# Algebraic MultigridΒΆ

Here we outline the basic principles behind the Algebraic Multigrid (AMG) method [BrMH85], [Stue99]. Consider a system of linear algebraic equations in the form

where \(A\) is an \(n \times n\) matrix. Multigrid methods are based on the recursive use of two-grid scheme, which combines

*Relaxation*, or*smoothing iteration*: a simple iterative method such as Jacobi or Gauss-Seidel; and*Coarse grid correction*: solving residual equation on a coarser grid. Transfer between grids is described with*transfer operators*\(P\) (*prolongation*or*interpolation*) and \(R\) (*restriction*).

A setup phase of a generic algebraic multigrid (AMG) algorithm may be described as follows:

- Start with a system matrix \(A_1 = A\).
- While the matrix \(A_i\) is too big to be solved directly:
- Introduce prolongation operator \(P_i\), and restriction operator \(R_i\).
- Construct coarse system using Galerkin operator: \(A_{i+1} = R_i A_i P_i\).

- Construct a direct solver for the coarsest system \(A_L\).

Note that in order to construct the next level in the AMG hierarchy, we only need to define transfer operators \(P\) and \(R\). Also, the restriction operator is often chosen to be a transpose of the prolongation operator: \(R=P^T\).

Having constructed the AMG hierarchy, we can use it to solve the system as follows:

- Start at the finest level with initial approximation \(u_1 = u^0\).
- Iterate until convergence (
*V-cycle*): - At each level of the grid hiearchy, finest-to-coarsest:
- Apply a couple of smoothing iterations (
*pre-relaxation*) to the current solution \(u_i = S_i(A_i, f_i, u_i)\). - Find residual \(e_i = f_i - A_i u_i\) and restrict it to the RHS on the coarser level: \(f_{i+1} = R_i e_i\).

- Apply a couple of smoothing iterations (

- Solve the corasest system directly: \(u_L = A_L^{-1} f_L\).
- At each level of the grid hiearchy, coarsest-to-finest:
- Update the current solution with the interpolated solution from the coarser level: \(u_i = u_i + P_i u_{i+1}\).
- Apply a couple of smoothing iterations (
*post-relaxation*) to the updated solution: \(u_i = S_i(A_i, f_i, u_i)\).

- Iterate until convergence (

More often AMG is not used standalone, but as a preconditioner with an iterative Krylov subspace method. In this case single V-cycle is used as a preconditioning step.

So, in order to fully define an AMG method, we need to choose transfer operators \(P\) and \(R\), and smoother \(S\).