Accelerated mirror descent in continuous and discrete time

We study accelerated mirror descent dynamics in continuous and discrete time. Combining the original continuous-time motivation of mirror descent with a recent ODE interpretation of Nesterov's accelerated method, we propose a family of continuous-time descent dynamics for convex functions with Lipschitz gradients, such that the solution trajectories converge to the optimum at a O(1/t2) rate. We then show that a large family of first-order accelerated methods can be obtained as a discretization of the ODE, and these methods converge at a O(1/k2) rate. This connection between accelerated mirror descent and the ODE provides an intuitive approach to the design and analysis of accelerated first-order algorithms

The Phase–Space Approach to time Evolution of Quantum States in Confined Systems: the Spectral Split–Operator Method

Using the phase space approach, we consider the quantum dynamics of a wave packet in an isolated confined system with three different potential energy profiles. We solve the Moyal equation of motion for the Wigner function with the highly efficient spectral split-operator method. The main aim of this study is to compare the accuracy of the employed algorithm through analysis of the total energy expectation value, in terms of deviation from its exact value. This comparison is performed for the second and fourth order factorizations of the time evolution operator