基于模型降阶的无网格方法及其在分数阶扩散方程反问题中的应用

Many problems in computational science and engineering are characterized by experimental design, measurement acquisition, and parameter estimation or prediction. This process involves mathematical modeling of the physical systems pertinent to the observations, as well as estimation of unknown model parameters from the acquired measurements by formulating and solving a large-scale inverse problem. Those problems are mathematically and numerically very challenging due to their inherent ill-posedness and nonlinearity. Much progress has been made in the field of large-scale inverse problems, but many challenges still remain for future research. The development of new numerical algorithms for large-scale inverse problems have received great attention recently in scientific computing. Based on the regularization theory, this project attempts to design some efficient and stable meshless methods, which are applied to solve several specific problems. Firstly, we explore the main challenge occur in the meshless methods to solve large-scale inverse problems, then to construct adaptive meshless method based on the reduced order modeling algorithm for large-scale inverse problems. Secondly, we discuss the application of the meshless for the high dimensional fractional diffusion equations, and consider design some efficiency numerical algorithm to reduce the storage amount and decrease computational complexity. The study of these issues mentioned above not only promote development and refinement of the meshless method for solving high-dimensional fractional diffusion equations, but also provide a new system of framework for solving large-scale inverse problems.

在计算科学和工程等问题中，经常涉及实验设计、测量采集及参数识别等问题。这些问题可以通过建立与测量有关的物理系统的大规模反问题模型进行描述求解。数值求解此类问题时面临着非线性及不适定性等特点。发展处理具有非线性及不适定性的大规模反问题的数值求解方法是近年来国际学术界讨论的一个热点问题。本课题旨在结合有效的正则化方法提出和发展新的高效稳定的无网格方法，并将其应用于具体问题的求解。内容包括：探讨无网格方法在求解大规模不适定问题中出现的各种难题，研究基于模型降阶的自适应无网格方法，发展该方法在非线性及大规模不适定问题求解中的有效应用；设计求解高维分数阶扩散方程反问题的高效无网格算法。在模型降阶的基础上，考虑数值算法的快速实现，提高无网格方法的有效性。这些问题的解决，不仅可以促进无网格方法在数值求解高维分数阶扩散方程反问题中的进一步发展和完善，也为其他大规模反问题的有效数值计算方法提供了有力工具。

在计算科学和工程等问题中，经常涉及算法设计、参数识别及函数逼近等问题。这些问题在实际求解过程，面临着非线性、不适定及算法的复杂性等特点。发展处理具有高效的算法及不适定反问题的数值求解方法是今年来国内外学术上的热点问题。本项目围绕无网格方法展开研究。一方面设计了双随机的快速算法（DSRBF），另一方面将此方法应用于分数阶方程反问题的数值求解。本文获得了以下重要成果：.1）设计了一种估计最优RBF形状参数的快速随机LOOCV算法，并利用统计结果，运用了卡方分布的RBF形状参数可以得到快速收敛速度和h-稳定的渐近误差分布。.2）构造了一种基于核学习的降阶模型，将DSRBF用于选择核函数中的超参数来降阶模型，即设计了一种基于非侵入式降阶模型的集合Kalman反演方法应用于分数阶方程反问题，从数值结果中我们可以看到，在保证计算效率的同时，可以有效保障数值精度。在本项目的支持下，已发表2篇SCI论文。