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The '''proper generalized decomposition''' ('''PGD''') is an [[iterative method|iterative]] [[numerical method]] for solving [[boundary value problem]]s (BVPs), that is, [[partial differential equation]]s constrained by a set of boundary conditions, such as the [[Poisson's equation]] or the [[Laplace's equation]].
The PGD algorithm computes an approximation of the 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. In principle, the more modes obtained, the closer the approximation is to its theoretical solution.
By selecting only the most relevant PGD modes, a [[reduced order model]] of the solution is obtained. Because of this, PGD is considered a [[dimensionality reduction]] algorithm.
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=== Greedy algorithm ===
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 '''X<sub>1</sub>'''(''x''<sub>1</sub>), ..., '''X<sub>d</sub>'''(''x''<sub>d</sub>), which ''enrich'' the approximation of the solution.
== Features ==
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