We consider the problem of solving linear least squares problems in a framework where only evaluations of the linear map are possible. We derive randomized methods that do not need any other matrix operations than forward evaluations, especially no evaluation of the adjoint map is needed. Our method is motivated by the simple observation that one can get an unbiased estimate of the application of the adjoint. We show convergence of the method and then derive a more efficient method that uses an exact linesearch. This method, called random descent, resembles known methods in other context and has the randomized coordinate descent method as special case. We provide convergence analysis of the random descent method emphasizing the dependence on the underlying distribution of the random vectors. Furthermore we investigate the applicability of the method in the context of ill-posed inverse problems and show that the method can have beneficial properties when the unknown solution is rough. We illustrate the theoretical findings in numerical examples. One particular result is that the random descent method actually outperforms established transposed-free methods (TFQMR and CGS) in examples.

翻译：我们考虑仅允许对线性映射进行求值的框架下求解线性最小二乘问题。我们推导了无需任何其他矩阵运算（特别是无需伴随映射求值）的随机化方法。该方法源于一个简单观察：可以无偏估计伴随映射的应用。我们证明了方法的收敛性，进而推导出使用精确线搜索的更高效方法。该方法称为随机下降法，与其他领域已知方法类似，并将随机坐标下降法作为特例。我们提供了随机下降法的收敛分析，重点阐述了其与随机向量底层分布的依赖性。此外，我们探讨了该方法在不适定反问题中的适用性，并表明当未知解较为粗糙时该方法具有有益特性。我们通过数值示例说明了理论结果。特别地，随机下降法在示例中实际优于经典无转置方法（TFQMR和CGS）。