Local linear convergence analysis of Primal–Dual splitting methods

In this paper, we study the local linear convergence properties of a versatile class of Primal–Dual splitting methods for minimizing composite non-smooth convex optimization problems. Under the assumption that the non-smooth components of the problem are partly smooth relative to smooth manifolds, we present a unified local convergence analysis framework for these methods. More precisely, in our framework, we first show that (i) the sequences generated by Primal–Dual splitting methods identify a pair of primal and dual smooth manifolds in a finite number of iterations, and then (ii) enter a local linear convergence regime, which is characterized based on the structure of the underlying active smooth manifolds. We also show how our results for Primal–Dual splitting can be specialized to cover existing ones on Forward–Backward splitting and Douglas–Rachford splitting/ADMM (alternating direction methods of multipliers). Moreover, based on these obtained local convergence analysis result, several practical acceleration techniques are discussed. To exemplify the usefulness of the obtained result, we consider several concrete numerical experiments arising from fields including signal/image processing, inverse problems and machine learning. The demonstration not only verifies the local linear convergence behaviour of Primal–Dual splitting methods, but also the insights on how to accelerate them in practice.     

Modelling pH-Dependent and Microstructure-Dependent Streaming Potential Coefficient and Zeta Potential of Porous Sandstones

In this paper, we have significantly modified an existing model for calculating the zeta potential and streaming potential coefficient of porous media and tested it with a large, recently published, high-quality experimental dataset. The newly modified model does not require the imposition of a zeta potential offset but derives its high salinity zeta potential behaviour from Stern plane saturation considerations. The newly modified model has been implemented as a function of temperature, salinity, pH, and rock microstructure both for facies-specific aggregations of the new data and for individual samples. Since the experimental data include measurements on samples of both detrital and authigenic overgrowth sandstones, it was possible to model and test the effect of widely varying microstructural properties while keeping lithology constant. The results show that the theoretical model represents the experimental data very well when applied to model data for a particular lithofacies over the whole salinity, from 10^?5 to 6.3 mol/dm³, and extremely well when modelling individual samples and taking individual sample microstructure into account. The new model reproduces and explains the extreme sensitivity of zeta and streaming potential coefficient to pore fluid pH. The low salinity control of streaming potential coefficient by rock microstructure is described well by the modified model. The model also behaves at high salinities, showing that the constant zeta potential observed at high salinities arises from the development of a maximum charge density in the diffuse layer as it is compressed to the thickness of one hydrated metal ion.