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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. 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. The more modes obtained, the closer the approximation is to its theoretical solution. 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.
The PGD can be considered as a generalized form of the [[Proper orthogonal decomposition|Proper Orthogonal Decomposition]].
== Description ==
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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. Note that due to the greedy nature of the algorithm, the term 'enrich' is used rather than 'improve'. 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 [[
== Features ==
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