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Computational methods for fractional differential equations exhibit essential instability. Even a minor modification of the coefficients or other entry data may switch good results to the divergent. The goal of this paper is to suggest the…
Direct collocation methods are widely used numerical techniques for solving optimal control problems. The discretization of continuous-time optimal control problems transforms them into large-scale nonlinear programming problems, which…
We are interested in high-order linear multistep schemes for time discretization of adjoint equations arising within optimal control problems. First we consider optimal control problems for ordinary differential equations and show loss of…
An adaptive direct collocation method is developed for solving optimal control problems constrained by parabolic partial differential equations. The partial differential equation is first reformulated in a variational setting, where the…
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…
We consider large linear and nonlinear fixed point problems, and solution with proximal algorithms. We show that there is a close connection between two seemingly different types of methods from distinct fields: 1) Proximal iterations for…
In dual decomposition, the dual to an optimization problem with a specific structure is solved in distributed fashion using (sub)gradient and recently also fast gradient methods. The traditional dual decomposition suffers from two main…
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…
To reduce complexity and achieve scalable performance in high-dimensional black-box settings, we propose a distributed method for nonconvex derivative-free optimization of continuous variables with an additively separable objective, subject…
In this paper, we introduce a novel category of central compact schemes inspired by existing cell-node and cell-centered compact finite difference schemes, that offer a superior spectral resolution for solving the dispersive wave equation.…
This contribution proposes a new formulation to efficiently compute directional derivatives of order one to fourth. The formulation is based on automatic differentiation implemented with dual numbers. Directional derivatives are particular…
Direct multiple shooting (DMS) and direct collocation (DC) are two common transcription methods for solving optimal control problems (OCP) in biomechanics and robotics. They have rarely been compared in terms of solution and speed. Through…
An adaptive mesh refinement and error estimation method for numerically solving optimal control problems is developed using Legendre-Gauss-Radau direct collocation. In regions of the solution where the desired accuracy tolerance has not…
Direct collocation is a widely used method for solving dynamic optimization problems (DOPs), but its implementation simplicity and computational efficiency are limited for challenging problems like those involving singular arcs. In this…
We propose a fast and scalable optimization method to solve chance or probabilistic constrained optimization problems governed by partial differential equations (PDEs) with high-dimensional random parameters. To address the critical…
Derivative-based algorithms are ubiquitous in statistics, machine learning, and applied mathematics. Automatic differentiation offers an algorithmic way to efficiently evaluate these derivatives from computer programs that execute relevant…
Estimation of density derivatives is a versatile tool in statistical data analysis. A naive approach is to first estimate the density and then compute its derivative. However, such a two-step approach does not work well because a good…
Optimal prediction methods compensate for a lack of resolution in the numerical solution of time-dependent differential equations through the use of prior statistical information. We present a new derivation of the basic methodology, show…
This study focuses on the numerical discretization methods for the continuous-time discounted linear-quadratic optimal control problem (LQ-OCP) with time delays. By assuming piecewise constant inputs, we formulate the discrete system…
This chapter presents some numerical methods to solve problems in the fractional calculus of variations and fractional optimal control. Although there are plenty of methods available in the literature, we concentrate mainly on approximating…