Electrical Impedance Tomography (EIT) is a nonlinear PDE-based imaging modality where a patient is probed with harmless electric currents, and the resulting surface voltages are measured. EIT image reconstruction is an ill-posed inverse problem, meaning very sensitive to noise in the data and modelling errors. However, one can use complex geometric optics (CGO) solutions and a nonlinear Fourier transform to do robust medical imaging; this is the so-called regularized D-bar method. A connection between EIT and X-ray tomography was found in [Greenleaf et al. 2018] using microlocal analysis. Fourier transform applied to the spectral parameter of CGO solutions produces virtual X-ray projections, enabling a novel filtered back-projection type nonlinear reconstruction algorithm for EIT. It is remarkable how this new approach decomposes the EIT image reconstruction process in several steps, where all ill-posedness is confined in two linear steps. Therefore, we can separate the nonlinearity and ill-posedness of the fundamental EIT problem. Furthermore, the new decomposition enables targeted machine learning approaches as only one or two (mathematically well-structured) steps in the imaging chain are solved using neural networks.

本講演では，リーマン多様体上におけるシュレディンガー方程式の基本解の滑らかさについて考察する．ユークリッド空間や双曲空間上では，具体計算によって基本解が滑らかであることが容易にわかる．より一般に，全ての測地線が無限遠に逃げていく場合には，基本解が滑らかになることが知られている．本講演では，測地線が無限遠に逃げるとは限らない（捕捉的な場合）状況で基本解の滑らかさについて議論する．具体的には，測地線の捕捉性が弱い場合には基本解は滑らかである一方，捕捉性が強い場合には基本解の滑らかさが失われることがある，という結果を紹介する．