Imaging and Computing Seminar

Yajun Zhou , Chemistry, Harvard

Robust Solution to Integral Equations via Topological Compactness: Applications in Chemical Kinetics and Electromagnetic Scattering

We study the inverse Laplace problem in chemical kinetics and the vector wave scattering problem in dielectric optics. We point out that topological compactness plays an essential role in the robust solutions to the integral equations associated with the two problems under investigation. By exploiting the $ w^*$-compactness theorem of Banach-Alaoglu, we devise a numerically stable method to extract the probability distribution of kinetic rate constants from noisy measurements in the time domain. By constructing a compact operator $\hat {\mathcal G}+2\hat{\mathcal G}^2 $ from the non-compact Green operator $\hat{\mathcal G} $ of light scattering, we demonstrate robust solutions to the scattering of electromagnetic waves on non-accretive homogeneous dielectric media, except for a critical susceptibility value $ \chi=\epsilon_r-1=-2$ that is independent of the dielectric shape.