An adaptive pruning algorithm for the discrete L-curve criterion

We describe a robust and adaptive implementation of the L-curve criterion. The algorithm locates the corner of a discrete L-curve which is a log–log plot of corresponding residual norms and solution norms of regularized solutions from a method with a discrete regularization parameter (such as truncated SVD or regularizing CG iterations). Our algorithm needs no predefined parameters, and in order to capture the global features of the curve in an adaptive fashion, we use a sequence of pruned L-curves that correspond to considering the curves at different scales. We compare our new algorithm to existing algorithms and demonstrate its robustness by numerical examples.     

A new L-curve for ill-posed problems

The truncated singular value decomposition is a popular method for the solution of linear ill-posed problems. The method requires the choice of a truncation index, which affects the quality of the computed approximate solution. This paper proposes that an L-curve, which is determined by how well the given data (right-hand side) can be approximated by a linear combination of the first (few) left singular vectors (or functions), be used as an aid for determining the truncation index.     

Cosmic rays of the highest energies

The article describes the study of cosmic rays above 10 19 eV. Following a brief introduction, the reasons for the current interest in this topic are discussed. The methods used to measure the properties of the very rare ultrahigh-energy cosmic rays are explained and the current data described. The data lead to an enigma and the many theories proposed to explain it are summarized. The properties and potential of new detectors are outlined.     

L-曲线估计确定正则参数的双网格迭代法

本文考虑对不适定问题离散化得到的大规模不适定线性方程组进行Tiknonov正则化,然后用双网格迭代法求解得到的Tikhonov正则化方程组,并用L-曲线估计法来确定正则参数.试验问题的数值结果表明双网格迭代法求解正则化后的对称正定线性方程组效果很好,且L-曲线估计法确定正则参数计算量很小.     

A new variant of L-curve for Tikhonov regularization

In this paper we introduce a new variant of L-curve to estimate the Tikhonov regularization parameter for the regularization of discrete ill-posed problems. This method uses the solution norm versus the regularization parameter. The numerical efficiency of this new method is also discussed by considering some test problems.     

An L-ribbon for large underdetermined linear discrete ill-posed problems

The L-curve is a popular aid for determining a suitable value of the regularization parameter when solving ill-conditioned linear systems of equations with a right-hand side vector, which is contaminated by errors of unknown size. However, for large problems, the computation of the L-curve can be quite expensive, because the determination of a point on the L-curve requires that both the norm of the regularized approximate solution and the norm of the corresponding residual vector be available. Recently, an approximation of the L-curve, referred to as the L-ribbon, was introduced to address this difficulty. The present paper discusses how to organize the computation of the L-ribbon when the matrix of the linear system of equations has many more columns than rows. Numerical examples include an application to computerized tomography.     

GMRES, L-Curves, and Discrete Ill-Posed Problems

The GMRES method is a popular iterative method for the solution of large linear systems of equations with a nonsymmetric nonsingular matrix. This paper discusses application of the GMRES method to the solution of large linear systems of equations that arise from the discretization of linear ill-posed problems. These linear systems are severely ill-conditioned and are referred to as discrete ill-posed problems. We are concerned with the situation when the right-hand side vector is contaminated by measurement errors, and we discuss how a meaningful approximate solution of the discrete ill-posed problem can be determined by early termination of the iterations with the GMRES method. We propose a termination criterion based on the condition number of the projected matrices defined by the GMRES method. Under certain conditions on the linear system, the termination index corresponds to the vertex of an L-shaped curve.     

基于 L-曲线参数优化的均匀声场重建算法

为了得到最小二乘法声场重建问题的稳定解,通常需要引入Tikhonov正则化方法。然而正则化程度取决于正则化参数的选择。针对这一问题,提出了一种基于L-曲线法参数选择的均匀声场重建算法。该算法根据重建误差与扬声器功率计算得到L-曲线,该曲线上曲率最大的点所对应的参数值作为Tikhonov正则化参数的选值。确定正则化参数后可进一步得到扬声器权系数以及重建均匀声场。针对不同正则化参数取值方法,对控制区域进行均匀声场重建以及重建性能仿真。仿真结果及实验表明,L-曲线法实现了重建误差与扬声器驱动信号功率之间的平衡。     

Regularization of Ill-Posed Problems by Envelope Guided Conjugate Gradients

Abstract

We propose a new way to iteratively solve large scale ill-posed problems by exploiting the relation between Tikhonov regularization and multiobjective optimization to obtain, iteratively, approximations to the Tikhonov L-curve and its corner. Monitoring the change of the approximate L-curves allows us to adjust the regularization parameter adaptively during a preconditioned conjugate gradient iteration, so that the desired solution can be reconstructed with a low number of iterations. We apply the technique to an idealized image reconstruction problem in positron emission tomography.     

L曲线方法在地基红外高光谱反演温度廓线中的应用

利用地基红外高光谱辐射数据可以反演得到高时间分辨率的边界层大气温度廓线。目前的AERIoe最优化反演算法相比于传统的“剥洋葱”算法有较大的改进,且对初值的依赖程度较低。但AERIoe算法中正则化算子的选择对反演结果的稳定性和反演时间有重要影响。目前主要采用经验的方法选择正则化算子,迭代步数较多,耗费大量的计算时间。提出了利用L曲线方法代替经验法选取正则化算子的改进方案,以提高AERIoe方法的反演速度。改进后的算法通过绘制解范数和残余范数的二维曲线图,取其拐点作为最优的正则化参数,相比于传统的经验法有着更好的理论基础。采用2011年美国大气辐射测量计划中SGP站点的晴空大气红外辐射数据进行反演实验。结果表明,利用该方法得到的反演结果具有很好的稳定性、收敛性和精度。相比于经验的方法,利用L曲线方法获得的正则化算子反演温度廓线时的收敛速度更快,迭代步数较少,可以节约大量的计算时间;在反演精度方面,L曲线方法在边界层中上层的反演精度更高,1～3 km高度上温度廓线的RMSE值提高了大约0.2 K。