models. This paper provides various operator splitting methods to obtain an effective and accurate solution to the Black-Scholes equation with appropriate boundary conditions for a European option pricing problem. Finally brief comparisons of option prices are given by different models.展开更多
By introducing the dimensional splitting(DS)method into the multiscale interpolating element-free Galerkin(VMIEFG)method,a dimension-splitting multiscale interpolating element-free Galerkin(DS-VMIEFG)method is propose...By introducing the dimensional splitting(DS)method into the multiscale interpolating element-free Galerkin(VMIEFG)method,a dimension-splitting multiscale interpolating element-free Galerkin(DS-VMIEFG)method is proposed for three-dimensional(3D)singular perturbed convection-diffusion(SPCD)problems.In the DSVMIEFG method,the 3D problem is decomposed into a series of 2D problems by the DS method,and the discrete equations on the 2D splitting surface are obtained by the VMIEFG method.The improved interpolation-type moving least squares(IIMLS)method is used to construct shape functions in the weak form and to combine 2D discrete equations into a global system of discrete equations for the three-dimensional SPCD problems.The solved numerical example verifies the effectiveness of the method in this paper for the 3D SPCD problems.The numerical solution will gradually converge to the analytical solution with the increase in the number of nodes.For extremely small singular diffusion coefficients,the numerical solution will avoid numerical oscillation and has high computational stability.展开更多
The Peaceman-Rachford splitting method is efficient for minimizing a convex optimization problem with a separable objective function and linear constraints.However,its convergence was not guaranteed without extra requ...The Peaceman-Rachford splitting method is efficient for minimizing a convex optimization problem with a separable objective function and linear constraints.However,its convergence was not guaranteed without extra requirements.He et al.(SIAM J.Optim.24:1011-1040,2014)proved the convergence of a strictly contractive Peaceman-Rachford splitting method by employing a suitable underdetermined relaxation factor.In this paper,we further extend the so-called strictly contractive Peaceman-Rachford splitting method by using two different relaxation factors.Besides,motivated by the recent advances on the ADMM type method with indefinite proximal terms,we employ the indefinite proximal term in the strictly contractive Peaceman-Rachford splitting method.We show that the proposed indefinite-proximal strictly contractive Peaceman-Rachford splitting method is convergent and also prove the o(1/t)convergence rate in the nonergodic sense.The numerical tests on the l 1 regularized least square problem demonstrate the efficiency of the proposed method.展开更多
This paper discusses the two-block large-scale nonconvex optimization problem with general linear constraints.Based on the ideas of splitting and sequential quadratic optimization(SQO),a new feasible descent method fo...This paper discusses the two-block large-scale nonconvex optimization problem with general linear constraints.Based on the ideas of splitting and sequential quadratic optimization(SQO),a new feasible descent method for the discussed problem is proposed.First,we consider the problem of quadratic optimal(QO)approximation associated with the current feasible iteration point,and we split the QO into two small-scale QOs which can be solved in parallel.Second,a feasible descent direction for the problem is obtained and a new SQO-type method is proposed,namely,splitting feasible SQO(SF-SQO)method.Moreover,under suitable conditions,we analyse the global convergence,strong convergence and rate of superlinear convergence of the SF-SQO method.Finally,preliminary numerical experiments regarding the economic dispatch of a power system are carried out,and these show that the SF-SQO method is promising.展开更多
This work is about a splitting method for solving a nonconvex nonseparable optimization problem with linear constraints,where the objective function consists of two separable functions and a coupled term.First,based o...This work is about a splitting method for solving a nonconvex nonseparable optimization problem with linear constraints,where the objective function consists of two separable functions and a coupled term.First,based on the ideas from Bregman distance and Peaceman–Rachford splitting method,the Bregman Peaceman–Rachford splitting method with different relaxation factors for the multiplier is proposed.Second,the global and strong convergence of the proposed algorithm are proved under general conditions including the region of the two relaxation factors as well as the crucial Kurdyka–Łojasiewicz property.Third,when the associated Kurdyka–Łojasiewicz property function has a special structure,the sublinear and linear convergence rates of the proposed algorithm are guaranteed.Furthermore,some preliminary numerical results are shown to indicate the effectiveness of the proposed algorithm.展开更多
We present a new family of fourth-order splitting methods with positive coefficients especially tailored for the time integration of linear parabolic problems and,in particular,for the time dependent Schrodinger equat...We present a new family of fourth-order splitting methods with positive coefficients especially tailored for the time integration of linear parabolic problems and,in particular,for the time dependent Schrodinger equation,both in real and imaginary time.They are based on the use of a double commutator and a modified processor,and are more efficient than other widely used schemes found in the literature.Moreover,for certain potentials,they achieve order six.Several examples in one,two and three dimensions clearly illustrate the computational advantages of the new schemes.展开更多
In this paper,we explore bound preserving and high-order accurate local discontinuous Galerkin(LDG)schemes to solve a class of chemotaxis models,including the classical Keller-Segel(KS)model and two other density-depe...In this paper,we explore bound preserving and high-order accurate local discontinuous Galerkin(LDG)schemes to solve a class of chemotaxis models,including the classical Keller-Segel(KS)model and two other density-dependent problems.We use the convex splitting method,the variant energy quadratization method,and the scalar auxiliary variable method coupled with the LDG method to construct first-order temporal accurate schemes based on the gradient flow structure of the models.These semi-implicit schemes are decoupled,energy stable,and can be extended to high accuracy schemes using the semi-implicit spectral deferred correction method.Many bound preserving DG discretizations are only worked on explicit time integration methods and are difficult to get high-order accuracy.To overcome these difficulties,we use the Lagrange multipliers to enforce the implicit or semi-implicit LDG schemes to satisfy the bound constraints at each time step.This bound preserving limiter results in the Karush-Kuhn-Tucker condition,which can be solved by an efficient active set semi-smooth Newton method.Various numerical experiments illustrate the high-order accuracy and the effect of bound preserving.展开更多
In this paper, we propose a dimensional splitting method for the three dimensional (3D) rotating Navier-Stokes equations. Assume that the domain is a channel bounded by two surfaces and is decomposed by a series of...In this paper, we propose a dimensional splitting method for the three dimensional (3D) rotating Navier-Stokes equations. Assume that the domain is a channel bounded by two surfaces and is decomposed by a series of surfaces i into several sub-domains, which are called the layers of the flow. Every interface i between two sub-domains shares the same geometry. After establishing a semi-geodesic coordinate (S-coordinate) system based on i, Navier-Stoke equations in this coordinate can be expressed as the sum of two operators, of which one is called the membrane operator defined on the tangent space on i, another one is called the bending operator taking value in the normal space on i. Then the derivatives of velocity with respect to the normal direction of the surface are approximated by the Euler central difference, and an approximate form of Navier-Stokes equations on the surface i is obtained, which is called the two-dimensional three-component (2D-3C) Navier-Stokes equations on a two dimensional manifold. Solving these equations by alternate iteration, an approximate solution to the original 3D Navier-Stokes equations is obtained. In addition, the proof of the existence of solutions to 2D-3C Navier-Stokes equations is provided, and some approximate methods for solving 2D-3C Navier-Stot4es equations are presented.展开更多
In this paper, we further generalize the technique for constructing the normal (or pos- itive definite) and skew-Hermitian splitting iteration method for solving large sparse non- Hermitian positive definite system ...In this paper, we further generalize the technique for constructing the normal (or pos- itive definite) and skew-Hermitian splitting iteration method for solving large sparse non- Hermitian positive definite system of linear equations. By introducing a new splitting, we establish a class of efficient iteration methods, called positive definite and semi-definite splitting (PPS) methods, and prove that the sequence produced by the PPS method con- verges unconditionally to the unique solution of the system. Moreover, we propose two kinds of typical practical choices of the PPS method and study the upper bound of the spectral radius of the iteration matrix. In addition, we show the optimal parameters such that the spectral radius achieves the minimum under certain conditions. Finally, some numerical examples are given to demonstrate the effectiveness of the considered methods.展开更多
The Douglas–Peaceman–Rachford–Varga operator splitting methods are a class ofefficient methods for finding a zero of the sum of two maximal monotone operatorsin a real Hilbert space;however, they are sometimes diff...The Douglas–Peaceman–Rachford–Varga operator splitting methods are a class ofefficient methods for finding a zero of the sum of two maximal monotone operatorsin a real Hilbert space;however, they are sometimes difficult or even impossible tosolve the subproblems exactly. In this paper, we suggest an inexact version in whichsome relative error criterion is discussed. The corresponding convergence propertiesare established, and some preliminary numerical experiments are reported to illustrateits efficiency.展开更多
We propose Poisson integrators for the numerical integration of separable Poisson systems.We analyze three situations in which Poisson systems are separated in threeways and Poisson integrators can be constructed by u...We propose Poisson integrators for the numerical integration of separable Poisson systems.We analyze three situations in which Poisson systems are separated in threeways and Poisson integrators can be constructed by using the splittingmethod.Numerical results show that the Poisson integrators outperform the higher order non-Poisson integrators in terms of long-termenergy conservation and computational cost.The Poisson integrators are also shown to be more efficient than the canonicalized sympletic methods of the same order.展开更多
In this paper,we discuss a splitting method for group Lasso.By assuming that the sequence of the step lengths has positive lower bound and positive upper bound(unrelated to the given problem data),we prove its Q-linea...In this paper,we discuss a splitting method for group Lasso.By assuming that the sequence of the step lengths has positive lower bound and positive upper bound(unrelated to the given problem data),we prove its Q-linear rate of convergence of the distance sequence of the iterates to the solution set.Moreover,we make comparisons with convergence of the proximal gradient method analyzed very recently.展开更多
The strictly contractive Peaceman-Rachford splitting method is one of effective methods for solving separable convex optimization problem, and the inertial proximal Peaceman-Rachford splitting method is one of its imp...The strictly contractive Peaceman-Rachford splitting method is one of effective methods for solving separable convex optimization problem, and the inertial proximal Peaceman-Rachford splitting method is one of its important variants. It is known that the convergence of the inertial proximal Peaceman- Rachford splitting method can be ensured if the relaxation factor in Lagrangian multiplier updates is underdetermined, which means that the steps for the Lagrangian multiplier updates are shrunk conservatively. Although small steps play an important role in ensuring convergence, they should be strongly avoided in practice. In this article, we propose a relaxed inertial proximal Peaceman- Rachford splitting method, which has a larger feasible set for the relaxation factor. Thus, our method provides the possibility to admit larger steps in the Lagrangian multiplier updates. We establish the global convergence of the proposed algorithm under the same conditions as the inertial proximal Peaceman-Rachford splitting method. Numerical experimental results on a sparse signal recovery problem in compressive sensing and a total variation based image denoising problem demonstrate the effectiveness of our method.展开更多
Consider the problem of minimizing the sum of two convex functions,one being smooth and the other non-smooth.In this paper,we introduce a general class of approximate proximal splitting(APS)methods for solving such mi...Consider the problem of minimizing the sum of two convex functions,one being smooth and the other non-smooth.In this paper,we introduce a general class of approximate proximal splitting(APS)methods for solving such minimization problems.Methods in the APS class include many well-known algorithms such as the proximal splitting method,the block coordinate descent method(BCD),and the approximate gradient projection methods for smooth convex optimization.We establish the linear convergence of APS methods under a local error bound assumption.Since the latter is known to hold for compressive sensing and sparse group LASSO problems,our analysis implies the linear convergence of the BCD method for these problems without strong convexity assumption.展开更多
In this paper,we propose a partially parallel prediction-correction splitting method for solving block-separable linearly constrained convex optimization problems with three blocks.Unlike the extended alternating dire...In this paper,we propose a partially parallel prediction-correction splitting method for solving block-separable linearly constrained convex optimization problems with three blocks.Unlike the extended alternating direction method of multipliers,the last two subproblems in the prediction step are solved parallelly,and a correction step is employed in the method to correct the dual variable and two blocks of the primal variables.The step size adapted in the correction step allows for major contribution from the latest solution point to the iteration point.Some numerical results are reported to show the effectiveness of the presented method.展开更多
In this paper,we propose two new explicit multi-symplectic splitting methods for the nonlinear Dirac(NLD)equation.Based on its multi-symplectic formulation,the NLD equation is split into one linear multi-symplectic sy...In this paper,we propose two new explicit multi-symplectic splitting methods for the nonlinear Dirac(NLD)equation.Based on its multi-symplectic formulation,the NLD equation is split into one linear multi-symplectic system and one nonlinear infinite Hamiltonian system.Then multi-symplectic Fourier pseudospectral method and multi-symplectic Preissmann scheme are employed to discretize the linear subproblem,respectively.And the nonlinear subsystem is solved by a symplectic scheme.Finally,a composition method is applied to obtain the final schemes for the NLD equation.We find that the two proposed schemes preserve the total symplecticity and can be solved explicitly.Numerical experiments are presented to show the effectiveness of the proposed methods.展开更多
基金Partly supported by the State Major Key Project for Basic Researches
文摘In this paper, an improved splitting method, based on the completely square-conservative explicit difference schemes, is established. Not only can the time-direction precision of this method be higher than that of the traditional splitting methods but also can the physical feature of mutual dependence of the fast and the slow stages that are calculated separately and splittingly be kept as well. Moreover, the method owns an universality, it can be generalized to other square-conservative difference schemes, such as the implicit and complete ones and the explicit and instantaneous ones. Good time benefits can be acquired when it is applied in the numerical simulations of the monthly mean currents of the South China Sea.
基金Partly supported by the State Major Key Project for Researches and Project 85-906-04.
文摘In this paper, equations of atmospheric and oceanic dynamics are reduced to a kind of evolutionary equation in operator form, based on which a conclusion that the separability of motion stages is relative is made and an issue that the tractional splitting methods established on the physical separability of the fast stage and the slow stage neglect the interaction between the two stages to some extent is shown. Also, three splitting patterns are summed up from the splitting methods in common use so that a comparison between them is carried out. The comparison shows that only the improved splitting pattern (ISP) can be in second order and keep the interaction well. Finally, the applications of some splitting methods on numerical simulations of typhoon tracks made clear that ISP owns the best effect and can save more than 80% CPU time.
文摘In this work, a new numerical technique is proposed for the resolution of a fluid model based on three Boltzmann moments. The main purpose of this technique is to calculate electric and physical properties in the non-equilibrium electric discharge at low pressure. The transport and Poisson's equations form a self-consistent model. This equation system is written in cylindrical coordinates following the geometric shape of a plasma reactor. Our transport equation system is discretized using the finite volume approach and resolved by the N-BEE explicit scheme coupled to the time splitting method. This programming structure reduces computation time considerably. The 2D code is carried out and tested by comparing our results with those found in literature.
基金Project supported by the National Natural Science Foundation of China(Grant Nos.10971226,91130013,and 11001270)the National Basic Research Program of China(Grant No.2009CB723802)+1 种基金the Research Innovation Fund of Hunan Province,China (Grant No.CX2011B011)the Innovation Fund of National University of Defense Technology,China(Grant No.B120205)
文摘We propose a multi-symplectic wavelet splitting equations. Based on its mu]ti-symplectic formulation, method to solve the strongly coupled nonlinear SchrSdinger the strongly coupled nonlinear SchrSdinger equations can be split into one linear multi-symplectic subsystem and one nonlinear infinite-dimensional Hamiltonian subsystem. For the linear subsystem, the multi-symplectic wavelet collocation method and the symplectic Euler method are employed in spatial and temporal discretization, respectively. For the nonlinear subsystem, the mid-point symplectic scheme is used. Numerical simulations show the effectiveness of the proposed method during long-time numerical calculation.
文摘The Operator Splitting method is applied to differential equations occurring as mathematical models in financial models. This paper provides various operator splitting methods to obtain an effective and accurate solution to the Black-Scholes equation with appropriate boundary conditions for a European option pricing problem. Finally brief comparisons of option prices are given by different models.
基金supported by the Natural Science Foundation of Zhejiang Province,China(Grant Nos.LY20A010021,LY19A010002,LY20G030025)the Natural Science Founda-tion of Ningbo City,China(Grant Nos.2021J147,2021J235).
文摘By introducing the dimensional splitting(DS)method into the multiscale interpolating element-free Galerkin(VMIEFG)method,a dimension-splitting multiscale interpolating element-free Galerkin(DS-VMIEFG)method is proposed for three-dimensional(3D)singular perturbed convection-diffusion(SPCD)problems.In the DSVMIEFG method,the 3D problem is decomposed into a series of 2D problems by the DS method,and the discrete equations on the 2D splitting surface are obtained by the VMIEFG method.The improved interpolation-type moving least squares(IIMLS)method is used to construct shape functions in the weak form and to combine 2D discrete equations into a global system of discrete equations for the three-dimensional SPCD problems.The solved numerical example verifies the effectiveness of the method in this paper for the 3D SPCD problems.The numerical solution will gradually converge to the analytical solution with the increase in the number of nodes.For extremely small singular diffusion coefficients,the numerical solution will avoid numerical oscillation and has high computational stability.
基金supported by the Natural Science Foundation of Jiangsu Province (Grant No.BK20210267)supported by the National Natural Science Foundation of China (Grant No.11971239)+1 种基金the Natural Science Foundation of the Higher Education Institutions of Jiangsu Province (Grant No.21KJA110002)supported by the National Natural Science Foundation of China (Grant Nos.12131004,11625105).
文摘The Peaceman-Rachford splitting method is efficient for minimizing a convex optimization problem with a separable objective function and linear constraints.However,its convergence was not guaranteed without extra requirements.He et al.(SIAM J.Optim.24:1011-1040,2014)proved the convergence of a strictly contractive Peaceman-Rachford splitting method by employing a suitable underdetermined relaxation factor.In this paper,we further extend the so-called strictly contractive Peaceman-Rachford splitting method by using two different relaxation factors.Besides,motivated by the recent advances on the ADMM type method with indefinite proximal terms,we employ the indefinite proximal term in the strictly contractive Peaceman-Rachford splitting method.We show that the proposed indefinite-proximal strictly contractive Peaceman-Rachford splitting method is convergent and also prove the o(1/t)convergence rate in the nonergodic sense.The numerical tests on the l 1 regularized least square problem demonstrate the efficiency of the proposed method.
基金supported by the National Natural Science Foundation of China(12171106)the Natural Science Foundation of Guangxi Province(2020GXNSFDA238017 and 2018GXNSFFA281007)the Shanghai Sailing Program(21YF1430300)。
文摘This paper discusses the two-block large-scale nonconvex optimization problem with general linear constraints.Based on the ideas of splitting and sequential quadratic optimization(SQO),a new feasible descent method for the discussed problem is proposed.First,we consider the problem of quadratic optimal(QO)approximation associated with the current feasible iteration point,and we split the QO into two small-scale QOs which can be solved in parallel.Second,a feasible descent direction for the problem is obtained and a new SQO-type method is proposed,namely,splitting feasible SQO(SF-SQO)method.Moreover,under suitable conditions,we analyse the global convergence,strong convergence and rate of superlinear convergence of the SF-SQO method.Finally,preliminary numerical experiments regarding the economic dispatch of a power system are carried out,and these show that the SF-SQO method is promising.
基金supported by the National Natural Science Foundation of China(No.12171106)the Natural Science Foundation of Guangxi Province(Nos.2020GXNSFDA238017 and 2018GXNSFFA281007).
文摘This work is about a splitting method for solving a nonconvex nonseparable optimization problem with linear constraints,where the objective function consists of two separable functions and a coupled term.First,based on the ideas from Bregman distance and Peaceman–Rachford splitting method,the Bregman Peaceman–Rachford splitting method with different relaxation factors for the multiplier is proposed.Second,the global and strong convergence of the proposed algorithm are proved under general conditions including the region of the two relaxation factors as well as the crucial Kurdyka–Łojasiewicz property.Third,when the associated Kurdyka–Łojasiewicz property function has a special structure,the sublinear and linear convergence rates of the proposed algorithm are guaranteed.Furthermore,some preliminary numerical results are shown to indicate the effectiveness of the proposed algorithm.
基金supported by Ministerio de Ciencia e Innovacion(Spain)through projects PID2019-104927GB-C21 and PID2019-104927GB-C22,MCIN/AEI/10.13039/501100011033,ERDF(“A way of making Europe”)the support of the Conselleria d’Innovacio,Universitats,Ciencia i Societat Digital from the Generalitat Valenciana(Spain)through project CIAICO/2021/180.
文摘We present a new family of fourth-order splitting methods with positive coefficients especially tailored for the time integration of linear parabolic problems and,in particular,for the time dependent Schrodinger equation,both in real and imaginary time.They are based on the use of a double commutator and a modified processor,and are more efficient than other widely used schemes found in the literature.Moreover,for certain potentials,they achieve order six.Several examples in one,two and three dimensions clearly illustrate the computational advantages of the new schemes.
文摘In this paper,we explore bound preserving and high-order accurate local discontinuous Galerkin(LDG)schemes to solve a class of chemotaxis models,including the classical Keller-Segel(KS)model and two other density-dependent problems.We use the convex splitting method,the variant energy quadratization method,and the scalar auxiliary variable method coupled with the LDG method to construct first-order temporal accurate schemes based on the gradient flow structure of the models.These semi-implicit schemes are decoupled,energy stable,and can be extended to high accuracy schemes using the semi-implicit spectral deferred correction method.Many bound preserving DG discretizations are only worked on explicit time integration methods and are difficult to get high-order accuracy.To overcome these difficulties,we use the Lagrange multipliers to enforce the implicit or semi-implicit LDG schemes to satisfy the bound constraints at each time step.This bound preserving limiter results in the Karush-Kuhn-Tucker condition,which can be solved by an efficient active set semi-smooth Newton method.Various numerical experiments illustrate the high-order accuracy and the effect of bound preserving.
基金Supported by the National High-Tech Research and Development Program of China (No. 2009AA01A135)the National Natural Science Foundation of China (Nos. 10971165, 11001216, 11071193, 10871156)the Foundation of AVIC Chengdu Aircraft Design and Research Institute
文摘In this paper, we propose a dimensional splitting method for the three dimensional (3D) rotating Navier-Stokes equations. Assume that the domain is a channel bounded by two surfaces and is decomposed by a series of surfaces i into several sub-domains, which are called the layers of the flow. Every interface i between two sub-domains shares the same geometry. After establishing a semi-geodesic coordinate (S-coordinate) system based on i, Navier-Stoke equations in this coordinate can be expressed as the sum of two operators, of which one is called the membrane operator defined on the tangent space on i, another one is called the bending operator taking value in the normal space on i. Then the derivatives of velocity with respect to the normal direction of the surface are approximated by the Euler central difference, and an approximate form of Navier-Stokes equations on the surface i is obtained, which is called the two-dimensional three-component (2D-3C) Navier-Stokes equations on a two dimensional manifold. Solving these equations by alternate iteration, an approximate solution to the original 3D Navier-Stokes equations is obtained. In addition, the proof of the existence of solutions to 2D-3C Navier-Stokes equations is provided, and some approximate methods for solving 2D-3C Navier-Stot4es equations are presented.
文摘In this paper, we further generalize the technique for constructing the normal (or pos- itive definite) and skew-Hermitian splitting iteration method for solving large sparse non- Hermitian positive definite system of linear equations. By introducing a new splitting, we establish a class of efficient iteration methods, called positive definite and semi-definite splitting (PPS) methods, and prove that the sequence produced by the PPS method con- verges unconditionally to the unique solution of the system. Moreover, we propose two kinds of typical practical choices of the PPS method and study the upper bound of the spectral radius of the iteration matrix. In addition, we show the optimal parameters such that the spectral radius achieves the minimum under certain conditions. Finally, some numerical examples are given to demonstrate the effectiveness of the considered methods.
基金This work was partially supported by the National Natural Science Foundations of China(Nos.11471102 and 11701150)the Key Basic Research Foundation of the Higher Education Institutions of Henan Province(No.16A110012).
文摘The Douglas–Peaceman–Rachford–Varga operator splitting methods are a class ofefficient methods for finding a zero of the sum of two maximal monotone operatorsin a real Hilbert space;however, they are sometimes difficult or even impossible tosolve the subproblems exactly. In this paper, we suggest an inexact version in whichsome relative error criterion is discussed. The corresponding convergence propertiesare established, and some preliminary numerical experiments are reported to illustrateits efficiency.
基金supported by the National Natural Science Foundation of China(Grant Nos.11901564 and 12171466).
文摘We propose Poisson integrators for the numerical integration of separable Poisson systems.We analyze three situations in which Poisson systems are separated in threeways and Poisson integrators can be constructed by using the splittingmethod.Numerical results show that the Poisson integrators outperform the higher order non-Poisson integrators in terms of long-termenergy conservation and computational cost.The Poisson integrators are also shown to be more efficient than the canonicalized sympletic methods of the same order.
基金This research was supported by the National Natural Science Foundation of China(No.61179033)Collaborative Innovation Center on Beijing Society-Building and Social Governance.
文摘In this paper,we discuss a splitting method for group Lasso.By assuming that the sequence of the step lengths has positive lower bound and positive upper bound(unrelated to the given problem data),we prove its Q-linear rate of convergence of the distance sequence of the iterates to the solution set.Moreover,we make comparisons with convergence of the proximal gradient method analyzed very recently.
基金Acknowledgements This work was supported by the National Natural Science Foundation of China (Grant Nos. 11671116, 11271107, 91630202) and the Natural Science Foundation of Hebei Province of China (No. A2015202365).
文摘The strictly contractive Peaceman-Rachford splitting method is one of effective methods for solving separable convex optimization problem, and the inertial proximal Peaceman-Rachford splitting method is one of its important variants. It is known that the convergence of the inertial proximal Peaceman- Rachford splitting method can be ensured if the relaxation factor in Lagrangian multiplier updates is underdetermined, which means that the steps for the Lagrangian multiplier updates are shrunk conservatively. Although small steps play an important role in ensuring convergence, they should be strongly avoided in practice. In this article, we propose a relaxed inertial proximal Peaceman- Rachford splitting method, which has a larger feasible set for the relaxation factor. Thus, our method provides the possibility to admit larger steps in the Lagrangian multiplier updates. We establish the global convergence of the proposed algorithm under the same conditions as the inertial proximal Peaceman-Rachford splitting method. Numerical experimental results on a sparse signal recovery problem in compressive sensing and a total variation based image denoising problem demonstrate the effectiveness of our method.
文摘Consider the problem of minimizing the sum of two convex functions,one being smooth and the other non-smooth.In this paper,we introduce a general class of approximate proximal splitting(APS)methods for solving such minimization problems.Methods in the APS class include many well-known algorithms such as the proximal splitting method,the block coordinate descent method(BCD),and the approximate gradient projection methods for smooth convex optimization.We establish the linear convergence of APS methods under a local error bound assumption.Since the latter is known to hold for compressive sensing and sparse group LASSO problems,our analysis implies the linear convergence of the BCD method for these problems without strong convexity assumption.
文摘In this paper,we propose a partially parallel prediction-correction splitting method for solving block-separable linearly constrained convex optimization problems with three blocks.Unlike the extended alternating direction method of multipliers,the last two subproblems in the prediction step are solved parallelly,and a correction step is employed in the method to correct the dual variable and two blocks of the primal variables.The step size adapted in the correction step allows for major contribution from the latest solution point to the iteration point.Some numerical results are reported to show the effectiveness of the presented method.
基金the open foundations of State Key Laboratory of High Performance Computing and State Key Laboratory of Aerodynamics.Y.C.gratefully acknowledges support from NUDT’s Innovation Foundation(Grant No.B110205)H.Z.was supported by the Natural Science Foundation of China(Grant No.11301525).
文摘In this paper,we propose two new explicit multi-symplectic splitting methods for the nonlinear Dirac(NLD)equation.Based on its multi-symplectic formulation,the NLD equation is split into one linear multi-symplectic system and one nonlinear infinite Hamiltonian system.Then multi-symplectic Fourier pseudospectral method and multi-symplectic Preissmann scheme are employed to discretize the linear subproblem,respectively.And the nonlinear subsystem is solved by a symplectic scheme.Finally,a composition method is applied to obtain the final schemes for the NLD equation.We find that the two proposed schemes preserve the total symplecticity and can be solved explicitly.Numerical experiments are presented to show the effectiveness of the proposed methods.