The Proper Generalized Decomposition (PGD) [1, 2] is an *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,

$$\begin{aligned} f(x_1, x_2, \ldots , x_n) = \sum _{m=1}^q F^m_{x_1}(x_1) F^m_{x_2}(x_2) \ldots F^m_{x_n}(x_n) = \sum _{m=1}^q \prod _{p=1}^{n} F^m_{x_p}(x_p). \end{aligned}$$

(1)

The key benefit of the use of a separable solution is to transform the multidimensional integrals arising in the weak form of the problem into products of single (or lower) dimensional integrals. This can be done as the integral of a separated function \(s(x,y,z)=f(x) g(y) h(z)\) can be written as,

$$\begin{aligned} \int _{\Omega _x} \int _{\Omega _y} \int _{\Omega _z} s(x,y,z)\, dx\, dy\,dx = \int _{\Omega _x} f(x) dx \; \int _{\Omega _y} g(y) dy\; \int _{\Omega _z} h(z) \,dx. \end{aligned}$$

(2)

The evaluation of 3 one-dimensional integrals of the right hand side requires smaller computational effort compared with its left hand side. The key idea is to apply the same separation strategy to the integrals arising in the weak form. A more detailed explanation of this procedure is given in Sect. “Problem statement and PGD solution for separated space dimensions”. However, as indicated in Eq. (2), not only the solution of the problem but all the functions involved in the operators must be separable. The typical functions present are material properties (thermal diffusivity, viscosity, density, etc.), initial or boundary conditions, and source terms, among others. In practice these functions usually do not admit an exact separable representation. The example used in this work consists in a Poisson problem including a non separable diffusivity function. Diffusivity could be defined by empirical laws or a fitting function of laboratory measurements; its expression then will be hardly separable. In those cases, in order to apply PGD it is usual to replace the non-separable function by a separable *approximation* to it. Section “Separation of the input data” presents one procedure to obtain separable approximations of known functions based on singular value decomposition.

Section “Results” shows results of a Poisson problem including the following non-separable diffusivity function:

$$\begin{aligned} k(x,y) = \sin \left( 0.5(x+y)^2\right) + 2. \end{aligned}$$

(3)

To apply PGD, *k* it is replaced by an approximation with the form

$$\begin{aligned} k(x,y) \approx k^{\text {sep}}(x,y) = \sum _{l=1}^{n_k} G^l_{x}(x) G^l_{y}(y). \end{aligned}$$

(4)

The separation procedures usually work with discrete (mesh based) versions of the function. Therefore, this separation introduces two new sources of errors that are not present in the traditional Finite Element approach. First, a truncation error is introduced due to the finite number of terms (\(n_k\)) used to describe \(k^{\text {sep}}\). Second, an interpolation error in the spatial representation of the functions \(G^l_{x}(x)\) and \(G^l_{y}(y)\) similar to the usual FE error is also included. The goal of this work is to study the relation between these errors and the accuracy of the PGD solution. This relation can be useful to determine the number of terms \(n_k\) required to achieve a certain accuracy by PGD. Moreover, this relation can also be used as stopping criteria for the PGD enrichment process, because the PGD solution will be at most as accurate as \(k^{\text {sep}}\).

### 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.

The first example was found while solving a BVP with parameterized geometry. In that case the shape of the domain, or the location of internal interfaces, depend on a set of parameters. Figure 1 shows a parameter dependant geometry for an airfoil and the objective is to find the air flow around it. The method applied was proposed in [3] and later extended in [4]. It is based on the idea of having a reference domain \(\mathcal {T}\) and a mapping function that relates all possible geometries to the reference domain. In practice, this mapping introduces some Jacobians depending on the parameters to the equation and, therefore, the dependence of the problem on the geometrical parameters becomes explicit. For example, the usual bilinear form for a Poisson problem reads

$$\begin{aligned} a(u,v) =\int _{\Omega ({\varvec{\mu }}) } \nabla u \cdot (k \nabla v) \,d\Omega = \int _{\mathcal {T}} \nabla _{\!\hat{\varvec{x}}} u \cdot \big (\underbrace{k \, | \mathbf{J ({\varvec{\mu }})} | \mathbf{J ({\varvec{\mu }})}^{-\textsf {T}} \mathbf{J ({\varvec{\mu }})}^{-1}}_\mathbf{D ({\varvec{\mu }})} \nabla _{\!\hat{\varvec{x}}} v\big ) \,d\hat{\varvec{x}} \end{aligned}$$

where \(\hat{\varvec{x}}\) are the reference coordinates. The matrix \(\mathbf D ({\varvec{\mu }})\), including all the Jacobians, accounts for the geometrical parameterization. The analytical expression of \(\mathbf D ({\varvec{\mu }})\) is known, but it is not separable. An approximation to it is therefore utilized in practice. The discretization used in the separation of *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 second example arises in a scheme proposed in [5] for the real-time integration of solid dynamics equations. The scheme combines Proper Orthogonal Decomposition (POD) and PGD approaches and it is based upon a parametric formulation depending on the initial conditions. It implements a direct time integrator that can be seen as a sort of black-box: it that takes the resulting displacement field of the current time step as input and (via POD) provides the result for the subsequent time step. In order to reduce the high dimensionality produced by the large amount of parameters describing the initial displacement field a reduced basis is obtained using PGD. This step introduces again a truncation error that will affect the final convergence of the proposed scheme. Figure 2 shows two results with 3 and 8 terms in the approximation of the input data (initial conditions). Note that the solution including only 3 terms presents phase errors that disappear for the 8 terms solution.

### A priori estimates for FE

Different sources of errors are present in the solution provided by PGD (see for example [6–8]. If *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

$$\begin{aligned} e = u-u_{H,M} = \underbrace{u - u_H}_{e_{FE}} + \underbrace{u_H - u_{H,M}}_{e_M} \end{aligned}$$

(5)

where the contribution of each type of error becomes explicit. Figure 3 shows schematically the relation between these errors. When the input data separation is required and functions are replaced by separable approximations, another source of errors is introduced. The replacement of function *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].

The standard error estimates for FE read

$$\begin{aligned} e_H = \Vert u-u_H\Vert \le C H^{\alpha }, \end{aligned}$$

for some value of alpha depending on the norm chosen, the element type and the regularity of the solution. For the sake of simplicity and in concordance with the proper measure for error expected in the separated approximation, in the following the norm under consideration, denoted by \(\Vert \cdot \Vert \), is the L\(_{2}\) norm.

If the diffusivity function *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:

$$\begin{aligned} e^\text {sep}_H = \left\| u-u^\text {sep}_H\right\| \le \left\| u-u_H\right\| + \left\| u_{H}-u^\text {sep}_H\right\| \le C H^{\alpha } + \text {Osc}, \end{aligned}$$

being \(\text {Osc} \propto \Vert k - k^\text {sep}\Vert \) in the case in which *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].

The final error of the PGD solution, therefore can be stated as

$$\begin{aligned} \left\| u-u^\text {sep}_{H,M}\right\| \le C H^\alpha + \tilde{C} F(M) + \text {Osc}. \end{aligned}$$

(6)

This bound expression shows that if Osc dominates over the truncation error, the error of the PGD solution cannot be reduced. On the other hand, if an estimation for Osc and for \(e^\text {sep}_H\) at enrichment step *i* are available, (6) can be used as stopping criteria of the enrichment process.