English
We examine the numerical solution of partial differential equations on random domains using isogeometric analysis, wherein the primary focus is on uncertainty quantification; specifically, quantities of interest, i.e., expectation and correlation, are considered for two types of partial differential equations on complex geometries. To handle such complex geometries, we use the Morse-Smale complex to convert triangulated surfaces into conforming quad NURBS surfaces. As an application of the NURBS surface reconstruction method, we develop a shape optimisation approach that utilises the Morse-Smale complex at each iteration to minimise geometric singularities. Further, we propose the efficient computation of the space-time correlation using an online singular value decomposition in case of the diffusion equation and employ the p-multilevel Monte Carlo method to compute quantities of interest for the Helmholtz equation.