Byeong-Ho Bahn: Parametric Semilinear Eigenvalue Problem : Parametric Holomorphy and Uncertainty Quantification

Parametric partial differential equations have been actively studied for the last few decades. It has been well known that a special type of holomorphic dependence of the solution on the parameters guarantees the applications of various numerical methods, such as DeepOnet, to solve the problems. However, to the best of the author's knowledge, methods used to show the desired holomorphy is strongly dependent on specific PDE problems and it is difficult to use those methods for parametric eigenvalue problems. In this talk, a framework to verify the desired parametric holomorphy with parametric linear and semilinear eigenvalue problems as examples will be presented and discussed. Also, if time permits, with the very same condition that is needed for showing the desired holomorphy, quasi-Monte Carlos application will be discussed.