Optimal Stable Nonlinear Approximation

Foundations of Computational Mathematics - While it is well-known that nonlinear methods of approximation can often perform dramatically better than linear methods, there are still questions on how...     

Meshfree Approximation for Stochastic Optimal Control Problems

In this work,we study the gradient projection method for solving a class of stochastic control problems by using a mesh free approximation ap-proach to implement spatial dimension approximation.Our main contribu-tion is to extend the existing gradient projection method to moderate high-dimensional space.The moving least square method and the general radial basis function interpolation method are introduced as showcase methods to demonstrate our computational framework,and rigorous numerical analysis is provided to prove the convergence of our meshfree approximation approach.We also present several numerical experiments to validate the theoretical re-sults of our approach and demonstrate the performance meshfree approxima-tion in solving stochastic optimal control problems.     

关于圆和椭圆逼近显示的最优算法(英文)

本文给出了一个逼近显示圆的新算法。该算法是通过相交多边形而不是内接多边形逼近圆。由于构造相交多边形时其面积等于圆面积 ,因此新算法是最优逼近。同时还推广到椭圆     

多元Kantorovich算子的最优逼近类

本文研究定义在单纯形上的多元Kantorovich算子逼近的正逆不等式与饱和定理,给出该算子在Lp(1≤p≤∞)空间的最优逼近类,即利用K-泛函的特征刻画分别满足‖Knf-f‖p=O(n-1)与‖Knf-f‖p=o(n-1)的函数类.     

多元Kantorovich算子的最优逼近类

本文研究定义在单纯形上的多元Kantorovich算子逼近的正逆不等式与饱和定理,给出该算子在Lp(1≤p≤∞)空间的最优逼近类,即利用K-泛函的特征刻画分别满足‖Knf-f‖p=O(n-1) 与‖Knf-f‖p=o(n-1)的函数类．     

Optimal Constant in Approximation by Bernstein Operators

We obtained that for any n N, C = 1 is the smallest constant for which the inequality ||B _n (f) - f|| C ₂(f, 1/n) holds on the class of continuous functions f, as well as on the class of bounded functions f, where B _n is the Bernstein operators of degree n, ₂ is the second order modulus and || || is the sup-norm.     

On Simultaneous Approximation by Optimal Algebraic Polynomials

In this paper the order of bestapproximation by algebraic polynomials is related to the order of weighted simultaneous approximation. A special case of a Markov-Bernstein type inequality proved by Ditzian and Totik in a very general fashion is used.     

单纯形上的q-Stancu多项式的最优逼近阶

构造了单纯形上的多元q-Stancu多项式,它是著名的Bernstein多项式和Stancu多项式的推广.建立该类多项式逼近连续函数的上、下界估计,进而给出其对连续函数的最优逼近阶(饱和阶)及其特征刻画.此外,还研究了该类多项式逼近连续函数的饱和类.     

线性流形上广义中心反对称矩阵的最佳逼近

本讨论在线性流形上广义反对称矩阵的最佳逼近，给出了若干有意义的结果。     

线性流形上双反对称矩阵的最佳逼近

本文讨论了线性流形上用双反对称矩阵构造给定矩阵的最佳逼近问题,给出问题解的表达式,最后给出求最佳逼近解的数值方法与数值算例.     

Almost Optimal Estimates for Best Approximation by Translates on a Torus

We investigate the best approximation of some classes of functions defined on a $d$-dimensional torus ${\mbox{\smallbf T}}^d$ by the transfer manifold $H_n(\varphi)=\{\xxsum_{i=1}^n b_i\varphi(\cdot -a_i)\}$ in the Hilbert space $L_2({\mbox{\smallbf T}}^d)$. For Sobolev classes of functions $\tilde{W}_2^r$ we obtain two-sided estimates of the deviation ${\rm dist}(\tilde{W}_2^r,H_n(\varphi))$. These results are applied to obtain estimates for radial basis approximation.     

谱约束下对称正交对称矩阵束的最佳逼近

讨论了对称正交对称矩阵的广义逆特征值问题，得到了通解表达式和最佳解的表达式。     

Pseudo Approximation Algorithms with Applications to Optimal Motion Planning

We introduce a technique for computing approximate solutions to optimization problems. If $X$ is the set of feasible solutions, the standard goal of approximation algorithms is to compute $x\in X$ that is an $\varepsilon$-approximate solution in the following sense: $$d(x) \leq (1+\varepsilon)\, d(x^*),$$ where $x^* \in X$ is an optimal solution, $d\colon\ X\rightarrow {\Bbb R}_{\geq 0}$ is the optimization function to be minimized, and $\varepsilon>0$ is an input parameter. Our approach is first to devise algorithms that compute pseudo $\varepsilon$-approximate solutions satisfying the bound $$d(x) \leq d(x_R^*) + \varepsilon R,$$ where $R>0$ is a new input parameter. Here $x^*_R$ denotes an optimal solution in the space $X_R$ of $R$-constrained feasible solutions. The parameter $R$ provides a stratification of $X$ in the sense that (1) $X_R \subseteq X_{R}$ for $R < R$ and (2) $X_R = X$ for $R$ sufficiently large. We first describe a highly efficient scheme for converting a pseudo $\varepsilon$-approximation algorithm into a true $\varepsilon$-approximation algorithm. This scheme is useful because pseudo approximation algorithms seem to be easier to construct than $\varepsilon$-approximation algorithms. Another benefit is that our algorithm is automatically precision-sensitive. We apply our technique to two problems in robotics: (A) Euclidean Shortest Path (3ESP), namely the shortest path for a point robot amidst polyhedral obstacles in three dimensions, and (B) $d_1$-optimal motion for a rod moving amidst planar obstacles (1ORM). Previously, no polynomial time $\varepsilon$-approximation algorithm for (B) was known. For (A), our new solution is simpler than previous solutions and has an exponentially smaller complexity in terms of the input precision.     

An Approximation Scheme for Uncertain Minimax Optimal Control Problems

In this work, we address an uncertain minimax optimal control problem with linear dynamics where the objective functional is the expected value of the supremum of the running cost over a time interval. By taking an independently drawn random sample, the expected value function is approximated by the corresponding sample average function. We study the epi-convergence of the approximated objective functionals as well as the convergence of their global minimizers. Then we define an Euler discretization in time of the sample average problem and prove that the value of the discrete time problem converges to the value of the sample average approximation. In addition, we show that there exists a sequence of discrete problems such that the accumulation points of their minimizers are optimal solutions of the original problem. Finally, we propose a convergent descent method to solve the discrete time problem, and show some preliminary numerical results for two simple examples.     

一类矩阵方程的反中心对称最佳逼近解

利用矩阵的正交相似变换和广义奇异值分解,讨论了矩阵方程 AXB=C具有反中心对称解的充要条件,得到了解的具体表达式.然后应用Frobenius范数正交矩阵乘积不变性,在该方程的反中心对称解解集合中导出了与给定相同类型矩阵的最佳逼近解的表达式.     

Finite Element Approximation of Elliptic Dirichlet Optimal Control Problems

We develop a priori error analysis for the finite element Galerkin discretization of elliptic Dirichlet optimal control problems. The state equation is given by an elliptic partial differential equation and the finite dimensional control variable enters the Dirichlet boundary conditions. We prove the optimal order of convergence and present a numerical example confirming our results.     

线性流形上广义中心对称矩阵的最佳逼近

讨论了线性流形上广义中心对称矩阵的最小二乘解,得到了解的一般表达式。对于任意给定的实对称矩阵A,在最小二乘解集中得到了A的最佳逼近解．