۱٫ Introduction

۲٫ Regularized Forms of the Kaczmarz’s algorithm

۳٫ Accelerated Regularized Forms of Kaczmarz’s Algorithm

۴٫ Numerical Experiments

۵٫ Conclusion

۶٫ References

In this paper suggested methods for acceleration Kaczmarz algorithm regularized modifications to solve the standard regularization problem of A.N. Tikhonov. As shown in numerical experiments, for certain classes of problems, such methods allow reducing both the number of iterations and the time for finding solutions. For the two-dimensional problem of seismic tomography proposed greedy forms of Kaczmarz algorithm regularized modifications can reduce the number of iterations up to 28 times.

Introduction

In the case of ill-conditioned of the matrix A and (or) in the presence of noise in the vector of the right side f , the computation of stable solutions of SLAE (1) requires the use of regularization methods. Similar problems arise, for example, when SLAE (1) is obtained as a result of discretization of the Fredholm integral equation of the first kind with a smooth kernel, in regression analysis in the presence of multicollinearity and in the theory of machine learning – “retraining” of regression models. In practice, the most common method for finding stable solutions of SLAE (1) is the regularization method of A. N. Tikhonov2 .