The double layer structure of two ionic liquids (ILs), 1-butyl-1-methylpyrrolidinium tris(pentafluoroethyl)trifluorophosphate ([Py1,4]FAP) and 1-ethyl-3-methylimidazolium tris(pentafluoroethyl)trifluorophosphate ([EMIm]FAP) at the polarized Au(111) electrode interface is probed using Atomic Force Microscopy force measurements. The force-separation profiles suggest a multilayered morphology is present at the electrified Au(111)−IL interface, with more near surface layers detected at higher potentials. At the (slightly negative) open circuit potential, multiple ion layers are present, and the innermost layer, in contact with the Au(111) surface, is enriched in the cation due to electrostatic adsorption. Upon applying negative electrode potentials (−1.0 V, −2.0 V), stronger IL near surface structure is detected: both the number of ion layers and the force required to rupture these layers increases. Positive electrode potentials (+1.0 V, +2.0 V) also enhance IL near surface structure, but not as much as negative potentials, because surface-adsorbed anions are less effective at templating structure in subsequent layers than cations. This interfacial structure is not consistent with a double layer in the Stern−Gouy−Chapman sense, as there is no diffuse layer. The structure is consistent with a capicitative double-layer model, with a very small separation distance between the planes of charge.

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We propose a general alternating minimization algorithm for nonconvex optimization problems with separable structure and nonconvex coupling between blocks of variables. To fix our ideas, we apply the methodology to the problem of blind ptychographic imaging. Compared to other schemes in the literature, our approach differs in two ways: (i) it is posed within a clear mathematical framework with practically verifiable assumptions, and (ii) under the given assumptions, it is provably convergent to critical points. A numerical comparison of our proposed algorithm with the current state-of-the-art on simulated and experimental data validates our approach and points toward directions for further improvement.

In this paper we present two Douglas-Rachford inspired iteration schemes which can be applied directly to N-set convex feasibility problems in Hilbert space. Our main results are weak convergence of the methods to a point whose nearest point projections onto each of the N sets coincide. For affine subspaces, convergence is in norm. Initial results from numerical experiments, comparing our methods to the classical (product-space) DouglasRachford scheme, are promising.

In this paper, we establish sublinear and linear convergence of fixed point iterations generated by averaged operators in a Hilbert space. Our results are achieved under a bounded Hölder regularity assumption which generalizes the well-known notion of bounded linear regularity. As an application of our results, we provide a convergence rate analysis for many important iterative methods in solving broad mathematical problems such as convex feasibility problems and variational inequality problems. These include Krasnoselskii-Mann iterations, the cyclic projection algorithm, forward-backward splitting and the Douglas-Rachford feasibility algorithm along with some variants. In the important case in which the underlying sets are convex sets described by convex polynomials in a finite dimensional space, we show that the Hölder regularity properties are automatically satisfied, from which sublinear convergence follows.

In this work, we propose a simple modification of the forward-backward splitting method for finding a zero in the sum of two monotone operators. Our method converges under the same assumptions as Tseng's forward-backward-forward method, namely, it does not require cocoercivity of the single-valued operator. Moreover, each iteration only uses one forward evaluation rather than two as is the case for Tseng's method. Variants of the method incorporating a linesearch, relaxation and inertia, or a structured three operator inclusion are also discussed.

In this work, we propose a new algorithm for finding a zero in the sum of two monotone operators where one is assumed to be single-valued and Lipschitz continuous. This algorithm naturally arises from a non-standard discretization of a continuous dynamical system associated with the Douglas-Rachford splitting algorithm. More precisely, it is obtained by performing an explicit, rather than implicit, discretization with respect to one of the operators involved. Each iteration of the proposed algorithm requires the evaluation of one forward and one backward operator.Keywords. monotone operator · operator splitting · Douglas-Rachford algorithm · dynamical systems MSC2010. 49M29, 90C25, 47H05, 47J20, 65K15

In this work, we study fixed point algorithms for finding a zero in the sum of n ≥ 2 maximally monotone operators by using their resolvents. More precisely, we consider the class of such algorithms where each resolvent is evaluated only once per iteration. For any algorithm from this class, we show that the underlying fixed point operator is necessarily defined on a d-fold Cartesian product space with d ≥ n − 1. Further, we show that this bound is unimprovable by providing a family of examples for which d = n−1 is attained. This family includes the Douglas-Rachford algorithm as the special case when n = 2. Applications of the new family of algorithms in distributed decentralised optimisation and multi-block extensions of the alternation direction method of multipliers (ADMM) are discussed.