Optimal Prediction for Non-Linear Problems

最优预测方法是近年来发展起来的。它给出了非线性方程的最佳近似，假设高阶模态是随机的。它适用于计算复杂的流，当有关初始数据的信息是不完整的。研究了应用于三次薛定谔方程的最优预测方法的收敛性。这样的方程出现在量子力学中。此外，该方法还将用于研究不可压缩和无粘流体的三维流动。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规模运动和许多小而快的漩涡组成。研究者将小尺度运动作为一个随机分量来处理，并近似地考虑其对大尺度运动的影响，而不是用单一的数值算法来计算流量。目标是为湍流开发更快、更健壮的计算方案。

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.