基于共扩张正交多项式的加速Landweber方法
Accelerated Landweber methods based on co-dilated orthogonal polynomials
AI总结：
通过修改ν-方法的三项递推系数引入加速Landweber方法，利用共扩张超球多项式提高残差多项式在原点处的衰减速度，保持相同收敛阶，并开发自适应参数选择方案。
摘要：
本文介绍并研究了通过修改ν-方法的三项递推系数得到的线性不适定问题的加速Landweber方法。所考虑的半迭代方法的残差多项式与一族共扩张超球多项式相关联。这种联系使得通过扩张参数增加残差多项式在原点处的衰减成为可能。当求解涉及算子谱聚集在原点的线性不适定方程时，这种增加的衰减具有优势。新的半迭代方法的收敛阶与原始ν-方法的收敛阶相同。新算法经过数值测试，并开发了一种简单的自适应方案，用于计算最优扩张参数。
英文摘要：
In this article, we introduce and study accelerated Landweber methods for linear ill-posed problems obtained by an alteration of the coefficients in the three-term recurrence relation of the ν-methods. The residual polynomials of the semi-iterative methods under consideration are linked to a family of co-dilated ultraspherical polynomials. This connection makes it possible to increase the decay of the residual polynomials at the origin by means of a dilation parameter. This increased decay has advantages when solving linear ill-posed equations in which the spectrum of the involved operators is clustered at the origin. The convergence order of the new semi-iterative methods turns out to be the same as the convergence order of the original ν-methods. The new algorithms are tested numerically and a simple adaptive scheme is developed in which an optimal dilation parameter is computed.
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