论文标题
部分可观测时空混沌系统的无模型预测
Stochastic mirror descent method for linear ill-posed problems in Banach spaces
论文作者
论文摘要
考虑系统$ a_i x = y_i $ for $ i = 1，\ cdots，p $的线性不良问题，其中每个$ a_i $都是一个有界的线性运算符，从Banach Space $ x $到Hilbert Space $ y_i $。如果$ p $是巨大的，则通过每次迭代步骤中的整个信息来解决问题，这是非常昂贵的，这是因为每次迭代的内存和过多的计算负载。为了有效地解决如此大规模的不足系统，我们在本文中开发了一种随机镜下降方法，该方法仅使用在每个迭代步骤中随机选择的一小部分方程式，并将凸正则化项结合到算法设计中。因此，我们的方法与问题大小相当得好，并且具有捕获寻求解决方案的特征的能力。该方法的收敛属性取决于步骤尺寸的选择。我们考虑选择步骤尺寸的各种规则，并在{\ it先验}早期停止规则下获得收敛结果。特别是，通过纳入差异原则的精神，我们提出了一个阶梯大小的选择规则，该规则可以有效地抑制迭代中的振荡并降低半偶然的效果。此外，当寻求的解决方案满足基准源条件时，我们建立了一个阶最佳收敛率。报告了各种数值模拟以测试该方法的性能。
Consider linear ill-posed problems governed by the system $A_i x = y_i$ for $i =1, \cdots, p$, where each $A_i$ is a bounded linear operator from a Banach space $X$ to a Hilbert space $Y_i$. In case $p$ is huge, solving the problem by an iterative regularization method using the whole information at each iteration step can be very expensive, due to the huge amount of memory and excessive computational load per iteration. To solve such large-scale ill-posed systems efficiently, we develop in this paper a stochastic mirror descent method which uses only a small portion of equations randomly selected at each iteration steps and incorporates convex regularization terms into the algorithm design. Therefore, our method scales very well with the problem size and has the capability of capturing features of sought solutions. The convergence property of the method depends crucially on the choice of step-sizes. We consider various rules for choosing step-sizes and obtain convergence results under {\it a priori} early stopping rules. In particular, by incorporating the spirit of the discrepancy principle we propose a choice rule of step-sizes which can efficiently suppress the oscillations in iterates and reduce the effect of semi-convergence. Furthermore, we establish an order optimal convergence rate result when the sought solution satisfies a benchmark source condition. Various numerical simulations are reported to test the performance of the method.
