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Algorithms and complexity for functions on general domains

机译：常规域上函数的算法和复杂性

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Error bounds and complexity bounds in numerical analysis and information-based complexity are often proved for functions that are defined on very simple domains, such as a cube, a torus, or a sphere. We study optimal error bounds for the approximation or integration of functions defined on D-d subset of R-d and only assume that D-d is a bounded Lipschitz domain. Some results are even more general. We study three different concepts to measure the complexity: order of convergence, asymptotic constant, and explicit uniform bounds, i.e., bounds that hold for all n (number of pieces of information) and all (normalized) domains.It is known for many problems that the order of convergence of optimal algorithms does not depend on the domain D-d subset of R-d We present examples for which the following statements are true:1. Also the asymptotic constant does not depend on the shape of D-d or the imposed boundary values, it only depends on the volume of the domain.2. There are explicit and uniform lower (or upper, respectively) bounds for the error that are only slightly smaller (or larger, respectively) than the asymptotic error bound. (C) 2020 Elsevier Inc. All rights reserved.
机译：数值分析中的错误界限和复杂性界限通常被证明是在非常简单的域中定义的功能，例如立方体，圆环或球体。我们研究了R-D的D-D子集上定义的函数的近似或集成的最佳误差界限，并且仅假定D-D是有界LipsChitz域。一些结果更加一般。我们研究了三种不同的概念来测量复杂性：收敛性，渐近常量和显式均匀界限，即保持所有n的界限（信息的数量）和所有（归一化）域。对于许多问题是已知的最佳算法的收敛顺序不依赖于RD的域DD子集，我们存在以下语句的示例：1。渐近常量也不依赖于D-D的形状或施加的边界值，它仅取决于域的音量。对于误差的误差，存在显式和均匀的下部（或上部）界限，其误差仅略小（或更大）而不是渐近误差绑定。 （c）2020 Elsevier Inc.保留所有权利。

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