Convergence analysis of the adjoint ensemble method in inverse source problems for advection-diffusion-reaction models with image-type measurements

Figure 1.  Exact source of $ NO $ a), $ NO $ concentration field in the final time moment b), $ O_{3} $ concentration field in the final time moment c)

Figure 2.  The relative error a) and the relative misfit b) for the different numbers $ \Xi $ of the projection functions

Figure 3.  Exact source of $ NO $ a), the projection of the initial guess error $ \mathbf{\bar{r} }^{(\ast)}-\mathbf{\bar{r}}^{(0)} $ to the orthogonal complement of $ Ker\mathbf{m_{\bar{U}}[\mathbf{\bar{r}}^{(\ast)}, \mathbf{\bar{r}}^{(\ast)}]} $ b), source identification result c)

Figure 4.  The relative error a) and the relative misfit b) for direct measurements, when the emitted substance concentration measurements are available ($ L_{src} = L_{meas} = \left\{ NO\right\} $) and indirect measurements, when other substance concentration measurements are available ($ L_{src} = \left\{ NO\right\} $, $ L_{meas} = \left\{ O_{3}\right\} $)

Figure 5.  The relative error a) and the relative misfit b) for Levenberg-Marquardt-type Algorithm 1 with $ \Theta = \Theta_{LM} $ (LM) and truncated SVD-based Algorithm 1 with $ \Theta = \Theta_{TSVD} $ (SVD)

Figure 6.  The relative error a) and the relative misfit b) with different noise levels $ \delta $

Figure 7.  Relative errors on the final iteration with different noise levels $ \delta $