Multigrid (MG) methods in numerical analysis are a group of algorithms for solving differential equations using a hierarchy of discretizations. They are an example of a class of techniques called multiresolution methods, very useful in (but not limited to) problems exhibiting multiple scales of behavior. For example, many basic relaxation methods exhibit different rates of convergence for short- and long-wavelength components, suggesting these different scales be treated differently, as in a Fourier analysis approach to multigrid. MG methods can be used as solvers as well as preconditioners.

The main idea of multigrid is to accelerate the convergence of a basic iterative method by global correction from time to time, accomplished by solving a coarse problem. This principle is similar to interpolation between coarser and finer grids. The typical application for multigrid is in the numerical solution of elliptic partial differential equations in two or more dimensions.

Multigrid methods can be applied in combination with any of the common discretization techniques. For example, the finite element method may be recast as a multigrid method. In these cases, multigrid methods are among the fastest solution techniques known today. In contrast to other methods, multigrid methods are general in that they can treat arbitrary regions and boundary conditions. They do not depend on the separability of the equations or other special properties of the equation. They are also directly applicable to more-complicated non-symmetric and nonlinear systems of equations, like the Lamé system of elasticity or the Navier-Stokes equations.

Various methods are not robust in the sense that convergence within reasonable computing time and storage. Poor convergence depends on spectral properties (eigenvalue distribution, condition of the eigensystem, etc.). Preconditioning tries to find some suitable operator M such that inverse(M)A has better spectral properties for a given Ax=b problem. An operator that is used for this aim is called a preconditioner. Preconditioners are not a black box and there is no general theory to construct efficient preconditioner. However a suitable preconditioner should have the following properties:

- Good approximation to system matrix A
- Easy to construct by means of computing and memory cost
- Easier to solve the preconditioned system than the original system
- Easy to parallelize

Large sparse linear systems occur in many scientific and engineering applications. Such systems are typically solved using either iterative or direct methods. Although highly parallel formulations of dense matrix factorization are well known, it had been a challenge to implement efficient sparse linear system solvers using direct methods. Fast and accurate graph partitioning algorithms which is the most commonly used method for computing fill reducing orderings of a sparse matrix are needed for direct as well iterative parallel sparse solvers.