- Research Article
- Open Access

# Effect of the separated approximation of input data in the accuracy of the resulting PGD solution

- Sergio Zlotnik
^{1}Email author, - Pedro Díez
^{1}, - David Gonzalez
^{2}, - Elías Cueto
^{2}and - Antonio Huerta
^{1}

**2**:28

https://doi.org/10.1186/s40323-015-0052-6

© Zlotnik et al. 2015

**Received:**11 May 2015**Accepted:**5 November 2015**Published:**25 November 2015

## Abstract

The proper generalized decomposition (PGD) requires separability of the input data (e.g. physical properties, source term, boundary conditions, initial state). In many cases the input data is not expressed in a separated form and it has to be replaced by some separable approximation. These approximations constitute a new error source that, in some cases, may dominate the standard ones (discretization, truncation...) and control the final accuracy of the PGD solution. In this work the relation between errors in the separated input data and the errors induced in the PGD solution is discussed. Error estimators proposed for homogenized problems and oscillation terms are adapted to asses the behaviour of the PGD errors resulting from approximated input data. The PGD is stable with respect to error in the separated data, with no critical amplification of the perturbations. Interestingly, we identified a high sensitiveness of the resulting accuracy on the selection of the sampling grid used to compute the separated data. The separation has to be performed on the basis of values sampled at integration points: sampling at the nodes defining the functional interpolation results in an important loss of accuracy. For the case of a Poisson problem separated in the spatial coordinates (a complex diffusivity function requires a separable approximation), the final PGD error is linear with the truncation error of the separated data. This relation is used to estimate the number of terms required in the separated data, that has to be in good agreement with the truncation error accepted in the PGD truncation (tolerance for the stoping criteria in the enrichment procedure). A sensible choice for the prescribed accuracy of the PGD solution has to be kept within the limits set by the errors in the separated input data.

## Keywords

- Proper generalized decomposition
- Error assessment
- Separable functions

## Background

*a priori*reduced basis technique designed to deal efficiently with highly-dimensional Boundary Value Problems (BVP). Differently from other discretisation techniques such as Finite Elements or Finite Differences, PGD avoids the exponential growth of the number of degrees of freedom with the number of dimensions. This is achieved by means of a separated representation of the solution. A

*separable function*

*f*with rank

*q*, separated on

*n*dimensions has the form,

*approximation*to it. Section “Separation of the input data” presents one procedure to obtain separable approximations of known functions based on singular value decomposition.

*k*it is replaced by an approximation with the form

### Motivation examples

Next sections present the effects of the separation of the input data in the case of an academic Poisson problem. Although, the motivation of this study comes after the authors faced PGD convergence problems in some more complex examples. Two of these problems are briefly described next.

*D*was found of to be of key importance in order keep the convergence of PGD. The truncation errors are also relevant and ultimately may control the final convergence that PGD could attain.

### A priori estimates for FE

*u*is the analytical solution of the BVP and \(u_{H,M}\) is the solution of PGD characterized by a mesh size

*H*and a number of terms

*M*, the PGD error is then defined by \(e:=u-u_{H,M}\). This error can be divided into several sources: first, an interpolation error, \(e_{FE}= u - u_{H}\), related with the space discretization, where \(u_H\) is the standard FE solution of the problem. Second, a truncation error \(e_M:=u_{H} - u_{H,M}\) that comes from the finite number of terms computed by PGD. The PGD error

*e*, therefore, can be written as

*k*by \(k^\text {sep}\) is assumed to affect similarly to the FE solution and the PGD solution (i.e. the truncation error is assumed to be independent of the error introduced by FE). If the error affects the source term, error estimators proposed for data oscillation could be used, for example [9].

*k*is the one separated, the ideas of homogenization theory (e.g. [10, 11]) can be recalled:

*k*can be understood as \(k = k^\text {sep} + \varepsilon \), being \(\varepsilon \) a highly oscillatory function with small amplitude compared to

*k*. Figure 4 shows the spatial variation of \(\varepsilon \) (computed as \(\Vert k^\text {sep} - k\Vert \)). Note that \(\varepsilon \) can be reduced by increasing the number of terms \(n_k\) in \(k^\text {sep}\). The problem, although, is inverse to the standard homogenization problem: the exact solution here is smooth and the high frequency terms are the errors introduced by the separation. The fact of replacing

*k*by \(k^\text {sep}\) produces the same error as the opposite. Thus, \(k^\text {sep}\) is seen as a

*de-regularization*of

*k*, where the high-frequency terms are truncated. This is the same effect produced in the homogenization, and therefore the error introduced by the homogenization is of the same type of the error produced in using a separated approximation of the material property. Thus, if oscillation terms are included (either by perturbations of

*k*or

*s*) an extra term appears:

*k*is replaced by \(k^\text {sep}\). The truncation error, \(e_M\), introduced by PGD is a function decreasing with the number of terms

*M*, so its norm is bounded by \(\Vert e_M\Vert \le \tilde{C} F(M)\). Note that, as mentioned above, for error affecting the source term

*s*the standard estimates for oscillation terms provide a similar expression for Osc [9].

*i*are available, (6) can be used as stopping criteria of the enrichment process.

## Problem statement and PGD solution for separated space dimensions

*u*, taking values in \(\Omega \), satisfies,

*s*, the prescribed values on the Dirichlet boundary \(u_{D}\), the prescribed flux on the Neumann value \(g_{N}\) and the diffusivity

*k*are the data set. The usual variational form for this problem reads: find \(u\in V\) such that

### Space-separated PGD algorithm

*k*(

*x*,

*y*) is also replaced by its separable approximation \(k^{\text {sep}}\), see Eq. (4), and therefore the operator \(a(\cdot ,\cdot )\) is somehow redefined. The problem then reads: find \(u^{\text {sep}}\) such that

*k*is not separable, this last step can not be done and the two integrals stay nested.

The definition of \(\ell (\cdot )\) remains unchanged.

The PGD solution is constructed one term at a time using the incremental procedure suggested in (10). The addition of a new term involves solving problem (10) with all the previously computed modes in their right hand side. Note that this problem is non linear because of the multiplication of the unknown functions \(F_x\) and \(F_y\). This non linearity is usually handled by an alternate-directions algorithm consisting in first solving for \(F_x\), assuming \(F_y\) is known, and then solving for \(F_y\), assuming \(F_x\) is known. These two (linear) subproblems are iterated until convergence.

*v*belong to \(V_0\) and they are written as \(v=\delta {F}_x F_{y} + F_x \delta {F}_{y} \). When solving the first subproblem, \(F_y\) is assumed to be fix and therefore \(\delta F_y\) vanishes. The test function

*v*, then, simplifies to \(v=\delta {F}_x F_{y}\). The first subproblem is stated as,

*x*and

*y*.

## Separation of the input data

Several procedures can be applied to obtain separable approximations of known functions. The proper orthogonal decomposition (POD) and the singular value decomposition (SVD) are the most common techniques when the separation is done in two dimensions. Many techniques have been proposed to extend SVD to higher number of dimensions. These techniques are usually called *higher-order*, as they were originally proposed to decompose higher-order tensors. An overview can be found, for example, at [12]. Some examples are the *higher-order SVD* (HOSVD) [13], the CANDECOMP/PARAFAC (CP) [14, 15] and the Tucker decomposition [16].

When the number of separated dimensions is two, the POD and the SVD are equivalent and they provide a optimal decomposition in the sense that they provide the minimum number of required to obtain an given accuracy. Unfortunately, for \(n>2\) this property is lost and usually there is no guarantee of the optimality of the separated tensor.

Recently in [17] a method based on PGD was proposed to perform efficiently separation of functions. This approach has the advantages of being equivalent to SVD when the separation is done in two-dimensions and it is trivial to extend it to higher dimensions. This technique produced decompositions having lower rank than HOSVD for all tested cases and it does not require to specify the order of the separated function before starting the process (as CP does).

*f*(

*x*,

*y*) supported on a finite element (FE) mesh, that is,

*f*is determined by a set of nodal values \(f_i\) for \(i=1, \ldots , n_t\). In this case the dimensions in which the function will be separated are the cartesian axis

*x*and

*y*. The form of the approximation is,

*r*and coefficients \(f(x_i,y_i)\), where \(x_i\) and \(y_i\) are the nodal locations. The SVD provides a factorisation \(\mathbf M \) in the form

*k*(

*x*,

*y*) (top right), the amplitude of the initial terms in the separated version of

*k*, that is, the diagonal coefficients of the matrix \(\mathbf S \) (top left), and the functions \(F^m\) and \(G^m\) for the four initial terms of \(k^{\text {sep}}(x,y)\). Note that with the initial 25 terms the function

*k*is approximated to machine precision. The meshes corresponding to

*F*and

*G*, both have 402 nodes.

### Influence of the sampling points

The first idea is sampling the input data (material parameters, source terms\(\ldots \)) on the nodes of the grid used for the space and parametric discretization. As it is shown in the next section, this choice is not particularly sensible because the values of these functions are required at the integration points of the FE mesh used to solve the weak form of the equation. This extends not only to the spatial coordinates but also to the parametric coordinates because the parametric modes are approximated in a least squares sense (Galerkin L\(_{2}\) projection). Thus, separation has to be performed on the basis of values sampled at integration points: sampling at the nodes defining the functional interpolation results in an important loss of accuracy.

## Results

The behaviour of the PGD scheme with respect to errors in the input data is studied next via a series of numerical experiments. The problem (7) is solved using PGD as described above in a square domain with size \([0,4]\times [0,4]\). It is closed with Dirichlet boundary conditions on the top and bottom sides with values one and zero respectively and homogeneous Neumann in the lateral sides. The separated diffusivity function (4) is used. The mesh is structured and regular and has 100 elements in each dimension.

The relative errors shown in convergence curves are computed as the \(H_1\) norm of the relative difference between the PGD solution and a reference Finite Element solution computed over the same mesh. Note that the FE solution is computed using the exact analytic expression for the diffusivity *k*.

### Input data sampling

The diffusivity function (3) is separated using the SVD approach described in Sect. “Separation of the input data”. To do that, the spatial grid to sample the function *k*(*x*, *y*) needs to be selected. The first choice taken here is to evaluate *k* in the same mesh that will be later used in the discretization of *u*. In this case, it is a regular grid with \(101\times 101\) nodes. This is an overkill mesh to represent the function *k* (see first panel of Fig. 5). The \(k^\text {sep}\) separated function described with 26 terms has an maximum nodal relative error of the order of machine tolerance (\(10^{-14}\)).

*k*.

In the example above, despite the nodal values of \(k^\text {sep}\) have errors that could be negligible, the interpolated values at the mid points of the elements have relative error of order \(10^{-2}\), coinciding with the maximum accuracy that PGD could provide.

*x*and

*y*. Same as in the previous grid, the nodal values of \(k^\text {sep}\) have errors comparable of machine tolerance but, in this case, the spatial interpolation is completely avoided. When this new \(k^\text {sep}\) is used, the limit imposed by the interpolation disappears and PGD recovers it normal convergence.

### Accuracy of \(k^\text {sep}\)

*k*function. All curves present a final flattening and a convergence to an error that is imposed by truncation error of \(k^\text {sep}\). In other words, at some point, the error Osc (that does not depends on the number of terms) dominates in (6) and therefore the PGD error cannot decrease. The better the description of \(k^\text {sep}\) (that is, the larger \(n_k\) and the smaller the Osc term), the smaller the final error achieved by PGD. For this example and when Osc dominates, the relation between the PGD error and \(k^\text {sep}\) truncation error is linear with slope close to one (as shown in Fig. 9).

*k*. To test the robustness of the result, the accuracy study is repeated using a new discontinuous function \(k_2\) defined as

*k*to reach nodal machine precision. The results are consistent with the previous and the final convergence of the PGD solution is controlled by the accuracy of \(k_2^{sep}\). Figure 11 shows convergence curves for \(k_2^{sep}\) having 10, 20, 30, 35 and 40 terms.

## Conclusions

The stability of PGD with respect to errors in the input data was studied by means of numerical experiments. These errors are in practice present due to the need of approximate input data by truncated separable expressions. Moreover, separation requires discretization introducing into the input data a “spatial” h-like error.

Results show that PGD is stable (it does not amplify errors). In the tested case of a boundary value problem governed by the Poisson equation, the errors introduced on the diffusivity function are linear with the final error that PGD commits. The grid in which the separated data is represented is crucial to the accuracy of PGD; to minimize interpolation error, the mesh for the input data should coincide with the integration points used for the solution of *u*.

The relation between the errors in the input data and the final error of PGD can be used to decide the accuracy required in the input data to get a certain accuracy on the PGD solution. In the example presented in this work, as the relation between these errors is linear, it is straightforward to determine, given the desired accuracy in the final solution, which is the accuracy required at the nodal values in the input data.

This relation can also be used as stoping criteria for the enrichment process of the PGD solution. If some error indicator is available (see for example [6]), the limit imposed by the separated input data can be used as the tolerance to end the enrichment process. In the case that no formal error indicator is computed, the relative amplitude of the last computed term, \(\alpha _n / \Vert u^{PGD}\Vert \), can be compared with the tolerance imposed by the separation of the input data. Note that this is an heuristic stopping criteria that cannot be translated into an estimation on error on the solution. Although, it provides a reference value for the tolerance as the relative amplitude of the terms is not expected to be much smaller than it.

## Declarations

### Authors' contributions

All authors discussed the content of the article based on their previous experiences using PGD. SD and PD prepare and run the numerical examples based on the Poisson problem. EC and DG prepare and run the numerical examples based on the XXX problem. All authors read and approved the final manuscript.

### Acknowledgements

This work has been partially supported by the Spanish Ministry of Science and Competitiveness, through Grant Number CICYT- DPI2014-51844-C2-2-R and DPI2014-51844-C2-1-R and by the Generalitat de Catalunya, Grant Number 2014-SGR-1471.

### Competing interests

The authors declare that they have no competing interests.

**Open Access**This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

## Authors’ Affiliations

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