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：本文探讨了在精细网格上给定离散噪声点评估时，不适定积分方程的误差与成本问题。标准求解方法通常采用由测量点直接诱导的离散化方案，因此其计算规模可能随评估点数量增加而劣化，导致计算效率低下。为解决该问题，我们提出了一种算法，在保持相同精度水平的同时显著降低计算成本。该方法通过初始平均化过程稀疏化底层网格。为简化论述，本文仅聚焦于具有足够光滑性的一维不适定积分方程，但该方法可通过适当修正推广至更复杂的二维与三维问题。