Optimal Prediction for Non-Linear Problems

最优预报方法是近年来发展起来的一种方法，它在假定高阶模态是随机的情况下，对非线性方程给出了最佳逼近，适用于计算复杂的流动，当关于初始数据的信息是不完整的。提议者将研究应用于三次薛定谔方程的最佳预测方法的收敛性。这种方程出现在量子力学中。此外，该方法将用于研究不可压缩和无粘性流体在三维空间中的流动。这种流体的复杂行为是湍流问题的一部分。湍流通常由缓慢的大尺度运动和许多小但快的漩涡组成。研究者不再用单一的数值算法来计算流动，而是将小尺度运动作为一个随机分量来处理，并近似地分析它对大尺度运动的影响。我们的目标是发展更快更强的湍流计算方案。

The method of optimal prediction has been developed recently.It gives the best approximation to non-linear equationsassuming that the higher modes are random.It is suited for computing complicated flows, whenthe information about the initial data is incomplete.The proposer will study the convergence of the optimal prediction methodapplied to the cubic Schrodinger equation.Such equations occur in quantum mechanics.In addition the method will be used to study the flowof incompressible and inviscid fluids in three dimensions.The complicated behavior of such fluids is part of theproblem of turbulence.Turbulent flows typically consist of a slow, large scale motion,and a lot of small, but fast, eddies. Instead of computing the flowby a single numerical algorithm, the investigator will treatthe small scale motion as a random component, and incorporateits influence on the large scale motion approximately. The goalis to develope faster and more robust computational schemesfor turbulent flows.