论文标题
线性逆问题多步一击方法的收敛分析
Convergence analysis of multi-step one-shot methods for linear inverse problems
论文作者
论文摘要
在这项工作中，我们对使用固定点方法对相应的正向问题进行迭代解决的一般线性反问题感兴趣。然后，可以应用一击方法，同时在正向问题解决方案和未知的逆问题上迭代。我们通过研究耦合迭代的块矩阵的特征值来分析所谓的多步量方法的两个变体，并在下降步骤中建立足够的条件。提供了一些数值实验，以说明与经典梯度下降相比，这些方法的收敛性。特别是，我们观察到，在正向问题上的内部迭代很少足以保证反转算法的良好收敛性。
In this work we are interested in general linear inverse problems where the corresponding forward problem is solved iteratively using fixed point methods. Then one-shot methods, which iterate at the same time on the forward problem solution and on the inverse problem unknown, can be applied. We analyze two variants of the so-called multi-step one-shot methods and establish sufficient conditions on the descent step for their convergence, by studying the eigenvalues of the block matrix of the coupled iterations. Several numerical experiments are provided to illustrate the convergence of these methods in comparison with the classical usual and shifted gradient descent. In particular, we observe that very few inner iterations on the forward problem are enough to guarantee good convergence of the inversion algorithm.
