Estimating turbulence and chaos using semidefinite programming

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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.