Proper generalized decomposition: Difference between revisions

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== Description ==

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|first=|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|volume=|pages=|via=}}</ref><ref>{{Cite journal|last=Amine Ammar, Béchir Mokdad, Francisco Chinesta, Roland Keunings|first=|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|volume=|pages=|via=}}</ref>

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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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.

In this case, the obtained approximation of the solution is called ''computational [[vademecum]]'': a general meta-model containing all the particular solutions for every possible value of the involved parameters.<ref>{{Cite journal|last=Francisco Chinesta, Adrien Leygue, Felipe Bordeu, Elías Cueto, David Gonzalez, Amine Ammar, Antonio Huerta|first=|date=2013|title=PGD-Based Computational Vademecum for Efficient Design, Optimization and Control|url=https://hal.archives-ouvertes.fr/hal-01515083/document|journal=Archives of Computational Methods in Engineering|volume=|pages=|via=}}</ref>

== Sparse Subspace Learning ==