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In its basic form, the reverse mode of computational differentiation yields the gradient of a scalar-valued function at a cost that is a small multiple of the computational work needed to evaluate the function itself. However, the corresponding memory requirement is proportional to the run-time of the evaluation program. Therefore, the practical applicability of the reverse mode in its original formulation is limited despite the availability of ever larger memory systems. This observation leads to the development of checkpointing schedules to reduce the storage requirements. This article presents the function revolve, which generates checkpointing schedules that are provably optimal with regard to a primary and a secondary criterion. This routine is intended to be used as an explicit "controller" for running a time-dependent applications program.

Abstract. This article considers the problem of evaluating all pure and mixed partial derivatives of some vector function defined by an evaluation procedure. The natural approach to evaluating derivative tensors might appear to be their recursive calculation in the usual forward mode of computational differentiation. However, with the approach presented in this article, much simpler data access patterns and similar or lower computational counts can be achieved through propagating a family of univariate Taylor series of a suitable degree. It is applicable for arbitrary orders of derivatives. Also it is possible to calculate derivatives only in some directions instead of the full derivative tensor. Explicit formulas for all tensor entries as well as estimates for the corresponding computational complexities are given.

Automatic, or algorithmic, differentiation (AD) is a chain rule‐based technique for evaluating derivatives of functions given as computer programs for their elimination. We review the main characteristics and application of AD and illustrate the methodology on a simple example.

This paper considers the numerical solution of optimal control problems based on ODEs. We assume that an explicit Runge-Kutta method is applied to integrate the state equation in the context of a recursive discretization approach. To compute the gradient of the cost function, one may employ Automatic Differentiation (AD). This paper presents the integration schemes that are automatically generated when differentiating the discretization of the state equation using AD. We show that they can be seen as discretization methods for the sensitivity and adjoint differential equation of the underlying control problem. Furthermore, we prove that the convergence rate of the scheme automatically derived for the sensitivity equation coincides with the convergence rate of the integration scheme for the state equation. Under mild additional assumptions on the coefficients of the integration scheme for the state equation, we show a similar result for the scheme automatically derived for the adjoint equation. Numerical results illustrate the presented theoretical results.

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T he computation of a sparse Hessian matrix H using automatic differentiation (AD) can be made efficient using the following four-step procedure: (1) Determine the sparsity structure of H, (2) obtain a seed matrix S that defines a column partition of H using a specialized coloring on the adjacency graph of H, (3) compute the compressed Hessian matrix B ≡ HS, and (4) recover the numerical values of the entries of H from B.The coloring variant used in the second step depends on whether the recovery in the fourth step is direct or indirect: a direct method uses star coloring and an indirect method uses acyclic coloring. In an earlier work, we had designed and implemented effective heuristic algorithms for these two NP-hard coloring problems. Recently, we integrated part of the developed software with the AD tool ADOL-C, which has recently acquired a sparsity detection capability. In this paper, we provide a detailed description and analysis of the recovery algorithms and experimentally demonstrate the efficacy of the coloring techniques in the overall process of computing the Hessian of a given function using ADOL-C as an example of an AD tool. We also present new analytical results on star and acyclic coloring of chordal graphs. The experimental results show that sparsity exploitation via coloring yields enormous savings in runtime and makes the computation of Hessians of very large size feasible. The results also show that evaluating a Hessian via an indirect method is often faster than a direct evaluation. This speedup is achieved without compromising numerical accuracy.

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