用半定规划分析扩散和流动驱动不稳定性
Analyzing Diffusion and Flow-driven Instability using Semidefinite Programming
摘要：
扩散和流动驱动的不稳定性，或运输驱动的不稳定性，是在空间分布的化学系统中产生非均匀浓度梯度的中心机制之一。然而，验证传输驱动的反应扩散平流系统的不稳定性，需要检查的雅可比特征值的无限多个傅立叶模式，这是计算上棘手的。为了克服这一局限性，本文提出了数学优化算法，确定反应扩散平流系统的稳定性/不稳定性的代数计算的有限步骤。具体而言，傅立叶模式的稳定性/不稳定性分析被配制为平方和（SOS）优化程序，这是一类凸优化，其求解器作为软件包被广泛使用。优化程序被进一步扩展，以便于计算不稳定的空间模式。这种扩展允许预测和设计浓度梯度的形状，而无需模拟控制方程。自组织模式形成的流线型分析过程演示了一个简单的说明性的反应模型与扩散和平流。
英文摘要：
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 (SOS) 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 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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
