Claussen Gunnar, Hartmann Alexander K
Institut für Physik, Carl von Ossietzky Universität Oldenburg, 26129, Oldenburg, Germany.
Sci Rep. 2020 Oct 15;10(1):17336. doi: 10.1038/s41598-020-74366-5.
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.
圆柱形光波导参数的确定，例如（半）透明光纤r层的直径[公式：见正文]，可以通过对单色照明下出现的散射强度进行反向评估来实现。通过优化测量散射图案与模拟散射图案之间的失配[公式：见正文]，可以解决这个反问题。全局最优值对应于正确的参数值。失配[公式：见正文]可以看作是作为直径函数的能量景观。在这项工作中，我们研究了复折射率[公式：见正文]取不同值时，对于r = 2和r = 3层的能量景观结构。我们发现，对于这两个r值，根据[公式：见正文]的值，分别存在两种非常不同类型的能量景观。一种类型由一个全局最小值主导，另一种类型则表现出大量的局部最小值。从算法的角度来看，这分别对应于容易阶段和困难阶段。我们的结果表明，这两个阶段由尖锐的相变线分隔开，并且这些线的形状可以用一个“临界”指数b来描述，该指数略微依赖于r。有趣的是，相同的指数也描述了局部最小值的数量对直径的依赖性。因此，我们的发现与先前关于基本组合优化或旅行商问题和可满足性等决策问题中容易 - 困难转变的理论研究相当。据我们所知，我们的结果首次表明对于具有技术相关性的实际优化问题存在容易 - 困难转变。
