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Recent theoretical work on automatic differentiation (autodiff) has focused on characteristics such as correctness and efficiency while assuming that all derivatives are automatically generated by autodiff using program transformation, with…
We consider Hadamard fractional derivatives and integrals of variable fractional order. A new type of fractional operator, which we call the Hadamard-Marchaud fractional derivative, is also considered. The objective is to represent these…
Fractional calculus generalizes the derivative and antiderivative operations of differential and integral calculus from integer orders to the entire complex plane. Methods are presented for using this generalized calculus with Laplace…
Inspired by Gauss-Newton-like methods, we study the benefit of leveraging the structure of deep learning objectives, namely, the composition of a convex loss function and of a nonlinear network, in order to derive better direction oracles…
Several machine learning applications involve the optimization of higher-order derivatives (e.g., gradients of gradients) during training, which can be expensive in respect to memory and computation even with automatic differentiation. As a…
Diffusive representations of fractional derivatives have proven to be useful tools in the construction of fast and memory efficient numerical methods for solving fractional differential equations. A common challenge in many of the known…
A study is conducted to evaluate four derivative estimation methods when solving a large sparse nonlinear programming problem that arises from the approximation of an optimal control problem using a direct collocation method. In particular,…
A new method that enables easy and convenient discretization of partial differential equations with derivatives of arbitrary real order (so-called fractional derivatives) and delays is presented and illustrated on numerical solution of…
The subdifferential of a function is a generalization for nonsmooth functions of the concept of gradient. It is frequently used in variational analysis, particularly in the context of nonsmooth optimization. The present work proposes…
Two approximations, derived from continuous expansions of Riemann-Liouville fractional derivatives into series involving integer order derivatives, are studied. Using those series, one can formally transform any problem that contains…
We focus here on a class of fourth-order parabolic equations that can be written as a system of second-order equations by introducing an auxiliary variable. We design a novel second-order fully discrete mixed finite element method to…
Many modern numerical methods in computational science and engineering rely on derivatives of mathematical models for the phenomena under investigation. The computation of these derivatives often represents the bottleneck in terms of…
There exist many applications where it is necessary to approximate numerically derivatives of a function which is given by a computer procedure. In particular, all the fields of optimization have a special interest in such a kind of…
Discrete gradient methods are a class of numerical integrators producing solutions with exact preservation of first integrals of ordinary differential equations. In this paper, we apply order theory combined with the symmetrized Itoh--Abe…
New families of fourth-order composition methods for the numerical integration of initial value problems defined by ordinary differential equations are proposed. They are designed when the problem can be separated into three parts in such a…
Without question regarding its pivotal significance, the computation of function derivatives carries substantial weight within a multitude of engineering and applied mathematical fields. These encompass optimization, the development of…
The known Complex Step Derivative (CSD) method allows easy and accurate differentiation up to machine precision of real analytic functions by evaluating them a small imaginary step next to the real number line. The current paper proposes…
Derivatives play a critical role in computational statistics, examples being Bayesian inference using Hamiltonian Monte Carlo sampling and the training of neural networks. Automatic differentiation is a powerful tool to automate the…
A new directional derivative and a new subdifferential for set-valued convex functions are constructed, and a set-valued version of the so-called 'max-formula' is proven. The new concepts are used to characterize solutions of convex…
Expressions for the derivatives with respect to order of modified Bessel functions evaluated at integer orders and certain integral representations of associated Legendre functions with modulus argument greater than unity are used to…