In this paper, we study the inertial manifolds for a class of the Kirchhoff-type equations with strongly damped terms and source terms. The inertial manifold is a finite dimensional invariant smooth manifold that cont...In this paper, we study the inertial manifolds for a class of the Kirchhoff-type equations with strongly damped terms and source terms. The inertial manifold is a finite dimensional invariant smooth manifold that contains the global attractor, attracting the solution orbits by the exponential rate. Under appropriate assumptions, we firstly exert the Hadamard’s graph transformation method to structure a graph norm of a Lipschitz continuous function, and then we prove the existence of the inertial manifold by showing that the spectral gap condition is true.展开更多
2020年Pham Ky Anh等人在Hilbert空间中提出了一种求解映射伪单调且Lipschitz连续的自适应投影算法(简记为PDNA).该算法无需知道映射的Lipschitz系数,且具有强收敛的结果.注意到算法的步长与其收敛速度密切相关,通常大步长的算法具有更...2020年Pham Ky Anh等人在Hilbert空间中提出了一种求解映射伪单调且Lipschitz连续的自适应投影算法(简记为PDNA).该算法无需知道映射的Lipschitz系数,且具有强收敛的结果.注意到算法的步长与其收敛速度密切相关,通常大步长的算法具有更好的收敛速度.Liu和Yang提出了一种求解拟单调变分不等式的自适应算法(简记为LYA),LYA的步长比PDNA中的步长长.本文提出了一种自适应的求解映射伪单调且Lipschitz连续的次梯度外梯度投影算法.新算法的步长比LYA长,且可以退化为LYA中的步长.在与PDNA相同的假设条件下证明了新算法的强收敛性.数值实验表明新算法有更好的数值实验结果.展开更多
A modified Gauss-type Proximal Point Algorithm (modified GG-PPA) is presented in this paper for solving the generalized equations like 0 ∈T(x), where T is a set-valued mapping acts between two different Bana...A modified Gauss-type Proximal Point Algorithm (modified GG-PPA) is presented in this paper for solving the generalized equations like 0 ∈T(x), where T is a set-valued mapping acts between two different Banach spaces X and Y. By considering some necessary assumptions, we show the existence of any sequence generated by the modified GG-PPA and prove the semi-local and local convergence results by using metrically regular mapping. In addition, we give a numerical example to justify the result of semi-local convergence.展开更多
This paper studies the exponential attractor for a class of the Kirchhoff-type equations with strongly damped terms and source terms. The exponential attractor is also called the inertial fractal set, which is an inte...This paper studies the exponential attractor for a class of the Kirchhoff-type equations with strongly damped terms and source terms. The exponential attractor is also called the inertial fractal set, which is an intermediate step between global attractors and inertial manifolds. Obtaining a set that attracts all the trajectories of the dynamical system at an exponential rate by the methods of Eden A. Under appropriate assumptions, we firstly construct an invariantly compact set. Secondly, showing the solution semigroups of the Kirchhoff-type equations is squeezing and Lipschitz continuous. Finally, the finite fractal dimension of the exponential attractor is obtained.展开更多
本文提出了求解多集分裂可行问题(Multiple-sets Split Feasibility Problem.简称MSFP)的一种新的松弛投影算法.已有求解MSFP的算法大多采用邻近函数p(x):=(1/2)sum from i=1 to t(α_i||x-P_(Ci)(x)||~2)+(1/2)sum from j=1 to r( λ_j...本文提出了求解多集分裂可行问题(Multiple-sets Split Feasibility Problem.简称MSFP)的一种新的松弛投影算法.已有求解MSFP的算法大多采用邻近函数p(x):=(1/2)sum from i=1 to t(α_i||x-P_(Ci)(x)||~2)+(1/2)sum from j=1 to r( λ_j||A_x-P_(Qj)(Ax)||~2)度量点到所有集合的距离并在迭代中直接利用其梯度方向,与此不同,本文引入了新的搜索方向,并基于此提出了新的算法.搜索方向的不同导致了算法的收敛性证明上的明显差异.初步的数值计算结果表明新算法对于不同的问题都能够有较快的收敛速度,且在问题维数增大时表现得越发明显.展开更多
文摘In this paper, we study the inertial manifolds for a class of the Kirchhoff-type equations with strongly damped terms and source terms. The inertial manifold is a finite dimensional invariant smooth manifold that contains the global attractor, attracting the solution orbits by the exponential rate. Under appropriate assumptions, we firstly exert the Hadamard’s graph transformation method to structure a graph norm of a Lipschitz continuous function, and then we prove the existence of the inertial manifold by showing that the spectral gap condition is true.
文摘2020年Pham Ky Anh等人在Hilbert空间中提出了一种求解映射伪单调且Lipschitz连续的自适应投影算法(简记为PDNA).该算法无需知道映射的Lipschitz系数,且具有强收敛的结果.注意到算法的步长与其收敛速度密切相关,通常大步长的算法具有更好的收敛速度.Liu和Yang提出了一种求解拟单调变分不等式的自适应算法(简记为LYA),LYA的步长比PDNA中的步长长.本文提出了一种自适应的求解映射伪单调且Lipschitz连续的次梯度外梯度投影算法.新算法的步长比LYA长,且可以退化为LYA中的步长.在与PDNA相同的假设条件下证明了新算法的强收敛性.数值实验表明新算法有更好的数值实验结果.
文摘A modified Gauss-type Proximal Point Algorithm (modified GG-PPA) is presented in this paper for solving the generalized equations like 0 ∈T(x), where T is a set-valued mapping acts between two different Banach spaces X and Y. By considering some necessary assumptions, we show the existence of any sequence generated by the modified GG-PPA and prove the semi-local and local convergence results by using metrically regular mapping. In addition, we give a numerical example to justify the result of semi-local convergence.
文摘This paper studies the exponential attractor for a class of the Kirchhoff-type equations with strongly damped terms and source terms. The exponential attractor is also called the inertial fractal set, which is an intermediate step between global attractors and inertial manifolds. Obtaining a set that attracts all the trajectories of the dynamical system at an exponential rate by the methods of Eden A. Under appropriate assumptions, we firstly construct an invariantly compact set. Secondly, showing the solution semigroups of the Kirchhoff-type equations is squeezing and Lipschitz continuous. Finally, the finite fractal dimension of the exponential attractor is obtained.
文摘本文提出了求解多集分裂可行问题(Multiple-sets Split Feasibility Problem.简称MSFP)的一种新的松弛投影算法.已有求解MSFP的算法大多采用邻近函数p(x):=(1/2)sum from i=1 to t(α_i||x-P_(Ci)(x)||~2)+(1/2)sum from j=1 to r( λ_j||A_x-P_(Qj)(Ax)||~2)度量点到所有集合的距离并在迭代中直接利用其梯度方向,与此不同,本文引入了新的搜索方向,并基于此提出了新的算法.搜索方向的不同导致了算法的收敛性证明上的明显差异.初步的数值计算结果表明新算法对于不同的问题都能够有较快的收敛速度,且在问题维数增大时表现得越发明显.
