Some Runge-Kutta Algorithms with Error Control and Variable Stepsizes for Solving a Class of Differential-Algebraic Equations

In this paper, we propose and discuss numerical algorithms for solving a class of nonlinear differential-algebraic equations (DAEs). These algorithms are based on half-explicit Runge-Kutta methods (HERK) that have been studied recently for solving strangeness-free DAEs. The main idea of this work is to use the half-explicit variants of some well-known embedded Runge-Kutta methods such as Runge-Kutta-Fehlberg and Dormand-Prince pairs. Thus, we can estimate local errors and choose suitable stepsizes accordingly to a given tolerance. The cases of unstructured and structured DAEs are investigated and compared. Finally, some numerical experiences are given for illustrating the efficiency of the algorithms.

A note on the accelerated proximal gradient method for nonconvex optimization

We improve a recent accelerated proximal gradient (APG) method in [Li, Q., Zhou, Y., Liang, Y. and Varshney, P. K., Convergence analysis of proximal gradient with momentum for nonconvex optimization, in Proceedings of the 34th International Conference on Machine Learning, Sydney, Australia, PMLR 70, 2017] for nonconvex optimization by allowing variable stepsizes. We prove the convergence of the APG method for a composite nonconvex optimization problem under the assumption that the composite objective function satisfies the Kurdyka-Łojasiewicz property.

An Efficient Dynamic Formulation for Solving Rigid and Flexible Multibody Systems Based on Semirecursive Method and Implicit Integration

This paper presents a method for solving the dynamic equations of multibody systems containing both rigid and flexible bodies. The proposed method uses independent coordinates and projects the dynamic equations on the constraint tangent manifold by means of a velocity transformation matrix. It can be used with a wide variety of integration formulae, considering both fixed and variable stepsizes. Topological semirecursive methods are used to take advantage of the relatively small number of parameters needed. An in depth implementation analysis is performed in order to evaluate the terms involved in the integration process. Numerical and stability issues are also discussed.