Tikhonov regularization for kinetic inverse source problems

Abstract

Reconstructing a spatial source in stationary kinetic transport equations is a complex inverse problem complicated by grazing boundary singularities. We introduce a zero-extension framework for boundary-vanishing sources that bypasses these singularities without artificial compact-support constraints. This yields a robust forward mapping with exact gradient trace control. To stably invert noisy boundary measurements, we formulate a variational Tikhonov regularization strategy governed by Morozov's discrepancy principle. Relying solely on the bounded linearity and injectivity of the forward operator, we analyze the regularization scheme and show that the approximations converge strongly to the exact source.