论文标题
通过基于时间离散化的迭代正则化方法解决严重的线性系统
Solving Severely Ill-Posed Linear Systems with Time Discretization Based Iterative Regularization Methods
论文作者
论文摘要
最近，逆问题吸引了越来越多的计算数学关注，并且在工程应用中变得越来越重要。离散化后，许多反问题将减少为线性系统。由于反问题的典型不良性，降低的线性系统通常是不适合的，尤其是当它们的尺度很大时。这带来了巨大的计算困难。特别是，在不良线性系统右侧的小扰动可能会导致溶液发生巨大变化。因此，应采用正则化方法来稳定解决方案。在本文中，采用了一种新的加速迭代正则化方法来解决这种大规模不足的线性系统。迭代方案仅在迭代早期终止时才成为正则化方法。莫罗佐夫的差异原则用于停止标准。与常规的陆地迭代相比，新方法具有加速效应，可以与众所周知的加速V-Method和Nesterov方法进行比较。从数值结果中可以看出，使用适当的离散方案，在与V-Method和Nesterov方法进行比较时，所提出的方法甚至具有更好的行为。
Recently, inverse problems have attracted more and more attention in computational mathematics and become increasingly important in engineering applications. After the discretization, many of inverse problems are reduced to linear systems. Due to the typical ill-posedness of inverse problems, the reduced linear systems are often ill-posed, especially when their scales are large. This brings great computational difficulty. Particularly, a small perturbation in the right side of an ill-posed linear system may cause a dramatical change in the solution. Therefore, regularization methods should be adopted for stable solutions. In this paper, a new class of accelerated iterative regularization methods is applied to solve this kind of large-scale ill-posed linear systems. An iterative scheme becomes a regularization method only when the iteration is early terminated. And a Morozov's discrepancy principle is applied for the stop criterion. Compared with the conventional Landweber iteration, the new methods have acceleration effect, and can be compared to the well-known accelerated v-method and Nesterov method. From the numerical results, it is observed that using appropriate discretization schemes, the proposed methods even have better behavior when comparing with v-method and Nesterov method.
