A Particle Method for Solving Fredholm Equations of the First Kind

Abstract

Fredholm integral equations of the first kind are the prototypical example of ill-posed linear inverse problems. They model, among other things, reconstruction of distorted noisy observations and indirect density estimation and also appear in instrumental variable regression. However, their numerical solution remains a challenging problem. Many techniques currently available require a preliminary discretization of the domain of the solution and make strong assumptions about its regularity. For example, the popular expectation-maximization smoothing (EMS) scheme requires the assumption of piecewise constant solutions which is inappropriate for most applications. We propose here a novel particle method that circumvents these two issues. This algorithm can be thought of as a Monte Carlo approximation of the EMS scheme which not only performs an adaptive stochastic discretization of the domain but also results in smooth approximate solutions. We analyze the theoretical properties of the EMS iteration and of the corresponding particle algorithm. Compared to standard EMS, we show experimentally that our novel particle method provides state-of-the-art performance for realistic systems, including motion deblurring and reconstruction of cross-sectional images of the brain from positron emission tomography.

Neural-network–based algorithm for the inverse problem of measuring K-shell ionization cross-sections of Si induced by 3–25 keV electrons and 4.5–9 keV positrons using the thick-target method

“…Therefore, the thick-target method is commonly employed to study atomic inner-shell ionization cross-sections induced by electron or position impacts. The formula employed for calculating atomic inner-shell ionization cross-sections from characteristic X-ray yields [6] falls within the category of class-I Fredholm integral equations, which represents an ill-posed inverse problem [7]. Several methods have been proposed to address this issue, including the direct comparison [8,9], yield differential [4,10,11], and Tikhonov regularization methods [5,6,12,13].…”