解非线性最小二乘问题的ABS方法

The ABS Methods for Nonlinear Least Squares Problems

在本文中,基于解非线性方程组的ABS方法的思想,我们对非线性最小二乘问题建立了一类新的算法。在类似于Gauss-Newton法的收敛条件下,我们证明了算法的局部收敛性。此外,在对算法结构进行深入分析的基础上,我们将新算法转化为一种近似Gauss-Newton法。并建立了它的Kantorovich型收敛定理。数值结果表明ABS算法是有效的,且在一定程度上优越于Gauss-Newton法。

In this paper, based on the idea of the ABS methods for solving systems of nonlinear equations, we establish a new class of algorithms for nonlinear least squares problems. The local convergence of the new algorithm are proved under the similar conditions as the Gauss-Newton method. Moreover, based on a careful analysis of the algorithm, structure, we convert the new algorithm into an approximate GaussNewton method, and establish the Kantorovich type Convergence theorem. The numerical results implies that the ABS method are satisfactory and superior to the Gauss-Newton method in certain extents.