Abstract: In this talk we present a survey of recent developments of direct sampling type methods (DSMs) for solving general linear and nonlinear inverse problems of partial differential equations. The sampling type methods were proposed earlier for solving inverse acoustic scattering problems with far-field or near-field data [1-2], then developed for inverse electromagnetic scattering problems [3], and further extended for several representative non-wave type inverse problems, including electric impedance tomography[4], diffusive optical tomography [5], inversion of Radon transform [6], as well as recovering moving inhomogeneous inclusions [7]. DSMs have also been demonstarted recently to be applicable to the simultaneous reconstruction of inhomogeneous inclusions of different physical nature [8]. The DSMs are computationally cheap, highly parallel, and robust against noise, particularly applicable to the cases when very limited data is available. Motivations, principles and justifications of DSMs are addressed in this survey talk. Numerical experiments are demonstrated for various inverse problems. There are intensive studies of this type of numerical methods, and some of the references for those most representative inverse problems are listed below.

These research projects were supported by Hong Kong RGC General Research Fund (Projects 14306921 and 14306719).

References
[1] Kazufumi Ito, Bangti Jin and Jun Zou, A direct sampling method to an inverse medium scattering problem, Inverse Problems, 28, 025003 (2012).
[2] Roland Potthast, A study on orthogonality sampling, Inverse Problems, 26, 074015 (2010).
[3] Kazufumi Ito, Bangti Jin and Jun Zou, A direct sampling method for inverse electromagnetic medium scattering, Inverse Problems 29, 095018 (2013).
[4] Yat Tin Chow, Kazufumi Ito and Jun Zou, A direct sampling method for electrical impedance tomography, Inverse Problems 30, 095003 (2014).
[5] Yat Tin Chow, Kazufumi Ito, Keji Liu and Jun Zou, Direct sampling method for diffusive optical tomography, SIAM J. Sci. Comput. 37, A1658-A1684 (2015).
[6] Yat Tin Chow, Fuqun Han and Jun Zou, A direct sampling method for the inversion of the Radon transform, SIAM J. Imaging Sci. 14, 1004-1038 (2021).
[7] Yat Tin Chow, Kazufumi Ito and Jun Zou, A time-dependent direct sampling method for recovering moving potentials in a heat equation, SIAM J. Sci. Comput. 40, A2720-A2748 (2018).
[8] Yat Tin Chow, Fuqun Han and Jun Zou, A direct sampling method for simultaneously recovering inhomogeneous inclusions of different nature, SIAM J. Sci. Comput., 43, A2161-A2189 (2021).

Biography: Jun Zou is currently Choh-Ming Li Chair Professor of Mathematics of The Chinese University of Hong Kong, and Head of Department of Mathematics. Before taking up his position at The Chinese University of Hong Kong in 1995, he had worked two years (93-95) in University of California at Los Angeles as a post-doctoral fellow and a CAM Assistant Professor, worked two and a half years (91-93) in Technical University of Munich as a Visiting Assistant Professor and an Alexander von Humboldt Research Fellow (Germany), and worked two years (89-91) in Chinese Academy of Sciences (Beijing) as an Assistant Professor. Jun Zou's research interests include numerical methods and analyses of direct and inverse problems of partial differential equations. He is currently vice president of Hong Kong Mathematical Society and president of East Asia Section of Inverse Problems International Association. He serves currently as the associate editor of 12 international mathematics journals, including SIAM Journal on Numerical Analysis, SIAM Journal on Scientific Computing and ESAIM: Mathematical Modelling and Numerical Analysis. Jun Zou was elected a SIAM Fellow in 2019 and an AMS Fellow in 2022.