带有随机参数输入的非线性双曲型方程的数值方法

The ultimate objective of numerical simulations is to predict physical events or the behaviors of engineering systems. In the past few decades, extensive efforts have been devoted to the development of accurate numerical algorithms, which make the simulation predictions reliable in the sense that numerical errors are well under control and understood. However, the mathematical models under considered are affected by many uncertain (stochastic) factors. The uncertainty may happen for parameter values, initial and boundary conditions and geometric domains and so on. In recent years, there have been growing interests in designing efficient numerical methods for mathematical models with stochastic parameters. The study of uncertainty quantification is to provide more reliable predictions for real-life problems. In this project, we will investigate uncertainty quantification for problems governed by hyperbolic partial differential equations, in particular for stochastic nonlinear hyperbolic equations that have many important applications in fluid mechanics,chemical reaction flows and porous media flows. The random effect and the fact that nonlinear hyperbolic equations admit discontinuous solutions yield great challenges for designing numerical algorithms and for the relevant numerical analysis. Thus, the numerical analysis and algorithm design for stochastic nonlinear hyperbolic equations are of great importance.

研究数值计算方法的最终目标是对于实际的物理过程或复杂系统进行准确预测。过去的很长一段时间里，研究工作者对于高效的数值计算方法进行了深入研究，这使得由数值算法产生的误差得到了很好的控制。然而，我们所求解的数学模型中存在许多的不确定（随机）因素，比如数学模型中的参数，不确定的初值或边界条件，求解区域的复杂性等等。近年来，对于带有随机参数输入的数学模型的数值算法研究受到了高度重视。其中一个重要原因是对这些不确定因素的定量化研究有助于对实际问题进行更加准确的预测。本课题将主要研究带有随机参数输入的非线性双曲型方程的数值算法及其理论分析。带有随机参数输入的非线性双曲型方程在流体动力学、化学反应过程、石油油藏模拟等领域有重要应用。随机因素的影响以及非线性双曲型问题固有的解间断性质使得精确的数值模拟具有很大的挑战性。因此，这一研究课题无论从计算理论还是从算法设计上都具有重要意义。

不确定性量化是近几年国际热门的研究课题之一。 本项目聚焦于不确定性量化研究的几个前沿问题，主要针对动态正交逼近方法，高效随机配置方法进行了深入的研究。 对于动态正交逼近方法，我们给出了收敛性证明，为该方法提供了严格的理论基础。对于离散投影方法以及压缩感知方法，我们设计了确定性样本以消除失败概率，并设计了基于平衡态测度抽样方法，大幅度降低了计算量。该项目共发表国际期刊论文11篇，其中大部分发表于计算数学领域著名期刊，如SIAM Journal on Scientific Computing 五篇，计算物理杂志Journal of Computational Physics 两篇等