10:30, Alexei Lozinski
Title: A method of fictitious domains with optimal convergence
Abstract: Fictive domain methods allow to discretize an EDP placed on a complex domain by using a simple background mesh (typically Cartesian) on a simpler domain (typically a rectangle). The classic variants of these methods are based on extending the solution to the entire fictitious domain, are very easy to implement, but converge slowly. Recently, several optimal convergence fictitious domains methods have been proposed following the XFEM or CutFEM paradigm. Unlike conventional approaches, weak formulation, and therefore finite element formulation, is based on the physical domain, although approximation spaces always live on the background mesh that can be arbitrarily cut by the real boundary. Non-trivial numerical integration is then used to calculate contributions to the finite element matrix on cut mesh elements, making implementation rather complex.
We will propose a method to bypass this technical complication by introducing an extension of the solution to a fictitious domain that is only slightly larger than the physical domain, namely the union of mesh elements having a non-empty intersection with the latter. In this respect, our method is a compromise between traditional fictitious domain methods and XFEM-CutFEM methods. This eliminates integration on the cut elements, but the integrals on the physical boundary remain always present.
At the end of the presentation we will present a modification of the above idea, based on the multiplication by the levelset, which leads to a method in which even the integrals on the physical boundary disappear (work in progress with Michel Duprez).
11h15, Andrew Papanicolaou
Title: Reduced Order Representation of Implied Volatility Surfaces
Abstract: We consider a Principal Component Analysis of implied vol surfaces (IVS) for US equities using data from OptionMetrics which is available through Wharton Research Data Services (WRDS) and specifically address a number of important questions about data of this nature using higher order decomposition methods for tensor data.
