In this talk, I will present some recent results on solving inverse boundary value problems for nonlinear PDEs, especially for a time-dependent Schrodinger equation with time-dependent potentials with partial boundary Dirichlet-to-Neumann map. After a higher order linearization step, the problem will be reduced to implementing special geometrical optics (GO) solutions to prove the uniqueness and stability of the reconstruction. This is a joint work with my PhD student Xuezhu Lu and Prof. Ru-Yu Lai.

Atmospheric pollution prevention and control is an important global issue[1]. Observing the emission of harmful trace gases and their atmospheric transport dynamics on a global scale is of great significance for in-depth study of major problems, such as climate change and ecological and environmental change[2, 3]. In recent years, the inverse problem of atmospheric contaminant source intensity distribution has attracted more and more attention from researchers[4]. Due to its mathematical ill-posedness and high computational costs, it is necessary to develop new computational tools[5]. Accurate, rapid, and stable inverse analysis of atmospheric contaminant source intensity distribution, and subsequently using high-resolution numerical simulation methods to predict the local or large-scale, short-term or long-term atmospheric environmental impacts caused by major sudden natural disasters and industrial pollution events, has important scientific significance and practical value.

By using limited information of satellite observation data obtained through NIS (Nonlinear Inverse Scattering) technology[7, 8, 9], we have established a high-throughput parallel computing framework for solving the mathematical and physical inverse problem of high-resolution spatiotemporal atmospheric contaminant sources distribution[10, 11, 12]. A real-time inverse analysis and transport simulation of atmospheric contaminant source intensity distribution with high resolution, stability, and reliability are realized. And thus the high-resolution reanalysis data and prediction information on a global scale are available, which can not be directly obtained by current satellite or optical radar measurement technologies. Considering the influence of complex physical and chemical processes, such as the transport of particles in wind fields and the scattering of particles due to light irradiation, the relationship between unknown source parameters and observed contaminant concentrations is usually nonlinear[2, 5, 7, 13, 14, 15]. Therefore, we comprehensively use numerical simulation, optimization methods, and statistical inference techniques[6, 16, 17]. Taking volcanic eruption and forest fire as examples, based on remote sensing data, we use the jointly developed Lagrangian transport model MPTRAC (Massive Parallel Trajectory Calculation) for forward simulation[11]. Combined with heuristic methods such as segmented strong constraint "product rule" proposed by us, the computational bottleneck of traditional serial regularization methods for solving such inverse problems is overcome[10, 12]. And a million-core supercomputing-based inverse calculation strategy is developed, which greatly reduces time costs while ensuring accuracy and reliability, meeting the needs of future real-time prediction tools.

This study provides practical application scenarios for NIS technology, and plays an important theoretical and practical role in ensuring aviation safety, exploring the mechanism of pollutant degradation, and revealing the causes of global climate change.

[1] N.-N. Zhang et al. Spatiotemporal trends in PM2. 5 levels from 2013 to 2017 and regional demarcations for joint prevention and control of atmospheric pollution in China, Chemosphere 210: 1176-1184, 2018.

[2] S. Huang et al. Inverse problems in atmospheric science and their application, Journal of Physics: Conference Series, IOP Publishing, 2005.

[3] W. Freeden et al. Handbook of geomathematics, 2nd edition. Springer Berlin Heidelberg, 2015.

[4] M. S. Zhdanov. Inverse Theory and Applications in Geophysics, 2nd edition. Elsevier Science, 2015.

[5] J. L. Mueller, S. Siltanen. Linear and nonlinear inverse problems with practical applications, SIAM, 2012.

[6] Y. Bai et al. Computational methods for applied inverse problems, Walter de Gruyter, 2012.

[7] D. Efremenko, A. Kokhanovsky. Foundations of Atmospheric Remote Sensing, Springer, 2021.

[8] W. C. Chew et al. Nonlinear diffraction tomography: The use of inverse scattering for imaging, Int J Imaging Syst Technol. 7(1):16-24, 1996.

[9] T. Hasegawa, T. Iwasaki. Microwave imaging by quasi-inverse scattering, Electron Comm Jpn Pt I. 87(5):52-61, 2004.

[10] Y. Heng et al. Inverse transport modeling of volcanic sulfur dioxide emissions using large-scale simulations, Geoscientific Model Development 9(4): 1627-1645, 2016.

[11] L. Hoffmann et al. Massive-Parallel Trajectory Calculations version 2.2 (MPTRAC-2.2): Lagrangian transport simulations on graphics processing units (GPUs), Geoscientific Model Development 15(7): 2731-2762, 2022.

[12] M. Liu et al. High-Resolution Source Estimation of Volcanic Sulfur Dioxide Emissions Using Large-Scale Transport Simulations, Computational Science – ICCS 2020: 60-73, 2020.

[13] K. Chadan et al. An introduction to inverse scattering and inverse spectral problems, Society for Industrial and Applied Mathematics, 1997.

[14] D. P. Winebrenner, J. Sylvester. Linear and nonlinear inverse scattering, SIAM Journal on Applied Mathematics, 59(2): 669-699, 1998.

[15] V. Isakov. Inverse problems for partial differential equations, Springer, 2006.

[16] E. Haber et al. On optimization techniques for solving nonlinear inverse problems, Inverse problems, 6(5): 1263, 2000.

[17] R. C. Aster et al. Parameter estimation and inverse problems, Elsevier, 2018.

This talk studies Carleman estimates and their applications for inverse problems of stochastic partial differential equations. By establishing new Carleman estimates for the stochastic parabolic and hyperbolic systems, conditional stability for inverse problems of these systems are proven. Based on the idea of Tikhonov method, regularized solutions are proposed. The analysis of the existence and uniqueness of the regularized solutions, and proofs for error estimate under an a priori assumption are presented. Numerical verification of the regularization, based on the idea of kernel- based learning method, including numerical algorithms and examples are also illustrated.

Inverse scattering problems aim to determine unknown scatterers with wave fields measured around the scatterers. However, from the practical point of views, we have only limited information, e.g., limited aperture data phaseless data and sparse data. In this talk, we introduce some data retrieval techniques and the applications in the inverse scattering problems. The theoretical and numerical methods for inverse scattering problems with multi-frequency spase measurements will also be mentioned.