Proper generalized decomposition

The proper generalized decomposition (PGD) is a numerical method for solving boundary value problems. It assumes that the solution of a multidimensional (or multiparametric) problem can be expressed in a separated representation of the form

\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}),

where the number of terms N, and the functions X are a priori unknown.

Since solving decoupled problems is computationally much less expensive than solving multidimensional problems, PGD is usually considered a dimensionality reduction algorithm.

References

Amine Ammar, B Mokdad, Francisco Chinesta, and Roland Keunings. A new family of solvers for some classes of multidimensional partial differential equations encountered in kinetic theory modeling of complex fluids. Journal of Non-Newtonian Fluid Mechanics, 139(3):153–176, 2006.
Amine Ammar, B Mokdad, Francisco Chinesta, and Roland Keunings. 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. Journal of Non-Newtonian Fluid Mechanics, 144(2):98–121, 2007.
F. Chinesta, R. Keunings, and A. Leygue. The Proper Generalized Decomposition for Advanced Numerical Simulations: A Primer. SpringerBriefs in Applied Sciences and Technology. Springer International Publishing, 2013.

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