We consider a linear ill-posed equation in the Hilbert space setting. Multiple independent unbiased measurements of the right hand side are available. A natural approach is to take the average of the measurements as an approximation of the right hand side and to estimate the data error as the inverse of the square root of the number of measurements. We calculate the optimal convergence rate (as the number of measurements tends to infinity) under classical source conditions and introduce a modified discrepancy principle, which asymptotically attains this rate.

翻译：在Hilbert空间设置中,我们考虑一个线性错误方程式。对右手侧可以进行多种独立的不偏倚测量。自然的方法是将测量平均值作为右手侧的近似值,并将数据误差估计为测量数字的平根的反面。我们计算出在传统源条件下的最佳趋同率(测量数量往往无限),并引入了经修改的差异原则,该原则不时达到这一速率。