A Succinct Multivariate Lazy Multivariate Tower AD for Weil Algebra Computation

Ishii, Hiromi

doi:10.48550/arxiv.2103.11615

Search citation statements

Order By: Relevance

Paper Sections

Select...

A Succinct Multivariate Lazy Tower Ad1

Citation Types

Supporting

Mentioning

Contrasting

Year Published

2021

Publication Types

Select...

Other1

Relationship

Self Cite1

Independent0

Authors

Journals

Cited by 1 publication

(1 citation statement)

References 5 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The method presented here is a mixture of Lazy Multivariate Tower [13] and nested Sparse Tower [9]. For details, we refer readers to the related paper by the author in RIMS Kôkyûroku [5].…”

Section: A Succinct Multivariate Lazy Tower Admentioning

confidence: 99%

Automatic Differentiation With Higher Infinitesimals, or Computational Smooth Infinitesimal Analysis in Weil Algebra

Ishii

2021

Preprint

Self Cite

View full text Add to dashboard Cite

We propose an algorithm to compute the C ∞ -ring structure of arbitrary Weil algebra. It allows us to do some analysis with higher infinitesimals numerically and symbolically. To that end, we first give a brief description of the (Forward-mode) automatic differentiation (AD) in terms of C ∞ -rings. The notion of a C ∞ -ring was introduced by Lawvere [10] and used as the fundamental building block of smooth infinitesimal analysis and synthetic differential geometry [11]. We argue that interpreting AD in terms of C ∞ -rings gives us a unifying theoretical framework and modular ways to express multivariate partial derivatives. In particular, we can "package" higher-order Forward-mode AD as a Weil algebra, and take tensor products to compose them to achieve multivariate higher-order AD. The algorithms in the present paper can also be used for a pedagogical purpose in learning and studying smooth infinitesimal analysis as well.

show abstract

Section: A Succinct Multivariate Lazy Tower Admentioning

confidence: 99%