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：本文讨论了在精细网格上给定离散含噪点评估时，不适定积分方程在误差与计算成本方面的问题。标准求解方法通常采用由测量点直接导出的离散化方案，因此其计算规模可能随评估点数量增加而呈不利增长，导致计算效率低下。为缓解这一问题，我们提出了一种算法，能够在保持相同精度的同时显著降低计算成本。该方法通过初始平均化过程对底层网格进行稀疏化处理。为简化论述，本文仅聚焦于具有充分光滑性的一维不适定积分方程，但该方法经过适当修改可推广至更复杂的二维与三维问题。