Estimating turbulence and chaos using semidefinite programming

使用半定规划估计湍流和混沌

• 批准号: RGPIN-2018-04263
• 负责人: Goluskin, David
• 金额: $ 2.7万
• 依托单位: University of Victoria
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2022
• 资助国家: 加拿大
• 起止时间: 2022-01-01 至 2023-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=758977
• 关键词: Estimating; turbulence; chaos; using; semidefinite

In every scientific and engineering discipline there are countless systems whose behavior is accurately modeled by differential equations. Usually the solutions to these equations are too complicated to be found exactly. A particularly fundamental yet challenging example is fluid turbulenceflow of a liquid or gas with intricate structure in space and time. When faced with differential equations governing such systems, one option is to seek a precise approximation of a single solution. This requires a lot of computation, often more than is possible even with a supercomputer. Another option is to infer properties of solutions by mathematical analysis without finding any particular solutions. This latter approach can produce numerous kinds of mathematical statements about solutions, but often they are too weak or imprecise to be useful. In a sense, the former approach uses too much information from the governing equations, while the latter approach uses too little. My research program lies in between: seeking statements about solutions, rather than the solutions themselves, but doing so with computer assistance. The resulting methods compromise between precision and computational cost in a controllable way: more computation affords more precision.The methods I am developing all rely on constructing functions that satisfy prescribed constraints. When such a function can be found, it implies a mathematical statement about solutions to the differential equations being studied. One type of statement constructed in this way, for instance, is a limit on the minimum or maximum value of a chosen quantity. Many of the constraints that are imposed require certain polynomial expressions to be equal to sums of squares of other polynomials. Such constraints are useful because they can be reformulated as semidefinite programsa type of convex optimization problem for which numerical solutions can be efficiently computed. In this way, solving semidefinite programs gives statements about solutions to differential equations.The goal of this proposal is twofold: to further improve methods based on semidefinite programming, and to use them to answer open questions about several fundamental models of fluid mechanics. Understanding of fluid turbulence in such systems helps inform the modeling of turbulence in aerodynamic and hydrodynamic engineering, atmospheric science, physical oceanography, and astrophysics. Moreover, researchers in many disciplines are interested in models governed by equations simpler than those governing fluids, so any methods that succeed for our purposes will be broadly useful well beyond fluid mechanics.

在每一个科学和工程学科中，都有无数的系统的行为被微分方程式准确地建模。通常，这些方程的解太复杂，找不到确切的答案。一个特别基本但具有挑战性的例子是在时空中具有复杂结构的液体或气体的流体湍流流动。当面对控制这类系统的微分方程式时，一种选择是寻求单一解的精确近似。这需要大量的计算，往往比使用超级计算机所能做到的还要多。另一种选择是通过数学分析来推断解的性质，而不找出任何特定的解。后一种方法可以产生许多关于解决方案的数学陈述，但它们通常太弱或不精确，没有用处。在某种意义上，前一种方法使用了太多的控制方程信息，而后一种方法使用的太少。我的研究项目介于两者之间：寻求关于解决方案的陈述，而不是解决方案本身，但要在计算机的帮助下完成。由此产生的方法以一种可控的方式在精度和计算成本之间进行了折衷：计算越多，提供的精度就越高。我正在开发的方法都依赖于构造满足规定约束的函数。当可以找到这样的函数时，它意味着关于所研究的微分方程解的数学陈述。例如，以这种方式构造的一种类型的陈述是对所选量的最小或最大值的限制。施加的许多约束要求某些多项式表达式等于其他多项式的平方和。这种约束是有用的，因为它们可以被重新表示为半定规划，这是一种可以有效地计算数值解的凸优化问题。通过这种方式，求解半定规划给出了关于微分方程解的陈述。这项建议的目的有两个：进一步改进基于半定规划的方法，并用它们来回答关于流体力学的几个基本模型的公开问题。对这类系统中流体湍流的了解有助于在空气动力学和流体力学工程、大气科学、物理海洋学和天体物理学中对湍流进行建模。此外，许多学科的研究人员对由方程控制的模型比那些控制流体的模型更感兴趣，所以任何成功的方法都将远远超出流体力学的范围。