Proper generalized decomposition

The proper generalized decomposition (PGD) is an iterative numerical method for solving boundary value problems (BVPs), that is, partial differential equations constrained by a set of boundary conditions. The PGD algorithm computes an approximation of the theoretical solution of the BVP by successive enrichment. This means that, in each iteration, a new component (or mode) is computed and added to the approximation, thus successively enriching it. By selecting only the first PGD modes, a reduced order model of the solution is obtained. Because of this, PGD is considered a dimensionality reduction algorithm.

Description

The proper generalized decomposition is a method characterized by a variational formulation of the problem, a discretization of the domain in the style of the finite element method and a numerical greedy algorithm that assumes the solution as a separated representation.

PGD assumes that the solution u of a multidimensional problem can be approximated as a separated representation u^N 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 functional products X₁(x₁), X₂(x₂), ..., X_d(x_d), each depending on a variable (or variables), are unknown beforehand.

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 X₁(x₁), ..., X_d(x_d), which are expected to improve the solution of the problem. 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. Unlike PCA, PGD modes are not necessarily orthogonal to each other.

PGD is suitable for solving high-dimensional problems, since it overcomes the limitations of classical approaches. In particular, PGD avoids the curse of dimensionality, as solving decoupled problems is computationally much less expensive than solving multidimensional problems. Because of this, PGD enables to re-adapt parametric problems into a multidimensional framework by setting the parameters of the problem as extra coordinates:

$\mathbf {u} \approx \mathbf {u} ^{N}(x_{1},\ldots ,x_{d};k_{1},\ldots ,k_{p})=\sum _{i=1}^{N}\mathbf {X_{1}} _{i}(x_{1})\cdots \mathbf {X_{d}} _{i}(x_{d})\cdot \mathbf {K_{1}} _{i}(k_{1})\cdots \mathbf {K_{p}} _{i}(k_{p}),$

where a series of functional products K₁(k₁), K₂(k₂), ..., K_p(k_p), each depending on a parameter (or parameters), has been incorporated to the equation.

In this case, the solution is called computational vademecum: a general meta-model containing all the particular solutions for every possible value of the involved parameters.

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.