Speaker:
Sarah Hamilton, University of Helsinki
Title:
D-bar Methods in Direct and Inverse Problems
Abstract:
The Inverse Scattering Transform is considered one of the most
important breakthroughs of mathematical physics in the 20th
century. Its origins lie in the explicit solution of the
Korteweg-de Vries equation for shallow water waves, but the
resulting approach can be applied to other nonlinear partial
differential equations as well. The method involves
transforming a nonlinear PDE (in the physical space) into a linear
spectral problem (in the frequency domain). In the 1980s,
Beals and Coifman considered framing the inverse scattering method
as the inversion of a D-bar equation, thus allowing generalizations
to multiple dimensions and the study of additional classes of
nonlinear PDEs. D-bar methods have proved very useful in
evolution equations as well as inverse problems by allowing the
rigorous study of the full nonlinear problems in the setting of
constructive proofs, thus providing a natural bridge towards
numerical implementations. In this talk, we explore the D-bar
method for an evolution equation (the Davey-Stewartson II defocusing
equation) and the inverse problem of Electrical Impedance Tomography
(via the Schrödinger equation). Numerical
evolutions/reconstructions are presented using simulated data.