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

Explore advanced adaptive multilevel approaches for parametric partial differential equations in this fourth lecture of a comprehensive series on uncertainty quantification and surrogate modeling. Delve into sophisticated numerical methods that construct approximation spaces specifically tailored to problem regularity, with particular emphasis on the crucial role of a posteriori error estimation. Learn how these advanced techniques build upon foundational concepts of spatially-varying uncertain PDE inputs, high-dimensional parametric PDEs, and Monte Carlo methods covered in earlier parts. Examine how adaptive multilevel methods improve upon basic surrogate modeling approaches such as stochastic collocation, reduced basis methods, and stochastic Galerkin methods by providing more efficient and accurate solutions for forward uncertainty quantification problems. Understand the mathematical framework for handling partial differential equations with uncertain material coefficients, boundary conditions, and source terms that are common in real-world physics-based models. Discover how these methods address the computational challenges of repeated PDE solutions required by naive sampling approaches, particularly when dealing with expensive high-fidelity finite element methods. Gain insights into the development and application of numerical schemes that have evolved over three decades to tackle both forward and inverse problems involving PDEs with uncertain inputs, focusing on creating functional approximations that can be efficiently evaluated for new parameter choices without additional computational overhead.