Inverse Problems and Imaging with Nonlinear Physics

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This project aims at studying imaging and inverse problems where nonlinear effects in physics play significant roles. Nonlinear physics could either help us image objects that could not be imaged with only linear physics or prevent us from imaging objects that we would have been able to image in the absence of such nonlinear effects. In either case, understanding the impact of such nonlinear effects would help us improve the quality of the underlying imaging modalities. The goal of the project is to perform detailed mathematical and numerical analysis to characterize the effects of nonlinearity on image reconstructions in such cases.The main research efforts in this project include, but are not limited to: (i) developing efficient reconstruction algorithms for imaging through nonlinear dispersive media; (ii) developing mathematical theory and computational methods for imaging two-photon absorption and second harmonic generation in optics; and (iii) analyzing quantitative photoacoustics with nonlinear absorption and developing corresponding computational image reconstruction algorithms. The mathematical ideas developed in this project are expected to provide insights on future studies of similar imaging and inverse problems related to nonlinear partial differential equations. The research is also expected to have impact on the development of practically useful methods for imaging through nonlinear media as well as imaging nonlinear physical properties of given media. This project involves an integrated educational component that aims at training advanced undergraduate, master and PhD students.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.