The goal of computational science is to develop models that predict phenomena observed in nature. However, these models are often based on parameters that are uncertain. In recent decades, main numerical methods for s...The goal of computational science is to develop models that predict phenomena observed in nature. However, these models are often based on parameters that are uncertain. In recent decades, main numerical methods for solving SPDEs have been used such as, finite difference and finite element schemes [1]-[5]. Also, some practical techniques like the method of lines for boundary value problems have been applied to the linear stochastic partial differential equations, and the outcomes of these approaches have been experimented numerically [7]. In [8]-[10], the author discussed mean square convergent finite difference method for solving some random partial differential equations. Random numerical techniques for both ordinary and partial random differential equations are treated in [4] [10]. As regards applications using explicit analytic solutions or numerical methods, a few results may be found in [5] [6] [11]. This article focuses on solving random heat equation by using Crank-Nicol- son technique under mean square sense and it is organized as follows. In Section 2, the mean square calculus preliminaries that will be required throughout the paper are presented. In Section 3, the Crank-Nicolson scheme for solving the random heat equation is presented. In Section 4, some case studies are showed. Short conclusions are cleared in the end section.展开更多
In this paper, we propose new variants of Newton’s method based on quadrature formula and power mean for solving nonlinear unconstrained optimization problems. It is proved that the order of convergence of the propos...In this paper, we propose new variants of Newton’s method based on quadrature formula and power mean for solving nonlinear unconstrained optimization problems. It is proved that the order of convergence of the proposed family is three. Numerical comparisons are made to show the performance of the presented methods. Furthermore, numerical experiments demonstrate that the logarithmic mean Newton’s method outperform the classical Newton’s and other variants of Newton’s method. MSC: 65H05.展开更多
A highly celebrated problem in dyadic harmonic analysis is the pointwise convergence of the Fejér （or （C, 1）） means of functions on unbounded Vilenkin groups. There are several papers of the author of this pa...A highly celebrated problem in dyadic harmonic analysis is the pointwise convergence of the Fejér （or （C, 1）） means of functions on unbounded Vilenkin groups. There are several papers of the author of this paper concerning this. That is, we know the a.e. convergence σnf → f （n → ∞） for functions f ∈ L^p, where p 〉 1 （Journal of Approximation Theory, 101（1）, 1-36, （1999）） and also the a.e. convergence σMnf → f （n → ∞） for functions f ∈ L^1 （Journal of Approximation Theory, 124（1）, 25-43, （2003））. The aim of this paper is to prove the a.e. relation limn→∞ σnf = f for each integrable function f on any rarely unbounded Vilenkin group. The concept of the rarely unbounded Vilenkin group is discussed in the paper. Basically, it means that the generating sequence m may be an unbounded one, but its ＂big elements＂ are not ＂too dense＂.展开更多
光腔衰荡方法是目前测量光学元件超高反射率(反射率>99.9%)的唯一方法。介绍了一种对光腔衰荡法中激光信号强度与时间关系的优化提取方法。设计了基于光腔衰荡法的光学元件超高反射比的测试系统,通过对采集的光腔衰荡曲线数据进行分...光腔衰荡方法是目前测量光学元件超高反射率(反射率>99.9%)的唯一方法。介绍了一种对光腔衰荡法中激光信号强度与时间关系的优化提取方法。设计了基于光腔衰荡法的光学元件超高反射比的测试系统,通过对采集的光腔衰荡曲线数据进行分段指数拟合,将光腔衰荡曲线数据分为5段,对每段指数拟合结果对应的R^(2)(R-square)和RMSE(root mean squared error)值进行对比分析,计算每段指数拟合的衰荡时间。实验结果表明:截取光腔衰荡曲线数据40%~60%部分拟合得到的结果最接近真实值,求得对应的腔镜的反射率为99.988 977%。最后通过与腔镜的自身反射率进行比较,表明该种数据拟合方法能有效地测量腔镜的反射率,并能减小实验数据本身带来的误差。展开更多
文摘The goal of computational science is to develop models that predict phenomena observed in nature. However, these models are often based on parameters that are uncertain. In recent decades, main numerical methods for solving SPDEs have been used such as, finite difference and finite element schemes [1]-[5]. Also, some practical techniques like the method of lines for boundary value problems have been applied to the linear stochastic partial differential equations, and the outcomes of these approaches have been experimented numerically [7]. In [8]-[10], the author discussed mean square convergent finite difference method for solving some random partial differential equations. Random numerical techniques for both ordinary and partial random differential equations are treated in [4] [10]. As regards applications using explicit analytic solutions or numerical methods, a few results may be found in [5] [6] [11]. This article focuses on solving random heat equation by using Crank-Nicol- son technique under mean square sense and it is organized as follows. In Section 2, the mean square calculus preliminaries that will be required throughout the paper are presented. In Section 3, the Crank-Nicolson scheme for solving the random heat equation is presented. In Section 4, some case studies are showed. Short conclusions are cleared in the end section.
文摘In this paper, we propose new variants of Newton’s method based on quadrature formula and power mean for solving nonlinear unconstrained optimization problems. It is proved that the order of convergence of the proposed family is three. Numerical comparisons are made to show the performance of the presented methods. Furthermore, numerical experiments demonstrate that the logarithmic mean Newton’s method outperform the classical Newton’s and other variants of Newton’s method. MSC: 65H05.
基金the Hungarian National Foundation for Scientific Research(OTKA),Grant No.M36511/2001 and T 048780
文摘A highly celebrated problem in dyadic harmonic analysis is the pointwise convergence of the Fejér （or （C, 1）） means of functions on unbounded Vilenkin groups. There are several papers of the author of this paper concerning this. That is, we know the a.e. convergence σnf → f （n → ∞） for functions f ∈ L^p, where p 〉 1 （Journal of Approximation Theory, 101（1）, 1-36, （1999）） and also the a.e. convergence σMnf → f （n → ∞） for functions f ∈ L^1 （Journal of Approximation Theory, 124（1）, 25-43, （2003））. The aim of this paper is to prove the a.e. relation limn→∞ σnf = f for each integrable function f on any rarely unbounded Vilenkin group. The concept of the rarely unbounded Vilenkin group is discussed in the paper. Basically, it means that the generating sequence m may be an unbounded one, but its ＂big elements＂ are not ＂too dense＂.
文摘光腔衰荡方法是目前测量光学元件超高反射率(反射率>99.9%)的唯一方法。介绍了一种对光腔衰荡法中激光信号强度与时间关系的优化提取方法。设计了基于光腔衰荡法的光学元件超高反射比的测试系统,通过对采集的光腔衰荡曲线数据进行分段指数拟合,将光腔衰荡曲线数据分为5段,对每段指数拟合结果对应的R^(2)(R-square)和RMSE(root mean squared error)值进行对比分析,计算每段指数拟合的衰荡时间。实验结果表明:截取光腔衰荡曲线数据40%~60%部分拟合得到的结果最接近真实值,求得对应的腔镜的反射率为99.988 977%。最后通过与腔镜的自身反射率进行比较,表明该种数据拟合方法能有效地测量腔镜的反射率,并能减小实验数据本身带来的误差。
