Proper generalized decomposition: Difference between revisions

Revision as of 15:14, 13 April 2020

Revision as of 15:29, 13 April 2020

Line 4:

== Description ==

The proper generalized decomposition is a method characterized by a [[variational formulation]] of the problem, a discretization of the [[Domain of a function|___domain]] in the style of the [[finite element method]] and a numerical [[greedy algorithm]] that assumes the solution as a separated representation.

The most implemented variational formulation in PGD is the [[Bubnov-Galerkin method]], although other implementations exist.

The discretization of the ___domain is a well defined set of procedures that cover (a) the creation of finite element meshes, (b) the definition of basis function on reference elements (also called shape functions) and (c) the mapping of reference elements onto the elements of the mesh.

PGD assumes that the solution '''u''' of a multidimensional problem can be approximated as a separated representation '''u'''^''N'' of the form

Line 11 ⟶ 15:

where the number of terms ''N'' and the functional products '''X₁'''(''x''₁), '''X₂'''(''x''₂), ..., '''X_d'''(''x''_d), each depending on a variable (or variables), are unknown beforehand.

The solution is sought by applying a [[greedy algorithm]], usually the [[fixed point algorithm]], to the [[weak formulation]] of the problem.

For each iteration ''i'' of the algorithm, a ''mode'' of the solution is computed. Each mode consists of a set of numerical values of the functional products '''X₁'''(''x''₁), ..., '''X_d'''(''x''_d), which are''enrich'' expectedthe toapproximation improveof the solution of(usually theby problemimproving it). The number of computed modes required to obtain an approximation of the solution below a certain error threshold depends on the stop criterium of the iterative algorithm. Unlike [[Principal Component Analysis|PCA]], PGD modes are not necessarily [[orthogonal]] to each other.

PGD is suitable for solving high-dimensional problems, since it overcomes the limitations of classical approaches. In particular, PGD avoids the [[curse of dimensionality]], as solving decoupled problems is computationally much less expensive than solving multidimensional problems. Because of this, PGD enables to re-adapt parametric problems into a multidimensional framework by setting the parameters of the problem as extra coordinates: