Analysing diffusion and flow-driven instability using semidefinite programming
Analysing diffusion and flow-driven instability using semidefinite programming
复制标题
使用半定规划分析扩散和流动驱动的不稳定性
DOI：
10.1098/rsif.2018.0586
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发表时间：
2019
影响因子：
3.9
通讯作者：
Hiroki Miyazako
中科院分区：
文献类型：
--
作者：
Yutaka Hori;Hiroki Miyazako
扩散和流动驱动的不稳定，或传输驱动的不稳定，是在空间分布的化学系统中产生不均匀浓度梯度的核心机制之一。然而，要验证反应-扩散-平流系统的输运驱动不稳定性，需要检查无穷多个傅立叶模的雅可比本征值，这在计算上是困难的。为了克服这一局限性，本文提出了通过有限步代数计算来确定反应扩散平流系统稳定性/不稳定性的数学优化算法。具体地说，傅里叶模式的稳定性/不稳定性分析被表示为平方和优化程序，这是一类凸优化，其求解程序以软件包的形式广泛提供。进一步扩展了优化程序，方便了失稳空间模式的计算。这种扩展允许在不模拟控制方程的情况下预测和设计浓度梯度的形状。用一个简单的考虑扩散和平流的反应模型演示了自组织花样形成的简化分析过程。
Diffusion and flow-driven instability, or transport-driven instability, is one of the central mechanisms to generate inhomogeneous gradient of concentrations in spatially distributed chemical systems. However, verifying the transport-driven instability of reaction–diffusion–advection systems requires checking the Jacobian eigenvalues of infinitely many Fourier modes, which is computationally intractable. To overcome this limitation, this paper proposes mathematical optimization algorithms that determine the stability/instability of reaction–diffusion–advection systems by finite steps of algebraic calculations. Specifically, the stability/instability analysis of Fourier modes is formulated as a sum-of-squares optimization program, which is a class of convex optimization whose solvers are widely available as software packages. The optimization program is further extended for facile computation of the destabilizing spatial modes. This extension allows for predicting and designing the shape of the concentration gradient without simulating the governing equations. The streamlined analysis process of self-organized pattern formation is demonstrated with a simple illustrative reaction model with diffusion and advection.