算子分裂方法及其若干计算问题的研究

The first order operator splitting methods are a class of important methods for solving the separable convex optimization problems, which have been widely concerned by communities in science and engineering in recent years. We will consider to apply and modify some operator splitting methods for solving linear inverse problems and fractional optimal control problems, which arise from the fields of physics, chemistry, medicine, financial, and etc., can be modeled as separable convex optimization problems. Then it becomes essential to design efficient algorithms to solve these models by exploiting their structures favorably. Based on our previous work on numerical studies of TV regularized inverse problems and fractional partial differential equations, we will develop some improved operator splitting methods to solve these two problems. Specifically, we will consider: (1) using finite element method to discretize TV regularized linear inverse problems, designing primal-dual methods with positive-indefinite proximal term, adaptively-chosen step sizes and its acceleration version, designing convergent ADMM in finite element space and its linearized acceleration algorithm, and then discussing their convergence analysis and error estimates; (2) using finite difference/element methods to discretize the optimal control problems of fractional Feynman-Kac equations, analyzing the error estimates, and using the ADMM based on fast algorithms to solve the problems efficiently.

一阶算子分裂方法是求解可分凸优化问题的一类重要方法，近年来受到科学与工程界的广泛关注。我们将利用并改进一些经典的算子分裂方法来求解线性反问题及分数阶偏微分方程的优化控制问题，它们来源于物理、化学、医药学、金融等领域，都可以归结为可分的凸优化问题。如何利用它们的特殊结构来设计高效的算法显得至关重要。基于申请人前期关于TV正则化反问题和分数阶方程数值计算方面的工作，拟在经典的算子分裂方法的基础上，设计出改进的算法来更有效地求解这两类问题。具体来说，我们将考虑：（1）利用有限元离散TV正则化的线性算子方程反问题，设计带不定邻近项和自适应步长的原始对偶方法及其加速算法、设计在有限元解空间中收敛的ADMM算法并讨论其线性化加速算法，进而分析算法的收敛性与误差估计;（2）利用有限差分/有限元离散分数阶Feynman-Kac方程的优化控制问题，分析误差估计，并利用基于快速算法的ADMM对其进行高效求解。

具有可分结构的凸优化问题广泛存在于科学与工程领域，一阶算子分裂方法是求解该类问题的重要方法之一。本项目考虑可分凸优化问题的一阶算子分裂方法研究，及其在不适定线性算子方程反问题及分数阶Feynman-Kac方程的二次优化控制问题中的应用。完成的研究结果有：（1）提出了可以自动选取最优迭代步长的原始对偶方法求解TV正则化问题；（2）设计了一个线性化原始对偶加速算法求解不适定线性算子方程的TV正则化问题，证明了其收敛性和收敛速度，分析了模型的正则化和扰动误差、有限元离散误差、优化算法的迭代误差以及数值解与原问题精确解之间的误差；（3）提出了一个求解非强凸的可分凸优化问题的（邻近）ADMM加速算法，分析了算法的收敛性和收敛速度；（4）得到了分数阶Feynman-Kac方程的二次优化控制问题的一阶最优性条件并建立了有限差分与有限元离散格式。