WE have obtained the convergence theorems of the iteration of Halley family by the point estimate of Smale. In the point estimate, map f which is desired to be solved is presumed to be analytic in some proper neighbor...WE have obtained the convergence theorems of the iteration of Halley family by the point estimate of Smale. In the point estimate, map f which is desired to be solved is presumed to be analytic in some proper neighborhood at the initial value z<sub>0</sub>. From the viewpoint of展开更多
The Krasnoselskii-Mann iteration plays an important role in the approximation of fixed points of nonexpansive mappings,and it is well known that the clas-sic Krasnoselskii-Mann iteration is weakly convergent in Hilber...The Krasnoselskii-Mann iteration plays an important role in the approximation of fixed points of nonexpansive mappings,and it is well known that the clas-sic Krasnoselskii-Mann iteration is weakly convergent in Hilbert spaces.The weak convergence is also known even in Banach spaces.Recently,Kanzow and Shehu pro-posed a generalized Krasnoselskii-Mann-type iteration for nonexpansive mappings and established its convergence in Hilbert spaces.In this paper,we show that the generalized Krasnoselskii-Mann-type iteration proposed by Kanzow and Shehu also converges in Banach spaces.As applications,we proved the weak convergence of generalized proximal point algorithm in the uniformly convex Banach spaces.展开更多
Aggregation equations are broadly used tomodel population dynamicswith nonlocal interactions,characterized by a potential in the equation.This paper considers the inverse problem of identifying the potential from a si...Aggregation equations are broadly used tomodel population dynamicswith nonlocal interactions,characterized by a potential in the equation.This paper considers the inverse problem of identifying the potential from a single noisy spatialtemporal process.The identification is challenging in the presence of noise due to the instability of numerical differentiation.We propose a robust model-based technique to identify the potential by minimizing a regularized data fidelity term,and regularization is taken as the total variation and the squared Laplacian.A split Bregman method is used to solve the regularized optimization problem.Our method is robust to noise by utilizing a Successively Denoised Differentiation technique.We consider additional constraints such as compact support and symmetry constraints to enhance the performance further.We also apply thismethod to identify time-varying potentials and identify the interaction kernel in an agent-based system.Various numerical examples in one and two dimensions are included to verify the effectiveness and robustness of the proposed method.展开更多
Factorization of the incompressible Stokes operator linking pressure and velocity is revisited.The main purpose is to use the inverse of the Stokes operator with a large time step as a preconditioner for Newton and Ar...Factorization of the incompressible Stokes operator linking pressure and velocity is revisited.The main purpose is to use the inverse of the Stokes operator with a large time step as a preconditioner for Newton and Arnoldi iterations applied to computation of steady three-dimensional flows and study of their stability.It is shown that the Stokes operator can be inversed within an acceptable computational effort.This inverse includes fast direct inverses of several Helmholtz operators and iterative inverse of the pressure matrix.It is shown,additionally,that fast direct solvers can be attractive for the inverse of the Helmholtz and Laplace operators on fine grids and at large Reynolds numbers,as well as for other problems where convergence of iterative methods slows down.Implementation of the Stokes operator inverse to time-steppingbased formulation of the Newton and Arnoldi iterations is discussed.展开更多
The existence, uniqueness and non-symmetric iterative approximation of solutions for a class of systems of mixed monotone operator equations are discussed. As an application, we utilize, the results presented in this ...The existence, uniqueness and non-symmetric iterative approximation of solutions for a class of systems of mixed monotone operator equations are discussed. As an application, we utilize, the results presented in this paper to study the existence and uniqueness problems of common solutions for a class of systems of functional equations arising in dynamic programming of multistage decision processes and a class of systems of nonlinear integral equation. The results obtained in this paper not only answer an open question suggested in [3] but also generalize the corresponding results of [1],[2].展开更多
文摘WE have obtained the convergence theorems of the iteration of Halley family by the point estimate of Smale. In the point estimate, map f which is desired to be solved is presumed to be analytic in some proper neighborhood at the initial value z<sub>0</sub>. From the viewpoint of
基金supported by the Students Innovation and Entrepreneurship Training Program Foundation of China West Normal University(No.201810638047)supported by the National Natural Science Foundation of China(Nos.11571178 and 11801455)Fundamental Research Funds of China West Normal University(Nos.17E084 and 18B031).
文摘The Krasnoselskii-Mann iteration plays an important role in the approximation of fixed points of nonexpansive mappings,and it is well known that the clas-sic Krasnoselskii-Mann iteration is weakly convergent in Hilbert spaces.The weak convergence is also known even in Banach spaces.Recently,Kanzow and Shehu pro-posed a generalized Krasnoselskii-Mann-type iteration for nonexpansive mappings and established its convergence in Hilbert spaces.In this paper,we show that the generalized Krasnoselskii-Mann-type iteration proposed by Kanzow and Shehu also converges in Banach spaces.As applications,we proved the weak convergence of generalized proximal point algorithm in the uniformly convex Banach spaces.
基金supported in part by Simons Foundation grant 282311 and 584960supported in part by NSF grant NSF-DMS 1818751 and NSF-DMS 2012652+1 种基金supported in part by HKBU 162784 and 179356supported in part by NSF grants DMS-1522585 and DMS-CDS&E-MSS-1622453.
文摘Aggregation equations are broadly used tomodel population dynamicswith nonlocal interactions,characterized by a potential in the equation.This paper considers the inverse problem of identifying the potential from a single noisy spatialtemporal process.The identification is challenging in the presence of noise due to the instability of numerical differentiation.We propose a robust model-based technique to identify the potential by minimizing a regularized data fidelity term,and regularization is taken as the total variation and the squared Laplacian.A split Bregman method is used to solve the regularized optimization problem.Our method is robust to noise by utilizing a Successively Denoised Differentiation technique.We consider additional constraints such as compact support and symmetry constraints to enhance the performance further.We also apply thismethod to identify time-varying potentials and identify the interaction kernel in an agent-based system.Various numerical examples in one and two dimensions are included to verify the effectiveness and robustness of the proposed method.
文摘Factorization of the incompressible Stokes operator linking pressure and velocity is revisited.The main purpose is to use the inverse of the Stokes operator with a large time step as a preconditioner for Newton and Arnoldi iterations applied to computation of steady three-dimensional flows and study of their stability.It is shown that the Stokes operator can be inversed within an acceptable computational effort.This inverse includes fast direct inverses of several Helmholtz operators and iterative inverse of the pressure matrix.It is shown,additionally,that fast direct solvers can be attractive for the inverse of the Helmholtz and Laplace operators on fine grids and at large Reynolds numbers,as well as for other problems where convergence of iterative methods slows down.Implementation of the Stokes operator inverse to time-steppingbased formulation of the Newton and Arnoldi iterations is discussed.
基金Supported by National Natural Science Foundation of China
文摘The existence, uniqueness and non-symmetric iterative approximation of solutions for a class of systems of mixed monotone operator equations are discussed. As an application, we utilize, the results presented in this paper to study the existence and uniqueness problems of common solutions for a class of systems of functional equations arising in dynamic programming of multistage decision processes and a class of systems of nonlinear integral equation. The results obtained in this paper not only answer an open question suggested in [3] but also generalize the corresponding results of [1],[2].
