Efficient solution of ill-posed integral equations through averaging
Michael Griebel, Tim Jahn
TL;DR
This work introduces an averaging-based data compression approach to efficiently solve ill-posed Fredholm integral equations from noisy point evaluations, achieving the same regularization accuracy on a coarser grid while reducing computational costs. For one-dimensional, smooth problems, the authors derive explicit error bounds showing that averaging can preserve the optimal rate $(\delta^2/m)^{4s/(5+4s)}$ up to discretization error, and they extend the method to general kernels via quadrature with comparable guarantees. A key contribution is the demonstration that a coarse, averaged design $m_o$ suffices for accurate reconstruction when paired with a carefully chosen truncation level $k$, yielding substantial cost savings. The paper also develops adaptive discrepancy-principle schemes across multiple discretization levels, validates the theory with numerical experiments on a Poisson-type kernel and classical inverse problems, and outlines clear avenues for extension to higher dimensions and alternative regularization methods.
Abstract
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.