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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]],<ref name=":0">{{Cite thesis|title=Proper generalised decompositions: theory and applications|url=http://orca.cf.ac.uk/73515/|publisher=Cardiff University|date=2015-04-09|degree=phd|language=en|first=Thomas Lloyd David|last=Croft}}</ref><ref>{{Cite book|last=Chinesta|first=Francisco|url=https://www.springer.com/gp/book/9783319028644|title=The Proper Generalized Decomposition for Advanced Numerical Simulations: A Primer|last2=Keunings|first2=Roland|last3=Leygue|first3=Adrien|date=2014|publisher=Springer International Publishing|isbn=978-3-319-02864-4|series=SpringerBriefs in Applied Sciences and Technology|language=en}}</ref> although other implementations exist.<ref>{{Cite web|url=https://hal.archives-ouvertes.fr/tel-01926078/document|title=Advanced strategies for the separated formulation of problems in the Proper Generalized Decomposition framework|last=Aguado|first=José Vicente|date=18 Nov 2018|website=|url-status=live|archive-url=|archive-date=|access-date=}}</ref><ref name=":0" />
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.
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