Parametric PDEs - Numerical Methods for Forward UQ and Surrogate Modelling - Part I

Learn numerical methods for forward uncertainty quantification and surrogate modeling in parametric partial differential equations through this comprehensive lecture. Explore how to handle physics-based models with uncertain inputs such as material coefficients, boundary conditions, and source terms by representing them as functions of random variables. Discover the challenges of naive sampling methods when dealing with expensive high-fidelity finite element solutions and understand why traditional approaches become infeasible for accurate uncertainty assessments. Examine appropriate modeling techniques for spatially-varying uncertain PDE inputs and grasp the concept of high-dimensional parametric PDEs. Review the basic Monte Carlo method as a foundation before diving into surrogate modeling approaches that create functional approximations for efficient evaluation without additional PDE solves. Study key methodologies including stochastic collocation methods, reduced basis methods, and the intrusive stochastic Galerkin method. Gain insights into advanced adaptive multilevel approaches that construct approximation spaces tailored to problem-specific regularity and understand the critical importance of a posteriori error estimation in these computational frameworks.