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

Explore advanced numerical methods for uncertainty quantification in parametric partial differential equations through this comprehensive lecture from the Hausdorff Center for Mathematics. Delve into the challenges of physics-based models where PDEs contain uncertain inputs such as material coefficients, boundary conditions, and source terms that vary in real-world applications. Learn how uncertainty quantification represents these uncertain inputs as functions of random variables, transforming PDEs into parametric equations on high-dimensional or infinite-dimensional parameter domains. Understand the limitations of naive sampling methods that require repeated numerical solutions for different input samples, particularly when high-fidelity finite element methods make single solutions computationally expensive. Discover how forward uncertainty quantification aims to understand the relationship between input uncertainty and solution uncertainty. Examine surrogate modeling techniques that create functional approximations for efficient evaluation of new input parameters without additional PDE solves. Master appropriate modeling approaches for spatially-varying uncertain PDE inputs and grasp the concept of high-dimensional parametric PDEs. Review the basic Monte Carlo method and explore fundamental surrogate modeling approaches including stochastic collocation and reduced basis methods. Investigate the intrusive stochastic Galerkin method and understand advanced adaptive multilevel approaches that construct approximation spaces tailored to problem-specific regularity. Appreciate the crucial role of a posteriori error estimation in developing robust numerical schemes for both forward and inverse problems involving PDEs with uncertain inputs.