Proper generalized decomposition: Difference between revisions

Revision as of 15:29, 13 April 2020

Revision as of 15:37, 13 April 2020

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The discretization of the ___domain is a well defined set of procedures that cover (a) the creation of finite element meshes, (b) the definition of basis function on reference elements (also called shape functions) and (c) the mapping of reference elements onto the elements of the mesh.

As for the numerical algorithm, PGD assumes that the solution '''u''' of a (multidimensional) problem can be approximated as a separated representation '''u'''^''N'' of the form

::<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>

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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 '''X₁'''(''x''₁), ..., '''X_d'''(''x''_d), which ''enrich'' the approximation of the solution, (usually by improving it). 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.

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, PGD enables to re-adapt parametric problems into a multidimensional framework by setting the parameters of the problem as extra coordinates:

== Features ==

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, PGD enables to re-adapt parametric problems into a multidimensional framework by setting the parameters of the problem as extra coordinates:

Therefore, PGD enables to re-adapt parametric problems into a multidimensional framework by setting the parameters of the problem as extra coordinates:

::<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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where a series of functional products '''K₁'''(''k''₁), '''K₂'''(''k''₂), ..., '''K_p'''(''k''_p), each depending on a parameter (or parameters), has been incorporated to the equation.

In this case, the obtained approximation of the solution is called ''computational [[vademecum]]'': a general meta-model containing all the particular solutions for every possible value of the involved parameters.

== References ==