Numerical Homogenization Based Fast Solver for Multiscale PDEs

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.

Intro
Model Problem
Numerical Resolution of Elliptic Operator
Low dimensionality of Solution Space
Bayesian Framework for Numerical Homogenization
Variational Formulation for the Basis Elements
Two Scale Decomposition and Optimality
Accuracy of Localization
RPS: An exponential decaying basis
A Hierachy of Exponential Decay Basis
Interpolation and restriction matrices/operators
Algorithm for Exact Gamblet transformation
Algorithm for Fast (Localized) Gamblet Transform
Gamblet: A Multiresolution Decomposition
Property of Exponential Decaying Basis
Properties of Gamblet Decomposition
Convergence of Gamblet based Multigrid