A new projection method for finding the closest point in the intersection of convex sets

In this paper we present a new iterative projection method for finding the closest point in the intersection of convex sets to any arbitrary point in a Hilbert space. This method, termed AAMR for averaged alternating modified reflections, can be viewed as an adequate modification of the Douglas--Rachford method that yields a solution to the best approximation problem. Under a constraint qualification at the point of interest, we show strong convergence of the method. In fact, the so-called strong CHIP fully characterizes the convergence of the AAMR method for every point in the space. We report some promising numerical experiments where we compare the performance of AAMR against other projection methods for finding the closest point in the intersection of pairs of finite dimensional subspaces

Enhanced CO2 Methanation by New Microstructured Reactor Concept and Design

A novel multichannel minireactor with channel internal diameter in the range of millimeters has been conceptually designed and constructed. Its configuration confers to the new concept reactor some features and advantages compared to conventional fixed-bed reactors g.e. better mass-transfer, avoid hot-spots formation, increased performance (10–20%). Consequently, this new reactor concept is ideal to be applied to exothermic reactions such as the Sabatier reaction that demands continuous removal of the heat produced to avoid hot-spot formation and the sintering of the catalyst. Thanks to its configuration, this reactor could control effectively the heat generated by the reaction and several tests were carried out to validate the reactor features. The results obtained demonstrate that the catalysts activity in the reaction is improved with the application of the novel reactor respect the conventional fixed- and fluidized-bed ones and neither catalyst sintering, nor pressure drop was appreciated during the catalytic tests. At the best reaction conditions, Tecnalia’s multichannel minireactor can handle 820 Nm3/h of CO2 per square meter of channel section showing the enormous potential of the new reactor concept.The authors gratefully acknowledge the Spanish Ministry of Economy and Competitiveness (MINECO) funding under the project RENOVAGAS (RTC-2014-2975-3). Likewise, the authors want to acknowledge the valuable contribution of other colleagues from other entities participating in the project

Global Behavior of the Douglas-Rachford Method for a Nonconvex Feasibility Problem

In recent times the Douglas-Rachford algorithm has been observed empirically to solve a variety of nonconvex feasibility problems including those of a combinatorial nature. For many of these problems current theory is not sufficient to explain this observed success and is mainly concerned with questions of local convergence. In this paper we analyze global behavior of the method for finding a point in the intersection of a half-space and a potentially non-convex set which is assumed to satisfy a well-quasi-ordering property or a property weaker than compactness. In particular, the special case in which the second set is finite is covered by our framework and provides a prototypical setting for combinatorial optimization problems

A feasibility approach for constructing combinatorial designs of circulant type

In this work, we propose an optimization approach for constructing various classes of circulant combinatorial designs that can be defined in terms of autocorrelations. The problem is formulated as a so-called feasibility problem having three sets, to which the Douglas-Rachford projection algorithm is applied. The approach is illustrated on three different classes of circulant combinatorial designs: circulant weighing matrices, D-optimal matrices, and Hadamard matrices with two circulant cores. Furthermore, we explicitly construct two new circulant weighing matrices, a $CW(126,64)$ and a $CW(198,100)$, whose existence was previously marked as unresolved in the most recent version of Strassler's table

How to generate pentagonal symmetry using Turing systems

We explore numerically the formation of Turing patterns in a confined circular domain with small aspect ratio. Our results show that stable fivefold patterns are formed over a well defined range of disk sizes, offering a possible mechanism for inducing the fivefold symmetry observed in early development of regular echinoids. Using this pattern as a seed, more complex biological structures can be mimicked, such as the pigmentation pattern of sea urchins and the plate arrangements of the calyxes of primitive camerate crinoids