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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 modeling physics-based systems where material coefficients, boundary conditions, and source terms contain inherent uncertainties that must be represented as functions of random variables. Learn how these uncertainties transform standard PDEs into parametric problems defined on high-dimensional or infinite-dimensional parameter domains, making traditional sampling approaches computationally prohibitive for expensive high-fidelity finite element solutions. Discover sophisticated surrogate modeling techniques that create functional approximations enabling rapid evaluation of new parameter choices without requiring additional PDE solves. Master fundamental concepts including appropriate modeling strategies for spatially-varying uncertain PDE inputs, high-dimensional parametric PDE formulations, and basic Monte Carlo methods. Examine key surrogate modeling approaches such as stochastic collocation methods, reduced basis techniques, and intrusive stochastic Galerkin methods for forward uncertainty quantification. Investigate cutting-edge adaptive multilevel approaches that construct approximation spaces specifically tailored to problem regularity, and understand the critical importance of a posteriori error estimation in ensuring solution accuracy. Gain practical insights into overcoming computational bottlenecks when dealing with uncertain model inputs in real-world engineering and scientific applications where repeated high-fidelity simulations are computationally infeasible.
