Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2502.10285

Evaluation of the Efficiency and Comparison of Different Numerical Differentiation Methods on Three Case Studies

Hamidreza Moradi·Hamideh Hossei
View original source ↗Read source document ↗Export metadata ↓

Abstract

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 nonlinear control systems, and the assessment of noisy time signals, among others. In this study, we have chosen three illustrative cases: the logistic model for population dynamics, temperature variation within buildings, and the determination of market equilibrium prices. The primary objective is to assess the effectiveness of various numerical differentiation techniques and conduct a comparative analysis of the outcomes for each of these case studies. To achieve this objective, we employed three distinct numerical differentiation techniques: The Forward, Backward, and Centered FiniteDifference methods, each executed in two different levels of precision, totaling six variations. Our findings clearly indicate that, for the initial case study, the methods characterized by lower computational costs (specifically, the Forward and Backward Finite-Difference methods) yield superior outcomes. In contrast, for the second case study, the Centered Finite-Difference method delivers better results. In the case of the third case study, our results reveal that none of the methods produce estimations that meet acceptable standards. It is noteworthy that the empirical equations for each case study have been validated against previous literature.

Explore related subjects

math.OCmath.AGmath.DG

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Hamidreza Moradi, Hamideh Hossei. 2025-07-11. Evaluation of the Efficiency and Comparison of Different Numerical Differentiation Methods on Three Case Studies. https://arxiv.org/abs/2502.10285

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Dynamic Programming-Compatible Uncertainty Sets in Robust Markov Decision Processes

In this paper, we investigate the compatibility of robust Markov Decision Processes (RMDPs) with dynamic programming under various assumptions on the uncertainty set, i.e., we investigate when one can solve an RMDP by solving a fixed point equation. We show that in all generality, s-rectangular and sa-rectangular uncertainty sets are the only models of uncertainty that are compatible with dynamic programming. Our analysis shows that existing non-rectangular models, including r-rectangularity, are only weakly compatible with dynamic programming, as they require the assumption that rewards do not depend on the next state. In this case, our results imply that one can always construct a rectangular uncertainty set that is equivalent, for both policy evaluation and optimization, to the dynamic programming-compatible non-rectangular model. This highlights a key limitation: dynamic-programming-compatible non-rectangular uncertainty sets, although practically relevant for uncertainty quantification, do not provide a genuinely distinct assessment of policy performance. Interestingly, our proof techniques rely on identifying a novel simultaneous solvability property, which we show is central to several important properties of RMDPs, including the existence of stationary optimal policies and dynamic programming-based formulations. The simultaneous solvability property enables a unified approach to studying all existing models of uncertainty, rectangular and non-rectangular alike.

math.OC↗
arXiv2026

A simple and practical adaptive trust-region method

We present an adaptive trust-region method for unconstrained optimization that allows inexact solutions to the trust-region subproblems. Our method is a simple variant of the classical trust-region method of Ssorensen. The method achieves the best possible convergence bound up to an additive logarithmic term for finding an $ε$-approximate stationary point, i.e., $O( Δ_f L^{1/2} ε^{-3/2}) + \tilde{O}(1)$ iterations, where $L$ is the Lipschitz constant of the Hessian, $Δ_f$ is the optimality gap, and $ε$ is the termination tolerance for the gradient norm. This improves over existing trust-region methods whose worst-case bound is at least a factor of $L$ worse. We compare our performance with state-of-the-art trust-region (TRU) and cubic regularization (ARC) methods from the GALAHAD library on the CUTEst benchmark problems with at least 100 variables. We also compare with the recently developed Universal trust-region (UTR) method, using the same subproblem solver. In terms of shifted geometric mean of wall-clock times our method's is between $1.2\times$ and $2\times$ faster. We report similar improvements for number of function evaluations, factorizations, gradient evaluations and Hessian evaluations. Compared to the conference version of this paper, our revised method includes several practical enhancements. These modifications dramatically improved performance, including almost an order of magnitude reduction in the shifted geometric mean of wall-clock times. We also show that it suffices for the function to be continuously twice-differentiable to guarantee that either the minimum gradient norm converges to zero or the objective value tends towards negative infinity, even when the iterates diverge.

math.OC↗
arXiv2026

Rough Stochastic Pontryagin Maximum Principle and an Indirect Shooting Method

We derive first-order Pontryagin optimality conditions for stochastic optimal control with deterministic controls for systems modeled by rough differential equations (RDE) driven by Gaussian rough paths. This Pontryagin Maximum Principle (PMP) applies to systems following stochastic differential equations (SDE) driven by Brownian motion, yet it does not rely on forward-backward SDEs and involves the same Hamiltonian as the deterministic PMP. The proof consists of first deriving various integrable error bounds for solutions to nonlinear and linear RDEs by leveraging recent results on Gaussian rough paths. The PMP then follows using standard techniques based on needle-like variations. As an application, we propose the first indirect shooting method for nonlinear stochastic optimal control and show that it converges 10x faster than a direct method on a stabilization task.

math.OC↗