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噪声点评估值时，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衍生的离散化方案，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。针对该问题，我们提出一种在保持同等精度的同时显著降低计算成本的算法。我们的方法通过初始平均流程对底层网格进行稀疏化处理。为简化阐述，本文仅聚焦于具有充分光滑性的一维不适定积分方程。然而，该方法经适当修改后可推广至更复杂的二维和三维问题。