This study investigates the iterative refinement method applied to the solution of linear discrete inverse problems by considering its application to the Tikhonov problem in mixed precision. Previous works on mixed precision iterative refinement methods for the solution of symmetric positive definite linear systems and least-squares problems have shown regularization to be a key requirement when computing low precision factorizations. For problems that are naturally severely ill-posed, we formulate the iterates of iterative refinement in mixed precision as a filtered solution using the preconditioned Landweber method with a Tikhonov-type preconditioner. Through numerical examples simulating various mixed precision choices, we showcase the filtering properties of the method and the achievement of comparable working accuracy of discrete inverse problems (i.e., to within a few decimal places in relative error) compared to results computed in double precision as well as another approximate iterative refinement method which we use as a benchmark.

翻译：本研究探讨了将迭代精化方法应用于求解线性离散反问题，具体考察其在混合精度下求解Tikhonov正则化问题的应用。先前关于对称正定线性系统和最小二乘问题的混合精度迭代精化方法的研究表明，在计算低精度分解时正则化是关键要求。对于自然严重不适定的问题，我们将混合精度迭代精化的迭代过程表述为采用Tikhonov型预条件子的预条件Landweber方法的滤波解。通过模拟多种混合精度选择的数值算例，我们展示了该方法的滤波特性，并实现了离散反问题可比较的工作精度（即相对误差达到小数点后数位），其效果与双精度计算结果以及作为基准的另一种近似迭代精化方法相当。