带随机Helmholtz方程约束控制优化问题的数值方法

确定性Helmholtz问题的理论与计算方法已广泛应用于地质勘探,军事科学等诸多科学技术领域，但对于更具有实际意义的带随机Helmholtz方程约束的控制优化问题还有待进一步研究。具体的说，我们拟针对带随机Helmholtz方程约束控制优化问题的数值方法进行研究，主要包括以下几类方法:基于稀疏网格的随机配置法，随机动态规划方法，广义多项式混沌展开法和多级模式展开方法。通过这几类数值算法的研究和比较，我们可以判别其各自优缺点以及适用范围，从而更好的模拟和设计无损检测，无人机等仪器。本项目具有鲜明的新颖性与挑战性，有广泛的应用前景，对更一般带SPDE约束控制优化问题的求解有着指导作用。

The theoretical and computational results of deterministic Helmholtz problems have important applications in many fields of science and technology, such as geophysical exploration, non-destructive testing, radar and sonar and other defense technology, etc.. In this project, we shall be concerned with several numerical methods for the optimal control problems constrained by random Helmholtz equations, which are of more practical significance and have received wide attentions recently in literature, consists of the stochastic collocation method based on sparse grid, the stochastic programming, the general polynomial chaos, and the multi-mode expansion. Through the studying and comparing, we can distinguish the advantages and disadvantages of each method, therefore enhance the efficiency of simulations and designs for the instruments such as non-destructive testing and unmanned aerial vehicle.The theoretical and computational results achieved in this project will deepen our knowledge about corresponding phenomena, and also have significant practical impacts for the optimal control problems constrained by general SPDEs.