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This article deals with a component of numerical methods. For coarse space in topology, see coarse structure. In numerical analysis, coarse problem is an auxiliary system of equations used in an iterative method for the solution of a given larger system of equations. A coarse problem is basically a version of the same problem at a lower ...
A continuous map f : X → Y remains continuous if the topology on Y becomes coarser or the topology on X finer. An open (resp. closed) map f : X → Y remains open (resp. closed) if the topology on Y becomes finer or the topology on X coarser. One can also compare topologies using neighborhood bases.
A coarse structure on a set is a collection of subsets of (therefore falling under the more general categorization of binary relations on ) called controlled set s, and so that possesses the identity relation, is closed under taking subsets, inverses, and finite unions, and is closed under composition of relations. Explicitly:
A partition α of a set X is a refinement of a partition ρ of X—and we say that α is finer than ρ and that ρ is coarser than α—if every element of α is a subset of some element of ρ. Informally, this means that α is a further fragmentation of ρ. In that case, it is written that α ≤ ρ.
A coarse mesh may provide an accurate solution if the solution is a constant, so the precision depends on the particular problem instance. One can selectively refine the mesh in areas where the solution gradients are high, thus increasing fidelity there. Accuracy, including interpolated values within an element, depends on the element type and ...
Granularity (also called graininess) is the degree to which a material or system is composed of distinguishable pieces, "granules" or "grains" (metaphorically). It can either refer to the extent to which a larger entity is subdivided, or the extent to which groups of smaller indistinguishable entities have joined together to become larger distinguishable entities.
(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, [ 20 ] but is often used simply because it avoids the coding necessary ...
In mathematics, coarse topology is a term in comparison of topologies which specifies the partial order relation of a topological structure to other one(s). Specifically, the coarsest topology may refer to: Initial topology, the most coarse topology in a certain category of topologies; Trivial topology, the most coarse topology possible on a ...