Proper generalized decomposition

The proper generalized decomposition (PGD) is an iterative numerical method for solving boundary value problems, meaning partial differential equations constrained by boundary conditions, such as Poisson's equation or Laplace's equation. It builds an approximation by successive enrichment: each iteration computes a new mode and adds it to the approximation, and unlike the principal components of proper orthogonal decomposition the modes need not be orthogonal to each other. Selecting only the most relevant modes gives a reduced order model, so PGD counts as a dimensionality reduction algorithm. The method rests on a variational formulation, most commonly Bubnov-Galerkin, a discretization of the domain in the style of the finite element method, the assumption that the solution can be written as a sum of products of functions of single variables, and a greedy algorithm, usually a fixed point iteration, to find each mode. Because it solves decoupled lower-dimensional problems it avoids the curse of dimensionality, and treating the parameters of a problem as extra coordinates yields a computational vademecum, a meta-model holding the solutions for every parameter value. The related Sparse Subspace Learning method uses hierarchical collocation to build non-intrusive approximations of parametric models.