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.

The focus of this tutorial is on algebraic multigrid (AMG). The tutorial will start with the basic principles of algebraic multigrid methods, followed by introducing two general methods: CF based algebraic multigrid and aggregation based algebraic multigrid. An overview of these methods, including common algorithms for their construction will be covered. The goal is to identify the key components of AMG. Advanced algebraic methods such as compatible relaxation, adaptive AMG, and element based AMG will also be briefly covered. In addition, an overview of some of the supporting theory will be given.

Machine learning and its cousin Scientific Machine Learning (SciML) have seen a dramatic rise in the last half decade. Yet the breadth of the work can make “self-guided tours” seem overwhelming. This talk will start by discussing neural networks and how to view them as adaptive basis generation algorithms. We will explore how these are used in physics-informed neural networks (PINNS) PDE solution approaches, operator regression scenarios, and growing interests in foundation models. We will discuss the strengths and weaknesses of these approximation approaches. Connections to the talks being presented at the meeting will be made through out. This talk will conclude with some speculation about the future direction and how we, as the multigrid community, can contribute to advancing this effort.