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== Description ==
The proper generalized decomposition is a method characterized by
The proper generalized decomposition is a method characterized by (1) a [[variational formulation]] of the problem, (2) a discretization of the [[Domain of a function|___domain]] in the style of the [[finite element method]], (3) the assumption that the solution can be approximated as a separated representation and (4) a numerical [[greedy algorithm]] to find the solution.<ref>{{Cite journal|last=Amine Ammar, Béchir Mokdad, Francisco Chinesta, Roland Keunings|date=2006|title=A New Family of Solvers for Some Classes of Multidimensional Partial Differential Equations Encountered in Kinetic Theory Modeling of Complex Fluids|url=https://hal.archives-ouvertes.fr/hal-01004909/document|journal=Journal of Non-Newtonian Fluid Mechanics}}</ref><ref>{{Cite journal|last=Amine Ammar, Béchir Mokdad, Francisco Chinesta, Roland Keunings|date=2007|title=A new family of solvers for some classes of multidimensional partial differential equations encountered in kinetic theory modelling of complex fluids. Part II: Transient simulation using space-time separated representations|url=https://hal.archives-ouvertes.fr/hal-01004910/document|journal=Journal of Non-Newtonian Fluid Mechanics}}</ref>▼
# a [[variational formulation]] of the problem,
# a discretization of the [[Domain of a function|___domain]] in the style of the [[finite element method]],
# the assumption that the solution can be approximated as a separate representation and
▲
=== Variational formulation ===
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}}</ref><ref name=":0" />
=== Domain discretization ===
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.
=== Separate representation ===
PGD assumes that the solution '''u''' of a (multidimensional) problem can be approximated as a ::<math> \mathbf{u} \approx \mathbf{u}^N(x_1, x_2, \ldots, x_d) = \sum_{i=1}^N \mathbf{X_1}_i(x_1) \cdot \mathbf{X_2}_i(x_2) \cdots \mathbf{X_d}_i(x_d), </math>
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where the number of addends ''N'' and the functional products '''X<sub>1</sub>'''(''x''<sub>1</sub>), '''X<sub>2</sub>'''(''x''<sub>2</sub>), ..., '''X<sub>d</sub>'''(''x''<sub>d</sub>), each depending on a variable (or variables), are unknown beforehand.
=== Greedy algorithm ===
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 '''X<sub>1</sub>'''(''x''<sub>1</sub>), ..., '''X<sub>d</sub>'''(''x''<sub>d</sub>), which ''enrich'' the approximation of the solution. Note that due to the greedy nature of the algorithm, the term 'enrich' is used rather than 'improve'. 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.
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