High order accuracy WENO methods for high dimensional problems on sparse grids

高阶精度数值方法对于求解包含复杂解结构的计算流体力学和计算生物学中的数学模型特别有效。当网格点数较大或问题的空间维数较高时，由于“维数灾难”的存在，计算成本会显著增加。如何利用高阶精度方法实现快速计算是一个非常重要的问题，特别是对于长时间的模拟。本研究计画旨在发展高空间维度问题之高效率高阶精度数值方法。这些新方法有可能应用于量子电子系统、分子马达、金融、生物学中的集体细胞运动、基因调控网络等更广泛的应用领域。分析并实现了求解双曲型或对流扩散型偏微分方程的新型高阶Krylov积分因子（IF）加权本质无振荡（韦诺）算法稀疏网格上的问题，通过使用稀疏网格组合技术来处理高维的挑战。稀疏网格方法是求解高维问题的一种有效的近似工具。它已成功地应用于许多科学和工程应用。稀疏网格上的离散化比单网格上的离散化涉及更少的自由度。有效的数值模拟这些高维系统将有助于研究有趣的生物学问题在这一领域。该研究将有助于解决“维数灾难”这一问题。一套强大的计算工具，解决高维非线性偏微分方程将开发。这些技术有望为生物和物理系统中复杂现象的计算机模拟做出积极贡献。拟议的活动还将为对数学，计算和应用界面研究感兴趣的研究生和本科生提供极好的培训和教育机会。

High order accuracy numerical methods are especially efficient for solving mathematical models in computational fluid dynamics and computational biology which contain complex solution structures. The computational cost increases significantly when the number of grid points is large or the spatial dimension of the problem is high, due to the "curse of dimensionality". How to achieve fast computations by high order accuracy methods is a very important question especially for long-time simulations. This research project aims to develop efficient high order accuracy numerical methods on sparse grids for high spatial dimensional problems. The new methods have the potential to be applied to a broader class of applications in quantum electronic systems, molecular motors, finance, collective cell motions in biology, gene regulatory network, etc. The PI will design, analyze and implement novel high order Krylov integration factor (IF) weighted essentially nonoscillatory (WENO) algorithms for solving hyperbolic or convection-diffusion partial differential equation (PDE) problems on sparse grids by using the sparse-grid combination technique to deal with the high dimensional challenge. The sparse-grid method is a powerful approximation tool for high dimensional problems. It has been successfully used in many scientific and engineering applications. Discretizations on sparse grids involve much fewer degrees of freedom than that on single grids. Efficient numerical simulations of these high dimensional systems will help in studying interesting biological questions in this area. The proposed research will contribute in the active area of dealing with the "curse of dimensionality". A suite of powerful computational tools for solving high dimensional nonlinear PDEs will be developed. These techniques are expected to make positive contributions to computer simulations of complicated phenomena in biological and physical systems. The proposed activity will also provide excellent training and education opportunities for both graduate and undergraduate students interested in research at the interface of mathematics, computation, and applications.