不确定性量化的高精度数值方法和理论
摘要
Uncertainty quantification (UQ) has been a hot research topic recently. UQ has a variety of applications, including hydrology, fluid mechanics, data assimilation, and weather forecasting. Among a large number of approaches, the high order numerical methods have become one of the important tools; and the relevant computational techniques and their mathematical theory have attracted great attention in recent years. This paper begins with a brief introduction of recent developments of high order numerical methods including Galerkin projection methods and stochastic collocation methods. The emphasis will be sample-based stochastic collocation methods, including random sampling, deterministic sampling and structured random sampling. The paper will review the recent progress on the discrete projection method and the compressed sensing approximation for the UQ research. In particular, we will discuss the relationship between the sample size <italic>M</italic> and the degree of freedom <italic>N</italic> for the basis function in the approximation space, by considering the stability and optimal convergence of the algorithms. Moreover, we will also discuss the interpolation method with arbitrary points in high dimensional spaces. A topic relevant to UQ computations, i.e., numerical methods for the forward backward stochastic differential equations, will be briefly introduced. To close this article, some challenging and open problems for the UQ research will be briefly discussed.