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1.
Sci Rep ; 10(1): 17336, 2020 Oct 15.
Artigo em Inglês | MEDLINE | ID: mdl-33060751

RESUMO

The determination of the parameters of cylindrical optical waveguides, e.g. the diameters [Formula: see text] of r layers of (semi-) transparent optical fibres, can be executed by inverse evaluation of the scattering intensities that emerge under monochromatic illumination. The inverse problem can be solved by optimising the mismatch [Formula: see text] between the measured and simulated scattering patterns. The global optimum corresponds to the correct parameter values. The mismatch [Formula: see text] can be seen as an energy landscape as a function of the diameters. In this work, we study the structure of the energy landscape for different values of the complex refractive indices [Formula: see text], for [Formula: see text] and [Formula: see text] layers. We find that for both values of r, depending on the values of [Formula: see text], two very different types of energy landscapes exist, respectively. One type is dominated by one global minimum and the other type exhibits a multitude of local minima. From an algorithmic viewpoint, this corresponds to easy and hard phases, respectively. Our results indicate that the two phases are separated by sharp phase-transition lines and that the shape of these lines can be described by one "critical" exponent b, which depends slightly on r. Interestingly, the same exponent also describes the dependence of the number of local minima on the diameters. Thus, our findings are comparable to previous theoretical studies on easy-hard transitions in basic combinatorial optimisation or decision problems like Travelling Salesperson and Satisfiability. To our knowledge our results are the first indicating the existence of easy-hard transitions for a real-world optimisation problem of technological relevance.

2.
Phys Rev E ; 94(5-1): 052120, 2016 Nov.
Artigo em Inglês | MEDLINE | ID: mdl-27967062

RESUMO

We study the polygons governing the convex hull of a point set created by the steps of n independent two-dimensional random walkers. Each such walk consists of T discrete time steps, where x and y increments are independent and identically distributed Gaussian. We analyze area A and perimeter L of the convex hulls. We obtain probability densities for these two quantities over a large range of the support by using a large-deviation approach allowing us to study densities below 10^{-900}. We find that the densities exhibit in the limit T→∞ a time-independent scaling behavior as a function of A/T and L/sqrt[T], respectively. As in the case of one walker (n=1), the densities follow Gaussian distributions for L and sqrt[A], respectively. We also obtained the rate functions for the area and perimeter, rescaled with the scaling behavior of their maximum possible values, and found limiting functions for T→∞, revealing that the densities follow the large-deviation principle. These rate functions can be described by a power law for n→∞ as found in the n=1 case. We also investigated the behavior of the averages as a function of the number of walks n and found good agreement with the predicted behavior.

3.
Artigo em Inglês | MEDLINE | ID: mdl-26066116

RESUMO

We study the convex hull of the set of points visited by a two-dimensional random walker of T discrete time steps. Two natural observables that characterize the convex hull in two dimensions are its perimeter L and area A. While the mean perimeter 〈L〉 and the mean area 〈A〉 have been studied before, analytically and numerically, and exact results are known for large T (Brownian motion limit), little is known about the full distributions P(A) and P(L). In this paper we provide numerical results for these distributions. We use a sophisticated large-deviation approach that allows us to study the distributions over a larger range of the support, where the probabilities P(A) and P(L) are as small as 10(-300). We analyze (open) random walks as well as (closed) Brownian bridges on the two-dimensional discrete grid as well as in the two-dimensional plane. The resulting distributions exhibit, for large T, a universal scaling behavior (independent of the details of the jump distributions) as a function of A/T and L/√[T], respectively. We are also able to obtain the rate function, describing rare events at the tails of these distributions, via a numerical extrapolation scheme and find a linear and square dependence as a function of the rescaled perimeter and the rescaled area, respectively.

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