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In this paper, we focus on the primal-dual hybrid gradient (PDHG) method, which is being widely used to solve a broad spectrum of saddle-point problems. Despite of its wide applications in different areas, the study of inexact versions of PDHG still seems to be in its infancy. We investigate how to design implementable inexactness criteria for solving the subproblems in PDHG scheme so that the convergence of an inexact PDHG can be guaranteed. We propose two specific inexactness criteria and accordingly some inexact PDHG methods for saddle-point problems. The convergence of both inexact PDHG methods is rigorously proved, and their convergence rates are estimated under different scenarios. Moreover, some numerical results on image restoration problems are reported to illustrate the efficiency of the proposed methods.

We discuss systematic extensions of the standard (Störmer–Verlet) method for integrating the differential equations of Hamiltonian mechanics. Our extensions preserve the symplectic geometry exactly, as well as all Nöether conservation laws caused by joint symmetries of the kinetic and potential energies (like angular momentum in rotation invariant systems). These extensions increase the accuracy of the integrator, which for the Störmer–Verlet method is of order τ² for a timestep of length τ, to higher-orders in τ. The schemes presented have, in contrast to most previous proposals, all intermediate timesteps real and positive. The schemes increase the relative accuracy to order τ^N (for N = 4, 6 and 8) for a large class of Hamiltonian systems.

A consistent BGK-type approach to reacting gas mixtures, according to Boltzmann-like kinetics for a bimolecular reversible chemical reaction has been recently introduced. In this paper we apply a numerical strategy based on time splitting techniques to simulate the reactive BGK equations. These techniques have the advantage of simplifying the problem by treating separately the convection step and collision step. Numerical results of the time-dependent Riemann problem for the reactive BGK system are presented.