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: ''This article deals with a component of numerical methods. For coarse space in topology, see
coarse structure In the mathematical fields of geometry and topology, a coarse structure on a set ''X'' is a collection of subsets of the cartesian product ''X'' × ''X'' with certain properties which allow the ''large-scale structure'' of metric spaces and topolog ...
.'' In
numerical analysis Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). It is the study of numerical methods t ...
, coarse problem is an auxiliary system of equations used in an
iterative method In computational mathematics, an iterative method is a mathematical procedure that uses an initial value to generate a sequence of improving approximate solutions for a class of problems, in which the ''n''-th approximation is derived from the pre ...
for the solution of a given larger system of equations. A coarse problem is basically a version of the same problem at a lower resolution, retaining its essential characteristics, but with fewer variables. The purpose of the coarse problem is to propagate information throughout the whole problem globally. In
multigrid method In numerical analysis, a multigrid method (MG method) is an algorithm for solving differential equations using a hierarchy of discretizations. They are an example of a class of techniques called multiresolution methods, very useful in problems exhi ...
s for
partial differential equation In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a multivariable function. The function is often thought of as an "unknown" to be solved for, similarly to ...
s, the coarse problem is typically obtained as a discretization of the same equation on a coarser grid (usually, in
finite difference method In numerical analysis, finite-difference methods (FDM) are a class of numerical techniques for solving differential equations by approximating derivatives with finite differences. Both the spatial domain and time interval (if applicable) are di ...
s) or by a Galerkin approximation on a subspace, called a coarse space. In
finite element method The finite element method (FEM) is a popular method for numerically solving differential equations arising in engineering and mathematical modeling. Typical problem areas of interest include the traditional fields of structural analysis, heat ...
s, the Galerkin approximation is typically used, with the coarse space generated by larger elements on the same
domain Domain may refer to: Mathematics *Domain of a function, the set of input values for which the (total) function is defined ** Domain of definition of a partial function ** Natural domain of a partial function ** Domain of holomorphy of a function * ...
. Typically, the coarse problem corresponds to a grid that is twice or three times coarser. Coarse spaces (coarse model,
surrogate model A surrogate model is an engineering method used when an outcome of interest cannot be easily measured or computed, so a model of the outcome is used instead. Most engineering design problems require experiments and/or simulations to evaluate design ...
) are the backbone of algorithms and methodologies exploiting the
space mapping The space mapping methodology for modeling and design optimization of engineering systems was first discovered by John Bandler in 1993. It uses relevant existing knowledge to speed up model generation and design optimization of a system. The know ...
concept for solving computationally intensive engineering modeling and design problems.J.W. Bandler and S. Kozie
"Advances in electromagnetics-based design optimization"
''IEEE MTT-S Int. Microwave Symp. Digest'' (San Francisco, CA, 2016).
In
space mapping The space mapping methodology for modeling and design optimization of engineering systems was first discovered by John Bandler in 1993. It uses relevant existing knowledge to speed up model generation and design optimization of a system. The know ...
, a fine or high fidelity (high resolution, computationally intensive) model is used to calibrate or recalibrate—or update on the fly, as in aggressive space mapping—a suitable coarse model. An updated coarse model is often referred to as
surrogate model A surrogate model is an engineering method used when an outcome of interest cannot be easily measured or computed, so a model of the outcome is used instead. Most engineering design problems require experiments and/or simulations to evaluate design ...
or mapped coarse model. It permits fast, but more accurate, harnessing of the underlying coarse model in the exploration of designs or in design optimization. In
domain decomposition method In mathematics, numerical analysis, and numerical partial differential equations, domain decomposition methods solve a boundary value problem by splitting it into smaller boundary value problems on subdomains and iterating to coordinate the solut ...
s, the construction of a coarse problem follows the same principles as in multigrid methods, but the coarser problem has much fewer unknowns, generally only one or just a few unknowns per subdomain or substructure, and the coarse space can be of a quite different type that the original finite element space, e.g. piecewise constants with averaging in
balancing domain decomposition In numerical analysis, the balancing domain decomposition method (BDD) is an iterative method to find the solution of a symmetric positive definite system of linear algebraic equations arising from the finite element method.J. Mandel, ''Balancing do ...
or built from energy minimal functions in
BDDC In numerical analysis, BDDC (balancing domain decomposition by constraints) is a domain decomposition method for solving large symmetric, positive definite systems of linear equations that arise from the finite element method. BDDC is used as a prec ...
. The construction of the coarse problem in FETI is unusual in that it is not obtained as a Galerkin approximation of the original problem, however. In Algebraic Multigrid Methods and in iterative aggregation methods in
mathematical economics Mathematical economics is the application of mathematical methods to represent theories and analyze problems in economics. Often, these applied methods are beyond simple geometry, and may include differential and integral calculus, difference ...
and
Markov chain A Markov chain or Markov process is a stochastic model describing a sequence of possible events in which the probability of each event depends only on the state attained in the previous event. Informally, this may be thought of as, "What happe ...
s, the coarse problem is generally obtained by the Galerkin approximation on a subspace. In mathematical economics, the coarse problem may be obtained by the aggregation of products or industries into a coarse description with fewer variables. In Markov chains, a coarse Markov chain may be obtained by aggregating states. The speed of convergence of multigrid and domain decomposition methods for
elliptic partial differential equation Second-order linear partial differential equations (PDEs) are classified as either elliptic, hyperbolic, or parabolic. Any second-order linear PDE in two variables can be written in the form :Au_ + 2Bu_ + Cu_ + Du_x + Eu_y + Fu +G= 0,\, wher ...
s without a coarse problem deteriorates with decreasing mesh step (or decreasing element size, or increasing number of subdomains or substructures), thus making a coarse problem necessary for a
scalable Scalability is the property of a system to handle a growing amount of work by adding resources to the system. In an economic context, a scalable business model implies that a company can increase sales given increased resources. For example, a ...
algorithm.


References

*
Jan Mandel Jan Mandel is a Czech- American mathematician. He received his PhD from the Faculty of Mathematics and Physics, Charles University in Prague and was a Senior Research Scientist there. Since 1986, he is professor of Mathematics at the University ...
and Bedrich Sousedik, "Coarse space over the ages", ''Nineteenth International Conference on Domain Decomposition'', Springer-Verlag, submitted, 2009
arXiv:0911.5725
* Olof B. Widlund, "The Development of Coarse Spaces for Domain Decomposition Algorithms", in: ''Domain Decomposition Methods in Science and Engineering XVIII'', Bercovier, M. and Gander, M.J. and Kornhuber, R. and Widlund, O. (eds.), Lecture Notes in Computational Science and Engineering 70, Springer-Verlag, 2009, Proceedings of 18th International Conference on Domain Decomposition, Jerusalem, Israel, January 2008
article
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See also

*
Multiscale modeling Multiscale modeling or multiscale mathematics is the field of solving problems which have important features at multiple scales of time and/or space. Important problems include multiscale modeling of fluids, solids, polymers, proteins, nucleic a ...
Domain decomposition methods