Explore a comprehensive lecture on numerical homogenization techniques for solving multiscale partial differential equations. Delve into the Bayesian reformulation of numerical homogenization, learning about methods that enable exponential decaying bases, localization, and optimal convergence rates. Discover how these techniques can be applied to construct efficient fine-scale fast solvers, including multi-resolution decomposition and multigrid solvers with bounded condition numbers. Examine the application of these methods to time-dependent problems and multi-scale eigenvalue problems. Gain insights into topics such as the numerical resolution of elliptic operators, low dimensionality of solution spaces, variational formulation for basis elements, and the properties of Gamblet decomposition. Understand the algorithms for exact and fast Gamblet transformation, and explore the convergence of Gamblet-based multigrid methods.