# American Institute of Mathematical Sciences

October  2020, 14(5): 757-782. doi: 10.3934/ipi.2020035

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

 pr. Akademika Lavrentjeva 6, Institute of Computational Mathematics and Mathematical Geophysics SB RAS, Novosibirsk, 630090, Russia

* Corresponding author: Alexey Penenko

Received  May 2019 Revised  March 2020 Published  July 2020

Fund Project: The author is supported by RSF grant 17-71-10184 and State Program 0315-2019-0004

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.

Citation: Alexey Penenko. Convergence analysis of the adjoint ensemble method in inverse source problems for advection-diffusion-reaction models with image-type measurements. Inverse Problems and Imaging, 2020, 14 (5) : 757-782. doi: 10.3934/ipi.2020035
##### References:
 [1] World Meteorological Organization (WMO), Guide to instruments and methods of observation, Measurement of Meteorological Variables, 1 (2018), 506–541. [2] A. B. Bakushinskij, M. Kokurin and N. A. Yusupova, Iterative Newton-type methods with projecting for solution of nonlinear ill-posed operator equations, Sibirskiĭ Zhurnal Vychislitel'noĭ Matematiki, 5 (2002), 101–111. [3] A. F. Bennett, Inverse Methods in Physical Oceanography (Cambridge Monographs on Mechanics), Cambridge University Press, 1992. doi: 10.1017/CBO9780511600807. [4] M. Bocquet, Data assimilation in atmospheric chemistry models: Current status and future prospects for coupled chemistry meteorology models, Atmospheric Chemistry and Physics Discussions, 14 (2014), 32233-32323.  doi: 10.5194/acpd-14-32233-2014. [5] V. A. Cheverda and V. I. Kostin, R-pseudoinverses for compact operators in Hilbert spaces: Existence and stability, Journal of Inverse and Ill-Posed Problems, 3 (1995), 131-148.  doi: 10.1515/jiip.1995.3.2.131. [6] F.-X. L. Dimet, I. Souopgui, O. Titaud, V. Shutyaev and M. Y. Hussaini, Toward the assimilation of images, Nonlinear Processes in Geophysics, 22 (2015), 15-32.  doi: 10.5194/npg-22-15-2015. [7] H. Elbern, A. Strunk, H. Schmidt and O. Talagrand, Emission rate and chemical state estimation by 4-dimensional variational inversion, Atmospheric Chemistry and Physics Discussions, 7 (2007), 1725-1783.  doi: 10.5194/acpd-7-1725-2007. [8] H. Engl, M. Hanke and A. Neubauer, Regularization of Inverse Problems, Kluwer, Dordrecht, 1996. [9] G. Evensen, Sequential data assimilation with a nonlinear quasi-geostrophic model using {M}onte Carlo methods to forecast error statistics, Journal of Geophysical Research, 99 (1994), 10143-10162.  doi: 10.1029/94JC00572. [10] J. Fischer, Global existence of renormalized solutions to entropy-dissipating reaction-diffusion systems, Archive for Rational Mechanics and Analysis, 218 (2015), 553-587.  doi: 10.1007/s00205-015-0866-x. [11] J. Fischer, Weak-strong uniqueness of solutions to entropy-dissipating reaction-diffusion equations, Nonlinear Analysis, 159 (2017), 181-207.  doi: 10.1016/j.na.2017.03.001. [12] J. J. Guerrette and D. K. Henze, Development and application of the WRFPLUS-chem online chemistry adjoint and WRFDA-chem assimilation system, Geoscientific Model Development, 8 (2015), 1857-1876.  doi: 10.5194/gmd-8-1857-2015. [13] J. Herman, A. Cede, E. Spinei, G. Mount, M. Tzortziou and N. Abuhassan, NO2column amounts from ground-based pandora and MFDOAS spectrometers using the direct-sun DOAS technique: Intercomparisons and application to OMI validation, Journal of Geophysical Research, 114 (2009), 1-20.  doi: 10.1029/2009JD011848. [14] M. A. Iglesias and C. Dawson, An iterative representer-based scheme for data inversion in reservoir modeling, Inverse Problems, 25 (2009), 1-34.  doi: 10.1088/0266-5611/25/3/035006. [15] J.-P. Issartel, Rebuilding sources of linear tracers after atmospheric concentration measurements, Atmospheric Chemistry and Physics, 3 (2003), 2111-2125.  doi: 10.5194/acp-3-2111-2003. [16] L. M. Judd, et al., The dawn of geostationary air quality monitoring: Case studies from Seoul and Los Angeles, Frontiers in Environmental Science, 6 (2018), 85 pp. doi: 10.3389/fenvs.2018.00085. [17] B. Kaltenbacher, A. Neubauer and O. Scherzer, Iterative Regularization Methods for Nonlinear Ill-Posed Problems, De Gruyter, Berlin, Germany, 2008. doi: 10.1515/9783110208276. [18] L. V. Kantorovich and G. P. Akilov, Functional Analysis, Pergamon Press, 1982. [19] A. L. Karchevsky, Reformulation of an inverse problem statement that reduces computational costs, Eurasian Journal Of Mathematical And Computer Applications, 1 (2013), 4-20.  doi: 10.32523/2306-3172-2013-1-2-4-20. [20] P. Laj, Measuring atmospheric composition change, Atmospheric Environment, 43 (2009), 5351-5414.  doi: 10.1016/j.atmosenv.2009.08.020. [21] A. V. Mamonov and Y.-H. R. Tsai, Point source identification in nonlinear advection-diffusion-reaction systems, Inverse Problems, 29 (2013), 1-26.  doi: 10.1088/0266-5611/29/3/035009. [22] G. Marchuk, On the formulation of some inverse problems, Reports of the USSR Academy of Sciences / Ed. Science, 156 (1964), 503-506. [23] K. Markakis, M. Valari, O. Perrussel, O. Sanchez and C. Honore, Climate-forced air-quality modeling at the urban scale: Sensitivity to model resolution, emissions and meteorology, Atmospheric Chemistry and Physics, 15 (2015), 7703-7723.  doi: 10.5194/acp-15-7703-2015. [24] A. Merlaud, The small whiskbroom imager for atmospheric compositioN monitorinG (SWING) and its operations from an unmanned aerial vehicle (UAV) during the AROMAT campaign, Atmospheric Measurement Techniques, 11 (2018), 551-567.  doi: 10.5194/amt-11-551-2018. [25] J. M. Ortega and W. C. Rheinboldt, Iterative Solution of Nonlinear Equations in Several Variables, Academic Press, New York, 1970. [26] A. V. Penenko, Consistent numerical schemes for solving nonlinear inverse source problems with gradient-type algorithms and Newton-Kantorovich methods, Numerical Analysis and Applications, 11 (2018), 73-88.  doi: 10.1134/S1995423918010081. [27] A. Penenko, S. Nikolaev, S. Golushko, A. Romashenko and I. Kirilova, Numerical algorithms for diffusion coefficient identification in problems of tissue engineering, Mathematical Biology and Bioinformatics, 11 (2016), 426-444.  doi: 10.17537/2016.11.426. [28] A. Penenko, V. Penenko, E. Tsvetova and Z. Mukatova, Consistent discrete-analytical schemes for the solution of the inverse source problems for atmospheric chemistry models with image-type measurement data, in Finite Difference Methods. Theory and Applications, Springer International Publishing, 2019,378–386. doi: 10.1007/978-3-030-11539-5_43. [29] A. Penenko, U. Zubairova, Z. Mukatova and S. Nikolaev, Numerical algorithm for morphogen synthesis region identification with indirect image-type measurement data, Journal of Bioinformatics and Computational Biology, 17 (2019), 1940002-1–1940002-18. doi: 10.1142/S021972001940002X. [30] A. Penenko, A Newton-Kantorovich method in inverse source problems for production-destruction models with time series-type measurement data, Numerical Analysis and Applications, 12 (2019), 51-69.  doi: 10.1134/S1995423919010051. [31] V. V. Penenko, A. V. Penenko, E. A. Tsvetova and A. V. Gochakov, Methods for studying the sensitivity of air quality models and inverse problems of geophysical hydrothermodynamics, Journal of Applied Mechanics and Technical Physics, 60 (2019), 392-399.  doi: 10.1134/S0021894419020202. [32] V. Penenko, E. Tsvetova and A. Penenko, Variational approach and Euler's integrating factors for environmental studies, Computers & Mathematics with Applications, 67 (2014), 2240–2256. doi: 10.1016/j.camwa.2014.04.004. [33] M. Schaap, M. Roemer, F. Sauter, G. Boersen, R. Timmermans and P. Builtjes, LOTOS-EUROS Documentation, Techreport B & O 2005/297, TNO report, 2005. [34] W. Stockwell and W. Goliff, Comment on "Simulation of a reacting pollutant puff using an adaptive grid algorithm" by R.K. Srivastava et al., Journal of Geophysical Research, 107 (2002), 4643-4650.  doi: 10.1029/2002jd002164. [35] G. M. Vainikko and A. Y. Veretennikov, Iterative procedures in ill-posed problems, Nauka, Moscow, 1986. [36] V. V. Vasin, Modified Newton-type processes generating fejér approximations of regularized solutions to nonlinear equations, Proceedings of the Steklov Institute of Mathematics, 284 (2014), 145-158.  doi: 10.1134/s0081543814020138. [37] A. D. Visscher, Air Dispersion Modeling: Foundations and Applications, John Wiley & Sons Inc., 2013. doi: 10.1002/9781118723098.

show all references

##### References:
 [1] World Meteorological Organization (WMO), Guide to instruments and methods of observation, Measurement of Meteorological Variables, 1 (2018), 506–541. [2] A. B. Bakushinskij, M. Kokurin and N. A. Yusupova, Iterative Newton-type methods with projecting for solution of nonlinear ill-posed operator equations, Sibirskiĭ Zhurnal Vychislitel'noĭ Matematiki, 5 (2002), 101–111. [3] A. F. Bennett, Inverse Methods in Physical Oceanography (Cambridge Monographs on Mechanics), Cambridge University Press, 1992. doi: 10.1017/CBO9780511600807. [4] M. Bocquet, Data assimilation in atmospheric chemistry models: Current status and future prospects for coupled chemistry meteorology models, Atmospheric Chemistry and Physics Discussions, 14 (2014), 32233-32323.  doi: 10.5194/acpd-14-32233-2014. [5] V. A. Cheverda and V. I. Kostin, R-pseudoinverses for compact operators in Hilbert spaces: Existence and stability, Journal of Inverse and Ill-Posed Problems, 3 (1995), 131-148.  doi: 10.1515/jiip.1995.3.2.131. [6] F.-X. L. Dimet, I. Souopgui, O. Titaud, V. Shutyaev and M. Y. Hussaini, Toward the assimilation of images, Nonlinear Processes in Geophysics, 22 (2015), 15-32.  doi: 10.5194/npg-22-15-2015. [7] H. Elbern, A. Strunk, H. Schmidt and O. Talagrand, Emission rate and chemical state estimation by 4-dimensional variational inversion, Atmospheric Chemistry and Physics Discussions, 7 (2007), 1725-1783.  doi: 10.5194/acpd-7-1725-2007. [8] H. Engl, M. Hanke and A. Neubauer, Regularization of Inverse Problems, Kluwer, Dordrecht, 1996. [9] G. Evensen, Sequential data assimilation with a nonlinear quasi-geostrophic model using {M}onte Carlo methods to forecast error statistics, Journal of Geophysical Research, 99 (1994), 10143-10162.  doi: 10.1029/94JC00572. [10] J. Fischer, Global existence of renormalized solutions to entropy-dissipating reaction-diffusion systems, Archive for Rational Mechanics and Analysis, 218 (2015), 553-587.  doi: 10.1007/s00205-015-0866-x. [11] J. Fischer, Weak-strong uniqueness of solutions to entropy-dissipating reaction-diffusion equations, Nonlinear Analysis, 159 (2017), 181-207.  doi: 10.1016/j.na.2017.03.001. [12] J. J. Guerrette and D. K. Henze, Development and application of the WRFPLUS-chem online chemistry adjoint and WRFDA-chem assimilation system, Geoscientific Model Development, 8 (2015), 1857-1876.  doi: 10.5194/gmd-8-1857-2015. [13] J. Herman, A. Cede, E. Spinei, G. Mount, M. Tzortziou and N. Abuhassan, NO2column amounts from ground-based pandora and MFDOAS spectrometers using the direct-sun DOAS technique: Intercomparisons and application to OMI validation, Journal of Geophysical Research, 114 (2009), 1-20.  doi: 10.1029/2009JD011848. [14] M. A. Iglesias and C. Dawson, An iterative representer-based scheme for data inversion in reservoir modeling, Inverse Problems, 25 (2009), 1-34.  doi: 10.1088/0266-5611/25/3/035006. [15] J.-P. Issartel, Rebuilding sources of linear tracers after atmospheric concentration measurements, Atmospheric Chemistry and Physics, 3 (2003), 2111-2125.  doi: 10.5194/acp-3-2111-2003. [16] L. M. Judd, et al., The dawn of geostationary air quality monitoring: Case studies from Seoul and Los Angeles, Frontiers in Environmental Science, 6 (2018), 85 pp. doi: 10.3389/fenvs.2018.00085. [17] B. Kaltenbacher, A. Neubauer and O. Scherzer, Iterative Regularization Methods for Nonlinear Ill-Posed Problems, De Gruyter, Berlin, Germany, 2008. doi: 10.1515/9783110208276. [18] L. V. Kantorovich and G. P. Akilov, Functional Analysis, Pergamon Press, 1982. [19] A. L. Karchevsky, Reformulation of an inverse problem statement that reduces computational costs, Eurasian Journal Of Mathematical And Computer Applications, 1 (2013), 4-20.  doi: 10.32523/2306-3172-2013-1-2-4-20. [20] P. Laj, Measuring atmospheric composition change, Atmospheric Environment, 43 (2009), 5351-5414.  doi: 10.1016/j.atmosenv.2009.08.020. [21] A. V. Mamonov and Y.-H. R. Tsai, Point source identification in nonlinear advection-diffusion-reaction systems, Inverse Problems, 29 (2013), 1-26.  doi: 10.1088/0266-5611/29/3/035009. [22] G. Marchuk, On the formulation of some inverse problems, Reports of the USSR Academy of Sciences / Ed. Science, 156 (1964), 503-506. [23] K. Markakis, M. Valari, O. Perrussel, O. Sanchez and C. Honore, Climate-forced air-quality modeling at the urban scale: Sensitivity to model resolution, emissions and meteorology, Atmospheric Chemistry and Physics, 15 (2015), 7703-7723.  doi: 10.5194/acp-15-7703-2015. [24] A. Merlaud, The small whiskbroom imager for atmospheric compositioN monitorinG (SWING) and its operations from an unmanned aerial vehicle (UAV) during the AROMAT campaign, Atmospheric Measurement Techniques, 11 (2018), 551-567.  doi: 10.5194/amt-11-551-2018. [25] J. M. Ortega and W. C. Rheinboldt, Iterative Solution of Nonlinear Equations in Several Variables, Academic Press, New York, 1970. [26] A. V. Penenko, Consistent numerical schemes for solving nonlinear inverse source problems with gradient-type algorithms and Newton-Kantorovich methods, Numerical Analysis and Applications, 11 (2018), 73-88.  doi: 10.1134/S1995423918010081. [27] A. Penenko, S. Nikolaev, S. Golushko, A. Romashenko and I. Kirilova, Numerical algorithms for diffusion coefficient identification in problems of tissue engineering, Mathematical Biology and Bioinformatics, 11 (2016), 426-444.  doi: 10.17537/2016.11.426. [28] A. Penenko, V. Penenko, E. Tsvetova and Z. Mukatova, Consistent discrete-analytical schemes for the solution of the inverse source problems for atmospheric chemistry models with image-type measurement data, in Finite Difference Methods. Theory and Applications, Springer International Publishing, 2019,378–386. doi: 10.1007/978-3-030-11539-5_43. [29] A. Penenko, U. Zubairova, Z. Mukatova and S. Nikolaev, Numerical algorithm for morphogen synthesis region identification with indirect image-type measurement data, Journal of Bioinformatics and Computational Biology, 17 (2019), 1940002-1–1940002-18. doi: 10.1142/S021972001940002X. [30] A. Penenko, A Newton-Kantorovich method in inverse source problems for production-destruction models with time series-type measurement data, Numerical Analysis and Applications, 12 (2019), 51-69.  doi: 10.1134/S1995423919010051. [31] V. V. Penenko, A. V. Penenko, E. A. Tsvetova and A. V. Gochakov, Methods for studying the sensitivity of air quality models and inverse problems of geophysical hydrothermodynamics, Journal of Applied Mechanics and Technical Physics, 60 (2019), 392-399.  doi: 10.1134/S0021894419020202. [32] V. Penenko, E. Tsvetova and A. Penenko, Variational approach and Euler's integrating factors for environmental studies, Computers & Mathematics with Applications, 67 (2014), 2240–2256. doi: 10.1016/j.camwa.2014.04.004. [33] M. Schaap, M. Roemer, F. Sauter, G. Boersen, R. Timmermans and P. Builtjes, LOTOS-EUROS Documentation, Techreport B & O 2005/297, TNO report, 2005. [34] W. Stockwell and W. Goliff, Comment on "Simulation of a reacting pollutant puff using an adaptive grid algorithm" by R.K. Srivastava et al., Journal of Geophysical Research, 107 (2002), 4643-4650.  doi: 10.1029/2002jd002164. [35] G. M. Vainikko and A. Y. Veretennikov, Iterative procedures in ill-posed problems, Nauka, Moscow, 1986. [36] V. V. Vasin, Modified Newton-type processes generating fejér approximations of regularized solutions to nonlinear equations, Proceedings of the Steklov Institute of Mathematics, 284 (2014), 145-158.  doi: 10.1134/s0081543814020138. [37] A. D. Visscher, Air Dispersion Modeling: Foundations and Applications, John Wiley & Sons Inc., 2013. doi: 10.1002/9781118723098.
Exact source of $NO$ a), $NO$ concentration field in the final time moment b), $O_{3}$ concentration field in the final time moment c)
The relative error a) and the relative misfit b) for the different numbers $\Xi$ of the projection functions
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)
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 ($L_{src} = \left\{ NO\right\}$, $L_{meas} = \left\{ O_{3}\right\}$)
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)
The relative error a) and the relative misfit b) with different noise levels $\delta$
Relative errors on the final iteration with different noise levels $\delta$
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
 $\delta$ 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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