Identification of Localized Pollution Sources in a Two-Dimensional Diffusion--Reaction Equation via the Conjugate Gradient Method with Iterative Regularization

Konstantin Pronin, Olga Krivorotko

Keywords: inverse source problem, diffusion-reaction equation, conjugate gradient method, iterative regularization, Morozov discrepancy principle, adjoint equations, Tikhonov regularization, spectral analysis, ill-posed problems, pollution source identification

TL;DR: Adjoint-based conjugate gradient method with Morozov early stopping recovers time-dependent pollution sources from noisy sensor data, matching optimally tuned Tikhonov regularization without parameter selection.

Abstract: The inverse problem of recovering time-dependent intensities of localized pollution sources from pointwise concentration measurements is considered. The forward model is a two-dimensional parabolic diffusion--reaction equation with homogeneous Neumann boundary conditions and a nonzero initial condition representing pre-existing contamination. Point sources are approximated by bilinear distributions on the computational grid, placing the problem within the standard L2-framework. Discrete adjoint equations are derived via the Lagrangian (discretize-then-optimize), yielding an explicit gradient requiring one forward and one adjoint solve. The inverse problem is solved by the Polak--Ribiere conjugate gradient method with exact line search; regularization is achieved by early stopping via the Morozov discrepancy principle. Spectral analysis of the discrete observation operator (σ_k ~ k^{-4.7}, condition number ~10^{15}, effective rank 16) quantitatively explains the reconstruction errors and the effectiveness of early stopping. Numerical experiments with 1--10% multiplicative noise confirm that the discrepancy principle terminates iterations correctly (ρ/δ ∈ [1.00, 1.07]), yielding errors of 36--45%. Iterative regularization matches optimally tuned Tikhonov regularization without parameter selection.

Submission Number: 136