be combined with multigrid methods. For example, one use of wavelets is to reformulate the finite element approach in terms of a multilevel method. Adaptive multigrid exhibits adaptive mesh refinement, that is, it adjusts the grid as the computation proceeds, in a manner dependent upon the computation itself. The idea is to increase resolution of the grid only in regions of the solution where it is needed. Algebraic multigrid (AMG) Practically important extensions of multigrid methods include techniques where no partial differential equation nor geometrical problem background is used to construct the multilevel hierarchy. Such algebraic multigrid methods (AMG) construct their hierarchy of operators directly from the system matrix. In classical AMG, the levels of the hierarchy are simply subsets of unknowns without any geometric interpretation. (More generally, coarse grid unknowns can be particular linear combinations of fine grid unknowns.) Thus, AMG methods become black-box solvers for certain classes of sparse matrices. AMG is regarded as advantageous mainly where geometric multigrid is too difficult to apply, but is often used simply because it avoids the coding necessary for a true multigrid implementation. While classical AMG was developed first, a related algebraic method is known as smoothed aggregation (SA). In an overview paper by Jinchao Xu and Ludmil Zikatanov, the "algebraic multigrid" methods are understood from an abstract point of view. They developed a unified framework and existing algebraic multigrid methods can be derived coherently. Abstract theory about how to construct optimal coarse space as well as quasi-optimal spaces was derived. Also, they proved that, under appropriate assumptions, the abstract two-level AMG method converges uniformly with respect to the size of the linear system, the coefficient variation, and the anisotropy. Their abstract framework covers most existing AMG methods, such as classical AMG, energy-minimization AMG, unsmoothed and smoothed aggregation AMG, and spectral AMGe. Multigrid in time methods Multigrid methods have also been adopted for the solution of initial value problems. Of particular interest here are parallel-in-time multigrid methods: in contrast to classical Runge–Kutta or linear multistep methods, they can offer concurrency in temporal direction. The well known Parareal parallel-in-time integration method can also be reformulated as a two-level multigrid in time. Multigrid for nearly singular problems Nearly singular problems arise in a number of important physical and engineering applications. Simple, but important example of nearly singular problems can be found at the displacement formulation of linear elasticity for nearly incompressible materials. Typically,