We propose a new selection rule for the coordinate selection in coordinate descent methods for huge-scale optimization. The efficiency of this novel scheme is provably better than the efficiency of uniformly random selection, and can reach the efficiency of steepest coordinate descent (SCD), enabling an acceleration of a factor of up to n, the number of coordinates. In many practical applicatio...

Central to the theoretical description of a chemical reaction is the reaction pathway. The intrinsic reaction coordinate is defined as the steepest descent path in mass weighted Cartesian coordinates that connects the transition state to reactants and products. In this work, a new integrator for the steepest descent pathway is presented. This method is a Hessian based predictor-corrector algori...

In recent years, there has been a series of results of Fokas and collaborators on boundary value problems for soliton equations (see [3] for a comprehensive review). The method of Fokas in [3] goes beyond existence and uniqueness. In fact, it reduces these problems to Riemann-Hilbert factorisation problems in the complex plane, thus generalising the existing theory which reduces initial value p...

Steepest descent drives both theory and practice of nonsmooth optimization. We study slight relaxations of two influential notions of steepest descent curves — curves of maximal slope and solutions to evolution equations. In particular, we provide a simple proof showing that lower-semicontinuous functions that are locally Lipschitz continuous on their domains — functions playing a central role ...

We have derived the electromagnetic fields for dipole antennas embedded in a two-layer dissipative medium by using the following techniques: (1) the brute-force numerical-integration method, (2) the modal approach by finding the residues for the integrals, (3) the image-source approach in conjunction with the steepest descent method, (4) and the hybrid technique with the combination of the two ...