Proper generalized decomposition

The Proper Generalized Decomposition (PGD) is a numerical method for solving boundary value problems that assumes that the solution of a multidimensional problem can be expressed in a separated representation of the form

$\mathbf {u} \approx \mathbf {u} ^{N}(x_{1},x_{2},\ldots ,x_{d})=\sum _{i=1}^{N}=\mathbf {X_{1}} _{i}(x_{1})\cdot \mathbf {X_{2}} _{i}(x_{2})\ldots \mathbf {X_{d}} _{i}(x_{d})$.

Since the solution of decoupled problems is computationally much less expensive than solving multidimensional problems, PGD is usually considered a model order reduction method.

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.