LEBESGUE CONSTANTS ARISING IN A CLASS OF COLLOCATION METHODS
LEBESGUE CONSTANTS ARISING IN A CLASS OF COLLOCATION METHODS
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一类搭配方法中出现的勒贝格常数
DOI：
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发表时间：
2015
期刊：
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通讯作者：
Anil V. Rao
中科院分区：
文献类型：
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作者：
W. Hager;Hongyan Hou;Anil V. Rao
得到了（i1，+1）上Gauss求积点被点i1增广的Lebesgue常数的估计，以及（i1，+1）或（i1，+1）上Radau求积点的Lebesgue常数的估计.证明了Lebesgue常数为O（pN），其中N为正交点数.这些点集出现在最近开发的正交配置方案的最优控制问题的残差估计。对于具有光滑解的问题，Lebesgue常数的估计可以意味着同位问题中的残差作为正交点数量的函数的指数衰减。为这些计划而设。该分析是基于一个边界的逆线性算子与离散化问题，和估计的残差时得到的连续问题的解决方案代入离散化问题。本文主要研究残差的估计问题。证明了上范数下的残差由连续问题解的导数与解的插值函数的导数之间的上范数距离有界。通过马尔可夫不等式（18），这个距离可以根据点集的勒贝格常数和最佳多项式逼近的误差来定界。杰克逊（17）的一个经典结果给出了最佳逼近误差的估计。我们需要分析的勒贝格常数对应于（-1，+1）上的Jacobi多项式的根，它被以下任一项增广：= +1或？=-1。Vertesi在（24）中分析了添加终点的影响。对于（-1，+1）上的高斯求积点，= +1或Radau求积点在（−1，+1）或（−1，+1）上，在（24，Thm. 2.1)对于Lebesgue常数，它是O（log（N）<$N），其中N是正交点的数目。我们把这个约束锐化为O（N）。为了激发Lebesgue常数与配置方法的相关性，让我们考虑标量一阶微分方程u
Estimates are obtained for the Lebesgue constants associated with the Gauss quadra- ture points on (i1, +1) augmented by the point i1 and with the Radau quadrature points on either (i1, +1) or (i1, +1). It is shown that the Lebesgue constants are O( p N), where N is the number of quadrature points. These point sets arise in the estimation of the residual associated with recently developed orthogonal collocation schemes for optimal control problems. For problems with smooth solutions, the estimates for the Lebesgue constants can imply an exponential decay of the residual in the collocated problem as a function of the number of quadrature points. tablished for these schemes. The analysis is based on a bound for the inverse of a linearized operator associated with the discretized problem, and an estimate for the residual one gets when substituting the solution to the continuous problem into the discretized problem. This paper focuses on the estimation of the residual. We show that the residual in the sup-norm is bounded by the sup-norm distance between the derivative of the solution to the continuous problem and the derivative of the inter- polant of the solution. By Markov's inequality (18), this distance can be bounded in terms of the Lebesgue constant for the point set and the error in best polynomial approximation. A classic result of Jackson (17) gives an estimate for the error in best approximation. The Lebesgue constant that we need to analyze corresponds to the roots of a Jacobi polynomial on (−1, +1) augmented by either ? = +1 or ? = −1. The effects of the added endpoints were analyzed by Vertesi in (24). For either the Gauss quadrature points on (−1, +1) augmented by ? = +1 or the Radau quadrature points on (−1, +1) or on (−1, +1), the bound given in (24, Thm. 2.1) for the Lebesgue constants is O(log(N) √ N), where N is the number of quadrature points. We sharpen this bound to O( √ N). To motivate the relevance of the Lebesgue constant to collocation methods, let us consider the scalar first-order differential equation u