Proper generalized decomposition: Difference between revisions

Revision as of 22:38, 22 October 2022

Revision as of 07:04, 23 December 2022

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# 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

# a numerical [[greedy algorithm]] to find the solution.<ref>{{Cite journal|last author1=Amine Ammar, | author2 = Béchir Mokdad, | author3 = Francisco Chinesta, | author4 = 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 author1=Amine Ammar, | author2 = Béchir Mokdad, | author3 = Francisco Chinesta, | author4 = 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>

=== 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| last1=Chinesta|first1=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 ===

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=== Separate representation ===

PGD assumes that the solution '''u''' of a (multidimensional) problem can be approximated as a separate representation of the form

::<math display="block"> \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>

::<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>

where the number of addends ''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.

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Therefore, PGD enables to re-adapt parametric problems into a multidimensional framework by setting the parameters of the problem as extra coordinates:

::<math display="block"> \mathbf{u} \approx \mathbf{u}^N(x_1, \ldots, x_d; k_1, \ldots, k_p) = \sum_{i=1}^N \mathbf{X_1}_i(x_1) \cdots \mathbf{X_d}_i(x_d) \cdot \mathbf{K_1}_i(k_1) \cdots \mathbf{K_p}_i(k_p),</math>

::<math> \mathbf{u} \approx \mathbf{u}^N(x_1, \ldots, x_d; k_1, \ldots, k_p) = \sum_{i=1}^N \mathbf{X_1}_i(x_1) \cdots \mathbf{X_d}_i(x_d) \cdot \mathbf{K_1}_i(k_1) \cdots \mathbf{K_p}_i(k_p),</math>

where a series of functional products '''K₁'''(''k''₁), '''K₂'''(''k''₂), ..., '''K_p'''(''k''_p), each depending on a parameter (or parameters), has been incorporated to the equation.