This paper discusses the error and cost aspects of ill-posed integral equations when given discrete noisy point evaluations on a fine grid. Standard solution methods usually employ discretization schemes that are directly induced by the measurement points. Thus, they may scale unfavorably with the number of evaluation points, which can result in computational inefficiency. To address this issue, we propose an algorithm that achieves the same level of accuracy while significantly reducing computational costs. Our approach involves an initial averaging procedure to sparsify the underlying grid. To keep the exposition simple, we focus only on one-dimensional ill-posed integral equations that have sufficient smoothness. However, the approach can be generalized to more complicated two- and three-dimensional problems with appropriate modifications.

翻译：本文讨论了当在精细网格上给定离散含噪点值评估时，不适定积分方程的误差与计算成本问题。标准求解方法通常采用由测量点直接导出的离散化方案。因此，这些方法的计算复杂度可能随评估点数量呈不利增长，从而导致计算效率低下。为解决此问题，我们提出一种算法，在保持相同精度水平的同时显著降低计算成本。我们的方法包含一个初始平均过程以稀疏化底层网格。为简化论述，我们仅关注具有充分光滑性的一维不适定积分方程。然而，该方法可通过适当修改推广至更复杂的二维和三维问题。