A computational model for multi-variable differential calculus
A computational model for multi-variable differential calculus
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多变量微分计算模型
DOI：
10.1016/j.ic.2012.11.006
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发表时间：
2005
期刊：
影响因子：
--
通讯作者：
D. Pattinson
中科院分区：
文献类型：
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作者：
A. Edalat;A. Lieutier;D. Pattinson
通过在有限维欧几里德空间上构造实值Lipschitz函数的有效给定连续Scott域，我们开发了多变量微分学的域理论计算模型，首次产生了分段可微或更一般的Lipschitz函数的数据类型。n变量的实值Lipschitz函数的模型通过将局部Lipschitz函数的一致信息及其l-导数或等价的Clarke梯度给出的微分性质（其值由Rn的非空、凸和紧子集给出）组合在一起，建立为两个域积的子域。为了得到一个计算上实用的框架，导数信息用Rn中最优拟合紧化超矩形来逼近。在这种情况下，我们证明了函数和导数信息的一致性可以通过将其简化为线性规划问题来确定。这提供了一种检验定义域上有理基元素一致性的算法，意味着该定义域可以具有有效的结构，并为多变量微分学提供了一个可计算的框架。我们还提出了一个域论的、区间值的线积分的概念，并证明了如果一个表示非空的、凸紧的值向量场的Scott连续函数是可积的，那么它在任何封闭的分段c1路径上的区间值积分包含零。在以紧致超矩形的形式给出导数信息的情况下，我们使用极小曲面理论的技术推导出相反的结果：如果超矩形的值向量场在任何分段c1路径上的区间值线积分包含零，则该值向量场是可积的。给出了路径积分基本定理的域论推广。最后，我们构造了由一对函数和超矩形导数信息得到的最小和最大分段线性函数。当这对是一致的，这提供了最小和最大的映射来见证一致性。
We develop a domain-theoretic computational model for multi-variable differential calculus, which for the first time gives rise to data types for piecewise differentiable or more generally Lipschitz functions, by constructing an effectively given continuous Scott domain for real-valued Lipschitz functions on finite dimensional Euclidean spaces. The model for real-valued Lipschitz functions of n variables is built as a sub-domain of the product of two domains by tupling together consistent information about locally Lipschitz functions and their differential properties as given by their L-derivative or equivalently Clarke gradient, which has values given by non-empty, convex and compact subsets of Rn. To obtain a computationally practical framework, the derivative information is approximated by the best fit compact hyper-rectangles in Rn. In this case, we show that consistency of the function and derivative information can be decided by reducing it to a linear programming problem. This provides an algorithm to check consistency on the rational basis elements of the domain, implying that the domain can be equipped with an effective structure and giving a computable framework for multi-variable differential calculus. We also develop a domain-theoretic, interval-valued, notion of line integral and show that if a Scott continuous function, representing a non-empty, convex and compact valued vector field, is integrable, then its interval-valued integral over any closed piecewise C1path contains zero. In the case that the derivative information is given in terms of compact hyper-rectangles, we use techniques from the theory of minimal surfaces to deduce the converse result: a hyper-rectangular valued vector field is integrable if its interval-valued line integral over any piecewise C1path contains zero. This gives a domain-theoretic extension of the fundamental theorem of path integration. Finally, we construct the least and the greatest piecewise linear functions obtained from a pair of function and hyper-rectangular derivative information. When the pair is consistent, this provides the least and greatest maps to witness consistency.