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首页> 外文期刊>SIAM journal on applied dynamical systems >Optimized Mixing by Cutting-and-Shuffling
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Optimized Mixing by Cutting-and-Shuffling

机译：通过切割和洗牌优化混合

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摘要

Mixing by cutting-and-shuffling can be understood and predicted using dynamical systems based tools and techniques. In existing studies, mixing is generated by maps that repeat the same cut-and-shuffle process at every iteration in a "fixed" manner. However, mixing can be greatly improved by varying the cut-and-shuffle parameters at each step using a "variable" approach. To demonstrate this approach, we show how to optimize mixing by cutting-and-shuffling on the one-dimensional line interval, known as an interval exchange transformation (IET). Mixing can be significantly improved by optimizing variable protocols, especially for initial conditions more complex than just a simple two-color line interval. While we show that optimal variable IETs can be found analytically for arbitrary numbers of iterations, for more complex cutting-and-shuffling systems, computationally expensive numerical optimization methods are required. Furthermore, the number of control parameters grows linearly with the number of iterations in variable systems. Therefore, optimizing over large numbers of iterations is generally computationally prohibitive. We demonstrate an ad hoc approach to cutting-and-shuffling that is computationally inexpensive and guarantees that the mixing metric is within a constant factor of the optimum. This ad hoc approach yields significantly better mixing than fixed IETs, which are known to produce weak-mixing, because cut pieces never reconnect. The heuristic principles of this method can be applied to more general cutting-and-shuffling systems.
机译：可以使用基于动态系统的工具和技术来理解和预测通过切割和洗牌混合。在现有研究中，通过以“固定”方式在每次迭代中重复相同切割和换动过程的映射产生混合。然而，通过使用“变量”方法在每个步骤中改变切割和加油参数，可以大大提高混合。为了展示这种方法，我们展示了如何通过在一维线间隔上切割和洗牌来优化混合，称为间隔交换变换（IET）。通过优化可变协议，可以显着改善混合，特别是对于初始条件比仅仅是简单的双色线间隔更复杂。虽然我们显示可以在分析上发现最佳变量IET，但对于任意数量的迭代，对于更复杂的切割和混洗系统，需要计算昂贵的数值优化方法。此外，控制参数的数量随着可变系统中的迭代次数而导致的。因此，大量迭代优化通常是在计算上的。我们展示了对切割和洗牌的临时方法，该方法是计算地廉价的，并且保证混合度量在最佳的恒定因子范围内。这种临时方法产生明显更好的混合，而不是固定的IET，这已知产生弱混合，因为切割件永远不会重新连接。这种方法的启发式原理可以应用于更一般的切割和混洗系统。

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