Constrained Regularization for Ill-Posed Inverse Problems: Non-Negative Least Squares for PGNAA Spectra

Preprint of a manuscript intended for submission to Inverse Problems. Prompt gamma neutron activation analysis (PGNAA) is widely used for industrial elemental analysis. We consider the problem of reconstructing the true spectrum from the observed spectrum distorted by the detector response. Formally, g^δ = Tg + ε, where T is the response matrix. We show that T is severely ill-conditioned: its singular values decay over 16 orders of magnitude (cond(T) ≈ 9.2 × 10^16), and naive inversion produces oscillations with amplitude 10^18 for a signal of order 10^5. Classical regularization — Tikhonov and truncated SVD — stabilizes the solution but leaves it physically unacceptable: in about 20% of channels the reconstructed spectrum takes negative values. We modify the Tikhonov functional by adding the non-negativity constraint g ≥ 0, reducing the problem to non-negative least squares (NNLS) with Tikhonov regularization. The regularized NNLS solution has essentially the same residual as the unconstrained Tikhonov solution but contains no negative values. The method is tested on 192 two-hour intervals (16 days). The regularization parameter α_opt = 0.028 ± 0.002 is stable, the discrepancy ratio ||Tg_α - g^δ||/δ = 1.00 ± 0.02, and peak areas are preserved within 95–105% with respect to the observed spectrum, indicating that regularization does not introduce significant additional distortion beyond the apparatus function T. The method is robust to the choice of the response model T: the correlation of α_opt between two different FWHM models is 0.82. We conclude that for ill-posed inverse problems with physical constraints, regularization must be constrained: non-negativity is not post-processing but a part of the method.