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::<math> \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), </math>
where the number of terms ''N'' and the functional products '''X<sub>1</sub>'''(''x''<sub>1</sub>), '''X<sub>2</sub>'''(''x''<sub>2</sub>), ..., '''X<sub>d</sub>'''(''x''<sub>d</sub>), each depending on a variable (or
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 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 [[Principal Component Analysis|PCA]], PGD modes are not necessarily [[orthogonal]] to each other.
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::<math> \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),</math>
where a series of functional products '''K<sub>1</sub>'''(''k''<sub>1</sub>), '''K<sub>2</sub>'''(''k''<sub>2</sub>), ..., '''K<sub>p</sub>'''(''k''<sub>p</sub>), each depending on a parameter (or
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.
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