Worked Out Problem 17: Uncertainty Quantification for a Stochastic Thermal Block

In this chapter, we consider a vector elliptic stochastic PDE modeling a steady heat conduction in a two-dimensional heterogeneous domain based on random input data. The aim is to obtain efficient evaluation of the statistics for the thermal field and the output of interest, defined as the average temperature over the whole domain. The solution is approximated using reduced order modeling techniques based on the POD-Galerkin method. The problem is described by presenting the physical domain, the goal, the parametrization, the boundary conditions, the governing PDE, and the corresponding weak formulation with its affine decomposition. Numerical results show the performance of the reduced approach with respect to the high-fidelity discretization, the quantity of interest, and the speedup. We provide the links to the source code, to the Jupyter notebook, and to the web-server ARGOS.