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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 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.
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,
::<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>
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