Multigrid is one of the few optimal methods for solving systems of equations arising from the discretization of partial differential equations as well as a wide variety of related problems on graphs. In this tutorial we will introduce the key ingredients of the multigrid method (smoothing and coarse grid correction), explain their complementarity (they don't work well alone), and describe the most common cycling strategies. We will present the concepts and motivating analysis using a simple geometric approach to solving the linear system arising from the discretization of the diffusion equation on structured orthogonal grids. Then we will highlight the elements of the algorithm that have been advanced to provide robustness and flexibility for more general problems (e.g., operator dependent interpolation, galerkin coarse grid operators, and algebraic methods), noting that these topics will be covered in more detail in the subsequent tutorials. Finally, we'll touch on the popular and powerful use of multigrid as a preconditioner for Krylov methods such as the conjugate gradient method.

This tutorial begins with foundational principles, examining different forms of regularization, Bayesian inference, kernel methods, and predictive distributions. Then, for nonlinear models such as those arising in scientific computing, we address the limitations of analytical solutions, motivating the need for simulation-based approaches and exploring Markov Chain Monte Carlo (MCMC) methods. The core of the tutorial focuses on Gaussian Process Regression (GPR), demonstrating its extension to complex, multi-output problems, such as modeling differential equations where capturing joint dependencies is critical. Finally, we discuss preconditioned iterative methods for solving the dense but structured kernel matrix systems that arise in these machine learning problems. Designed for students and researchers eager to move beyond black-box models, this tutorial aims to give attendees intuition using simple examples that illustrate the complex behavior of machine learning models, while also stating ideas in a mathematically precise way.