Wang J Y, Goulard R
Appl Opt. 1975 Apr 1;14(4):862-71. doi: 10.1364/AO.14.000862.
Several aspects of the behavior of Fredholm integral equations are examined in this paper. It is shown that collocation methods are better in general than least squares methods in linear approaches. The amplification of random noise inherent to the numerical inversion of the equation puts an upper limit to the information content of an ill-conditioned system. An estimation based on the magnitude of SNR is proposed for a system that lacks statistical information to determine the information content and to reconstruct the solution profile. To reduce the numerical instability of matrix inversion, some specific kernel transformations are discussed. Illustrative examples are given and compared to results of other approaches. An alternative linear approach that orthonormalizes the kernels-is also proposed. The linear approach was then employed in solving the radiative transfer equation with temperature-independent kernels. The necessary variable separation in linear inversions was examined. Iteration refinement was found necessary to accommodate the strong nonlinearity of high temperature sensing.
本文研究了弗雷德霍姆积分方程行为的几个方面。结果表明，在线性方法中，配置法通常比最小二乘法更好。方程数值反演中固有的随机噪声放大对病态系统的信息含量设置了上限。对于缺乏统计信息以确定信息含量和重构解剖面的系统，提出了一种基于信噪比大小的估计方法。为降低矩阵求逆的数值不稳定性，讨论了一些特定的核变换。给出了示例并与其他方法的结果进行了比较。还提出了一种使核正交归一化的替代线性方法。然后将该线性方法用于求解具有与温度无关核的辐射传输方程。研究了线性反演中必要的变量分离。发现迭代细化对于适应高温传感的强非线性是必要的。
