基于Landweber迭代的秩亏非线性最小二乘问题算法研究

ALGORITHM BASED ON LANDWEBER ITERATION FOR SOLVING RANK DEFICIENCY NONLINEAR LEAST SQUARES PROBLEM

对非线性Landweber迭代格式xkδ+1=xkδ-f′(xkδ)*(f(xδk)-yδ)进行分析,给出其导出方式。基于逆矩阵和矩阵伴随算子的换算关系,并用1/ω代替(B′(xk)B(xk)),构造出一个新的求解秩亏非线性最小二乘问题的Landweber迭代格式,xkδ+1=xkδ-ω(B′(xkδ)B(xkδ))*B′(xkδ)(f(xkδ)-yδ),从而避免了在迭代过程中由于迭代矩阵的秩亏和病态而产生的不适定现象。数值实验结果得出,此Landweber迭代格式适用于非线性秩亏自由网平差和秩亏非线性最小二乘问题的解算。

Abstract More methods such as the gauss-newton method or modified gauss-newton method will failure when the iteration matrix is rank-deficient or very ill-conditioned in solving ill-posed nonlinear least squares problem.Nonlinear Landweber iteration formula x δ k＋1=x δ k-f′（x δ k） ＊（f（x δ k）-y δ） is analyzed and a new method is derived.On the basis of the conversion relation of inverse matrix and adjoint matrix, by using 1/ω instead of ｜（B′（x k）B（x k））｜ a new Landweber iteration formula x δ k＋1=x δ k-ω（B′（x δ k）B（x δ k）） ＊B′（x δ k）（f（x δ k）-y δ） is constructed for solving rank deficiency nonlinear least squares problem, with which the phenomenon that leads to ill-posed problem because the iteration matrix is rank-deficient and very ill-conditioned in numerical iterative process is avoided. The numerical experiment showes that the new Landweber iteration formula is accurate and of applicability for nonlinear adjustment of free networks with rank deficiency and rank deficiency nonlinear least squares problems.