Solving Fredholm integral equation of the first kind using Gaussian process regression

Renjun Qiu, Liang Yan and Xiaojun Duan

Applied Mathematics and Computation, 2022, vol. 425, issue C

Abstract: Fredholm integral equation of the first kind is a typical ill-posed problem, and it is usually difficult to obtain a stable numerical solution. In this paper, a new method is proposed to solve Fredholm integral equation using Gaussian process regression (GPR). The key to this method is that the right-hand term of the original integral equation is reconstructed by the GPR model to obtain a new integral equation in a reproducing kernel Hilbert spaces (RKHS). We present an analytical approximate solution of the new equation and prove that it converges to the exact minimal-norm solution of the original equation under the L2-norm. Especially, for the degenerate kernel equation, we obtain an explicit formula of the exact minimal-norm solution. Finally, the proposed method is verified to be very effective in solution accuracy by multiple examples.

Keywords: Fredholm integral equation of the first kind; Degenerate kernel; Ill-posed problem; Gaussian process regression; Moore-Penrose inverse; Reproducing kernel Hilbert spaces (search for similar items in EconPapers)
Date: 2022
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Citations: View citations in EconPapers (1)

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Persistent link: https://EconPapers.repec.org/RePEc:eee:apmaco:v:425:y:2022:i:c:s0096300322001187

DOI: 10.1016/j.amc.2022.127032

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