Tensor Train Algorithms for Stochastic PDE Problems - Sergey Dolgov, University of Bath

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.

Intro
Stochastic partial differential equation
(tailored) Bayesian inverse problem
Bayesian inversion - solution approaches
Separation of variables
2 variables: low-rank matrices
Cross approximation methods
Cross interpolation
Maximum Volume principle
Cross approximation alternating iteration
Cross approximation algorithm
Tensor Train (TT) decomposition
How to compute a TT decomposition?
TT for inverse PDE problems
TT for inverse problems
Conditional probability factorisation
Conditional distribution sampling method
TT-CD sampling
Even better samples: mapped QMC
Two-level control variate QMC-MCMC algorithm
Inverse diffusion equation
MCMC chain and accuracy of the density function
Computation of the posterior Qol
Shock absorber: problem setting
Shock absorber: quadrature error
Conclusion