Explore tensor train algorithms for stochastic PDE problems in this 50-minute lecture by Sergey Dolgov from the University of Bath. Delve into the challenges of approximating high-dimensional functions from limited information and learn how modern approaches overcome the curse of dimensionality. Discover structural assumptions like low intrinsic dimensionality, partial separability, and sparse representations in a basis. Examine the mathematical foundations of multivariate approximation theory, high-dimensional integration, and non-parametric regression. Cover topics such as stochastic partial differential equations, Bayesian inverse problems, separation of variables, low-rank matrices, cross approximation methods, Tensor Train decomposition, conditional probability factorisation, and QMC-MCMC algorithms. Apply these concepts to real-world examples like inverse diffusion equations and shock absorber problems.