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

A Succinct Multivariate Lazy Multivariate Tower AD for Weil Algebra Computation

“…Our implementation performs particularly well in multivariate cases and gives pleasing improvements both in time and space when applied to Weil algebra computation as described in [5]. For univariate case, however, there is much room for improvements.…”

“…In particular, Sparse presents a steep quadratic growth both in time and space, our proposed method peroforms almost linearly. Total degrees: (0, 1), (2, 0), (2, 1), (2, 2), (3, 2) (4, 2), (3,4), (5,3), (3,6) Total degrees: (0, 0, 1), (1, 0, 1), (0, 1, 2), (1, 2, 1), (0, 3, 2), (2, 2, 2) (3, 2, 2), (3, 4, 1), (5, 3, 1), (2,3,5) Total degrees: (0, 1), (2, 0), (2, 1), (2, 2), (3, 2) (4, 2), (3,4), (5,3), (3,6), (6,4) Allocation (bytes) sin xe y Total degrees: (0, 0, 1), (1, 0, 1), (0, 1, 2), (1, 2, 1), (0, 3, 2), (2, 2, 2) (3, 2, 2), (3, 4, 1), (5,3,1), (2,3,5), (5,4,2), (3,4,5) Allocation (bytes) sin xe y 2 +z Sparse diffs our method In summary, our method performs really well both in terms of time and space in the multivariate case, although we need more optimisation in univariate cases.…”

“…In this subsection, we compare the performances of Tower AD applied to Weil algebra computation as described in [5].…”

“…Automatic Differentiation (AD) is known as a powerful technique to compute differential coefficients of a given (piecewise) smooth function efficiently and accurately. In the upcoming paper [5], the author proposed to use C ∞ -rings and Weil algebras to provide a modular and exprresive framework for forward-mode automatic difrerentiation. There, compute the C ∞ -structure of an arbitrary Weil algebra as a quotient of that of the formal power series ring R X .…”

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We propose a functional implementation of Multivariate Tower Automatic Differentiation. Our implementation is intended to be used in implementing C ∞ -structure computation of an arbitrary Weil algebra, which we discussed in [5].

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