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：本文考虑在仅能对线性映射进行求值的框架下求解线性最小二乘问题。我们推导出的随机方法除了正向求值外，不需要任何其他矩阵运算，尤其不需要对伴随映射进行求值。该方法的动机源于一个简单观察：可以在伴随映射的应用中获得无偏估计。我们证明了方法的收敛性，并进一步推导出一种采用精确线搜索的更高效方法。这种方法称为随机下降法，类似于其他上下文中的已知方法，并以随机坐标下降法作为特例。我们提供了随机下降法的收敛性分析，重点强调了随机向量本征分布对方法的影响。此外，我们探讨了该方法在病态逆问题中的适用性，并表明当未知解较为粗糙时，该方法可以具备有益特性。我们通过数值算例对理论发现进行了验证。其中一个特殊结果是：随机下降法在算例中实际上优于已建立的免转置方法（TFQMR和CGS）。