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Simulation research on improved regularized solution of the inverse problem in spectral extinction measurements

机译：光谱消光测量中反问题的改进正则解的仿真研究

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摘要

We present further results of the simulation research on the constrained regularized least squares (CRLS) solution of the ill-conditioned inverse problem in spectral extinction (turbidimetric) measurements, which we originally presented in this journal [Appl. Opt. 49, 4591 (2010)]. The inverse problem consists of determining the particle size distribution (PSD) function of a particulate system on the basis of a measured extinction coefficient as a function of wavelength. In our previous paper, it was shown that under assumed conditions the problem can be formulated in terms of the discretized Fredholm integral equation of the first kind. The CRLS method incorporates two constraints, which the PSD sought will satisfy: nonnegativity of the PSD values and normalization of the PSD to unity when integrated over the whole range of particle size, into the regularized least squares (RLS) method. This leads to the quadratic programming problem, which is solved by means of the active set algorithm within the research. The simulation research that is the subject of the present paper is a continuation and extension of the research described in our previous paper. In the present research, the performance of the CRLS method variants is compared not only to the corresponding RLS method variants but also to other regularization techniques: the truncated generalized singular value decomposition and the filtered generalized singular value decomposition, as well as nonlinear iterative algorithms: The Twomey algorithm and the Twomey-Markowski algorithm. Moreover, two methods of selecting the optimum value of the regularization parameter are considered: The L-curve method and the generalized cross validation method. The results of our simulation research provide even stronger proof that the CRLS method performs considerably better with reconstruction of PSD than other inversing methods, in terms of better fidelity and smaller uncertainty.
机译：我们提供了光谱消光（比浊）测量中病态逆问题的约束正则最小二乘（CRLS）解的模拟研究的进一步结果，该研究最初在本杂志上发表。选择。 49，4591（2010）。反问题包括根据测得的消光系数作为波长的函数确定颗粒系统的粒径分布（PSD）函数。在我们以前的论文中，表明在假定条件下，可以用第一类离散Fredholm积分方程来表示问题。 CRLS方法将两个要满足的PSD要求满足：将PSD值的非负性和在整个粒度范围内积分时PSD归一化到正则最小二乘（RLS）方法中。这导致了二次规划问题，该问题通过研究中的主动集算法得以解决。作为本文主题的仿真研究是对我们先前论文中描述的研究的延续和扩展。在本研究中，CRLS方法变体的性能不仅与相应的RLS方法变体进行了比较，而且还与其他正则化技术进行了比较：截断的广义奇异值分解和滤波的广义奇异值分解以及非线性迭代算法： Twomey算法和Twomey-Markowski算法。此外，考虑了选择正则化参数的最佳值的两种方法：L曲线方法和广义交叉验证方法。我们的仿真研究结果提供了更强的证据，即在保真度和不确定性较小方面，CRLS方法在重构PSD方面比其他反演方法表现更好。

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