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

  • * Corresponding author: Alexey Penenko

    * Corresponding author: Alexey Penenko
The author is supported by RSF grant 17-71-10184 and State Program 0315-2019-0004
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  • The inverse source problems for nonlinear advection-diffusion-reaction models with image-type measurement data are considered. The use of the sensitivity operators, constructed of the ensemble of adjoint problem solutions, allows transforming the inverse problems stated as the systems of nonlinear PDE to a family of operator equations depending on the given set of functions in the space of measurement results. The tangential cone conditions for the resulting operator equations are studied. Newton-Kantorovich type methods are applied for the solution of the operator equations. The algorithms are numerically evaluated on an inverse source problem of atmospheric chemistry.

    Mathematics Subject Classification: Primary: 47J06; Secondary: 35R30.


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  • 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 $

    Table 1.  The relative measurement error levels $ J_{m}(\delta) $ for different $ \delta $

    $ \delta $ 0 0.005 0.01 0.02 0.05 0.1
    $ J_{m}(\delta) $ 0 0.0028 0.0057 0.011 0.028 0.057
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