This paper presents a novel approach to construct regularizing operators for severely ill-posed Fredholm integral equations of the first kind by introducing parametrized discretization. The optimal values of discretization and regularization parameters are computed simultaneously by solving a minimization problem formulated based on a regularization parameter search criterion. The effectiveness of the proposed approach is demonstrated through examples of noisy Laplace transform inversions and the deconvolution of nuclear magnetic resonance relaxation data.

翻译：本文提出一种通过引入参数化离散化构造严重病态第一类Fredholm积分方程正则化算子的新方法。基于正则化参数搜索准则构建极小化问题，同步求解离散化参数与正则化参数的最优值。通过含噪拉普拉斯变换反演与核磁共振松弛数据反卷积实例，验证了所提方法的有效性。