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: 2018
• 资助国家: 加拿大
• 起止时间: 2018-01-01 至 2019-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=652122
• 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 turbulence—flow 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 programs—a 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，它们的行为可以用微分方程精确地建模。通常这些方程的解都太复杂，很难精确地找到。一个特别基本但具有挑战性的例子是流体湍流——在空间和时间上具有复杂结构的液体或气体的流动。当面对控制这类系统的微分方程时，一种选择是寻求单一解的精确近似值。这需要大量的计算，甚至比一台超级计算机所能做的还要多。另一种选择是通过数学分析来推断解的性质，而不需要找到任何特定的解。后一种方法可以产生许多种关于解的数学陈述，但它们通常太弱或不精确而无用。从某种意义上说，前一种方法使用了太多来自控制方程的信息，而后一种方法使用的信息太少。我的研究计划介于两者之间：寻找关于解决方案的陈述，而不是解决方案本身，但要借助计算机辅助。由此产生的方法以可控的方式在精度和计算成本之间折衷：更多的计算提供更高的精度。******我正在开发的方法都依赖于构造满足规定约束的函数。当这样一个函数可以被找到时，它就意味着一个关于所研究的微分方程解的数学表述。例如，以这种方式构造的一种语句是对所选数量的最小值或最大值的限制。许多强加的约束要求某些多项式表达式等于其他多项式的平方和。这些约束是有用的，因为它们可以被重新表述为半定规划——一种可以有效计算数值解的凸优化问题。用这种方法，求解半定规划给出微分方程解的表述。******本提案的目标是双重的：进一步改进基于半确定规划的方法，并使用它们来回答关于流体力学几个基本模型的开放问题。对此类系统中流体湍流的理解有助于为气动和流体动力工程、大气科学、物理海洋学和天体物理学中的湍流建模提供信息。此外，许多学科的研究人员对由比控制流体的方程更简单的方程控制的模型感兴趣，因此任何成功达到我们目的的方法都将广泛适用于流体力学以外的领域。