Proper Generalized Decomposition computational methods on a benchmark problem: introducing a new strategy based on Constitutive Relation Error minimization

The benchmark problem

We consider a transient problem defined on a one dimensional domain \(\Omega = [0, L]\), with boundary \(\partial \Omega = \{0;L\}\), over the time interval \(\mathcal {I}= [0,T]\). It is assumed that a prescribed zero temperature is applied on \(\partial \Omega\) and that the domain is subjected to a time-dependent thermal loading that consists of a thermal source \(f_d(x,t)\) in \(\Omega\). This loading can be seen as a moving flame/laser beam such that \(f_d(x,t) = \delta (x - vt)\), a Dirac distribution, where v is the velocity of the flame/laser beam (see Figure 1).

For the sake of simplicity, the initial conditions are set to zero. The material that composes \(\Omega\) is assumed to be homogeneous and fully known where c (resp. \(\mu\)) represents the heat capacity (resp. conductivity) of the material.

In the following, in order to be consistent with other linear problems encountered in Mechanics (linear elasticity for instance), we reform the variable \(\varphi \rightarrow -\varphi\) which leads, in particular, to the new constitutive relation \(\varphi = \mu \frac{\mathrm \partial ^{}{u}}{\mathrm \partial {x}^{}}\).

Optimal reduced solution

Let us suppose that the full discrete approximated solution \(u_h\) is known (under the form \(\mathbb {U}\)), and one looks to extract basis functions such as \(u_h\) can be approximated under the separated representation form (1).

For physical problems, the energy norm \(\Vert \cdot \Vert _E\) is the natural norm in which measuring the error between \(u_{ex}\) and \(u_h\). Therefore, to compare SVD and PGD algorithms, we choose the associated energy scalar product, denoted \(\left\langle \!\left\langle \cdot , \cdot \right\rangle \!\right\rangle _E\), that verifies the variable separation property:

$$\begin{aligned} \left\langle \!\left\langle u, {w} \right\rangle \!\right\rangle _E = \left\langle \sqrt{\left\langle u(t), {w}(t) \right\rangle _{\mathcal {V}}}, \sqrt{\left\langle u(t), {w}(t) \right\rangle _{\mathcal {V}}} \right\rangle _{\mathcal {T}} \end{aligned}$$

(7a)

$$\begin{aligned} \left\langle u(x,t), {w}(x,t) \right\rangle _{\mathcal {T}} = \int \limits _{\mathcal {I}}^{}{u(x,t) {w}(x,t)}\,\text {d}{t} \end{aligned}$$

(7b)

$$\begin{aligned} \left\langle u(x,t), {w}(x,t) \right\rangle _{\mathcal {V}} = \int \limits _{\Omega }^{}{\frac{\mathrm \partial ^{}{u}}{\mathrm \partial {x}^{}}\!(x,t) \frac{1}{\mu } \frac{\mathrm \partial ^{}{{w}}}{\mathrm \partial {x}^{}}\!(x,t) }\,\text {d}{x} \end{aligned}$$

(7c)

The implied discrete inner products \(\left\langle \cdot , \cdot \right\rangle _{\mathcal {V}_h}\) and \(\left\langle \cdot , \cdot \right\rangle _{\mathcal {T}_h}\) can be written:

$$\begin{aligned} \left\langle u, {w} \right\rangle _{\mathcal {V}_h} = \varvec{u}^{\dagger } \mathbb {K} \varvec{{w}}, \quad \left\langle u, {w} \right\rangle _{\mathcal {T}_h} = \varvec{u} \mathbb {M} \varvec{{w}}^{\dagger } \end{aligned}$$

where \(\mathbb {K} \in \mathbb {R}^{N_x^2}\) (resp. \(\mathbb {M} \in \mathbb {R}^{N_t^2}\)) is the finite element stiffness matrix (resp. the finite element mass matrix in time) and \(\varvec{u},\varvec{w} \in \mathbb {R}^{N_x \times N_t}\) the FEM solution. Employing a Cholesky factorization, the inner product \(\langle \!\langle \cdot , \cdot \rangle \!\rangle _E\) can be transformed to the standard Euclidean inner product as:

$$\begin{aligned} \Vert u \Vert _{\mathcal {V}_h} = \Vert \sqrt{\mathbb {K}} \varvec{u} \Vert _{2}, \quad \Vert u \Vert _{\mathcal {T}_h} = \Vert \varvec{u} \sqrt{\mathbb {M}} \Vert _{2} \end{aligned}$$

(8)

To avoid the use of the Euclidean norm in (6) and thanks to the transformation (8), ones can compute the reduced SVD according to the energy norm through the regular SVD algorithm on the matrix \(\sqrt{\mathbb {K}}\mathbb {U}\sqrt{\mathbb {M}}\) rather than on the matrix \(\mathbb {U}\). Therefore, the separated variable decomposition

obtained from the regular SVD

verifies:

Matrices

and

that contain the reduced basis functions \(\psi _i\) and \(\lambda _i\) introduced in (1), have to verify the following relations:

Classical PGD methods

Usually, the brute force solution \(u_h\) is out of reach. In this section, classical PGD strategies are reviewed, aiming to approximate the solution u under the form \(u_m\) that verifies (1) without knowledge of the solution u. In these methods, the reduced basis is computed on the fly as the problem is solved.

Galerkin PGD

First, a PGD computational method based on Galerkin orthogonality [11] and well-suited for diffusion-dominated problems is presented. By injecting the separated variable form (9) in (4), the new couple \((\psi ,\lambda ) \in \mathcal {V}\times \mathcal {T}\) is the optimal one that verifies the Galerkin orthogonality:

$$\begin{aligned} B\!\left( u_m + \psi \lambda , \psi ^*\lambda + \psi \lambda ^*\right) = L\!\left( \psi ^*\lambda + \psi \lambda ^*\right) \quad \forall \psi ^*\in \mathcal {V},\lambda ^* \in \mathcal {T}\end{aligned}$$

(10)

That equation naturally leads to the following problems :

• the weak formulation of a partial differential equation (PDE) in space, usually approximated by FEM:

  $$\begin{aligned} B\!\left( u_m + \psi \lambda , \psi ^*\lambda \right) = L\!\left( \psi ^*\lambda \right) \quad \forall \psi ^* \in \mathcal {V}\end{aligned}$$

  (11)
• the weak formulation of ordinary differential equation (ODE) in time:

  $$\begin{aligned} B\!\left( u_m + \psi \lambda , \psi \lambda ^* \right) = L\!\left( \psi \lambda ^*\right) \quad \forall \lambda ^* \in \mathcal {T}\end{aligned}.$$

  (12)

A couple \((\psi ,\lambda ) \in \mathcal {V}\times \mathcal {T}\) then verifies (10) if and only if \(\psi\) verifies (11) and \(\lambda\) verifies (12), which is a non-linear problem. As this problem can be interpreted as a pseudo-eigenproblem [22], a natural algorithm to capture the dominant eigenfunction consists of a power iterations strategy. Starting from an initial random time function \(\lambda\) (verifying initial conditions), each iteration of the power algorithm verifies a sequence of lower dimensional problems (11–12), leading to Algorithm 1.

Remark 2

The key property of Galerkin computational method is that the error is orthogonal to the approximation spaces. Therefore, \(u_{ex} - u_m\) is orthogonal to \(\mathcal {V}_h \otimes \mathcal {T}_h\) according to \(B(\cdot ,\cdot )\) scalar product.

Remark 3

Normalization of space \(\psi\) or time \(\lambda\) functions is preferable for stability reason of the power iterations algorithm. In Algorithm 1, we arbitrarily choose to normalize the space function \(\psi\).

Remark 4

One could check the convergence of \(\psi \lambda\) to stop the power iterations. As explained in [22], a coarse criterion is sufficient to obtain good approximation as convergence is reached quickly. In practice, we rather prefer to fix a number of sub-iterations \(k_{max} = 4\), letting the next PGD mode to correct the previous one if necessary.

Improvement of the decomposition

A post-processing of the computed functions can be made to improve the decomposition, like orthogonalization of space basis functions. A simple modification to the power iterations algorithm consists in updating the whole set of time functions \(\Lambda _m = \left\{ \lambda _i\right\} _{i=1 \ldots m}\) after every n-th new PGD mode. These functions are defined by a system of ODEs of dimension m analogous to (12). A mapping application T is defined taking the whole set of space functions \(\Psi _m = \left\{ \psi \right\} _{i=1 \ldots m} \in \mathcal {V}^{m}\) into time functions \(\Lambda _m = T(\Psi _m) \in \mathcal {T}^{m}\) defined by:

$$\begin{aligned} B(\Psi _m \cdot \Lambda _m,\Psi _m \cdot \Lambda _m^*) = L(\Psi _m \cdot \Lambda _m^*) \quad \forall \Lambda _m^* \in \mathcal {T}^m \end{aligned}$$

(13)

This post-processing improves the quality of the progressive PGD algorithm, although the obtained decomposition is not the optimal one. If the update step is done after each new PGD mode, it is added after the power iterations step, as shown in Algorithm 2. Known as update step [26], it can be applied to all PGD computational methods.

Minimal residual PGD

The next technique is a PGD strategy based on a minimal residual criterion [21]. This construction presents monotonic convergence of the decomposition in the sense of residual norm.

The residual \(\mathcal {R}(u) \in \mathcal {V}\otimes \mathcal {T}\) is defined under the form:

$$\begin{aligned} \left\langle \!\left\langle {w}, \mathcal {R}(u) \right\rangle \!\right\rangle = L({w}) - B(u,{w}) = \langle \!\langle {w}, \delta - \underbrace{\left( k\frac{\mathrm \partial ^{}{u}}{\mathrm \partial {x}^{}}+c\frac{\mathrm \partial ^{}{u}}{\mathrm \partial {t}^{}}\right) }_{\mathcal {B}(u)} \rangle \!\rangle \end{aligned}$$

Therefore, an optimal couple \((\psi ,\lambda ) \in \mathcal {V}\times \mathcal {T}\) is defined as the one that minimizes the residual norm:

$$\begin{aligned} (\psi ,\lambda ) = \mathop {{{\mathrm{\arg \min }}}}\limits _{\psi ,\lambda } \Vert \mathcal {R}(u_{m}+\psi \lambda )\Vert ^2 \end{aligned}$$

or equivalently:

$$\begin{aligned} (\psi ,\lambda ) = \mathop {{{\mathrm{\arg \min }}}}\limits _{\psi ,\lambda } \frac{1}{2} \left\langle \!\left\langle \mathcal {B}(\psi \lambda ), \ \mathcal {B}(\psi \lambda ) \right\rangle \!\right\rangle - \left\langle \!\left\langle \mathcal {R}(u_m), \ \mathcal {B}(\psi \lambda ) \right\rangle \!\right\rangle \end{aligned}$$

(14)

In a continuous framework, (14) leads to a non-classical formulation as it requires the introduction of a more refined functional space of the space domain \(\mathcal {V}\) in order to guaranty existence and uniqueness of solutions (by introducing \(H^2(\Omega ) \otimes H^1(\mathcal {I})\) in place of \(H^1(\Omega ) \otimes L^2(\mathcal {I})\) for our benchmark problem). In practice, a minimal residual formulation is applied to the discretized problem (4) after introducing approximation spaces \(\mathcal {V}_h\) and \(\mathcal {T}_h\).

Therefore, by introducing the discrete \(N_xN_t \times N_xN_t\) matrix \(\mathbb {B}\) (resp. \(N_xN_t\) vector \(\varvec{L}\)) of the bilinear operator \(B(\cdot ,\cdot )\) (resp. the linear operator \(L(\cdot )\)), the discrete \(N_xN_t\) residual vector \(\varvec{R}\) of \(\mathcal {R}(\cdot .)\) takes the form:

$$\begin{aligned} \varvec{R}(u) = \mathbb {B}\varvec{u} - \varvec{L}, \quad \varvec{u} = \begin{bmatrix} \varvec{u}(t_1) \\ \varvec{u}(t_2) \\ \vdots \end{bmatrix} \end{aligned}$$

such that an optimal couple \((\psi ,\lambda ) \in \mathcal {V}_h \times \mathcal {T}_h\) minimizes the discretized residual norm (Euclidean norm, since the matrix \(\mathbb {B}\) is not invertible):

$$\begin{aligned} (\psi ,\lambda ) = \mathop {{{\mathrm{\arg \min }}}}\limits _{\psi ,\lambda } \Vert \varvec{R}(u_m + \psi \lambda )\Vert _{2}^2 \end{aligned}$$

By introducing notation \(\varvec{u}_{m+1} = \varvec{u}_{m} + \varvec{\psi }\cdot \varvec{\lambda }\), the minimization leads to the problem:

$$\begin{aligned} \left( \varvec{\psi }^*\cdot \varvec{\lambda }+\varvec{\psi }\cdot \varvec{\lambda }^*\right) ^T\left[ \mathbb {B}^T\mathbb {B}\left( \varvec{u}_m + \varvec{\psi }\cdot \varvec{\lambda }\right) - \mathbb {B}^T\varvec{L}\right] = 0 \end{aligned}$$

(15)

Let us define the \(N_x \times N_x\) matrices \(\mathbb {C}_{ij}\) and the \(N_x\) vectors \(\varvec{F}_i\) such that:

$$\begin{aligned} \mathbb {B}^T\mathbb {B} = \begin{bmatrix} \mathbb {C}_{11}&\mathbb {C}_{12}&\cdots \\ \mathbb {C}_{21}&\ddots&\\ \vdots&&\ddots \\ \end{bmatrix}, \quad \mathbb {B}^T\varvec{L} = \begin{bmatrix} \varvec{F}_1 \\ \varvec{F}_2 \\ \vdots \end{bmatrix} \end{aligned}$$

The discrete minimization (15) leads to the following reduced problems:

• a discretized space problem:

  $$\begin{aligned} \left( \sum _{i,j} \varvec{\lambda }_i\varvec{\lambda }_j \mathbb {C}_{ij}\right) \varvec{\psi } = \sum _{i}\varvec{\lambda }_i\varvec{F}_i \end{aligned}$$

  (16)
• a discretized time problem:

  $$\begin{aligned} \begin{bmatrix} \varvec{\psi }^T\mathbb {C}_{11}\varvec{\psi }&\varvec{\psi }^T\mathbb {C}_{12}\varvec{\psi }&\cdots \\ \varvec{\psi }^T\mathbb {C}_{21}\varvec{\psi }&\ddots&\\ \vdots&&\ddots \end{bmatrix}\varvec{\lambda } = \begin{bmatrix} \varvec{\psi }^T\varvec{F}_1 \\ \varvec{\psi }^T\varvec{F}_2 \\ \vdots \end{bmatrix} \end{aligned}$$

  (17)

Remark 5

As for all Galerkin based methods, the error \(u_{ex}-u_m\) is orthogonal to the approximation space \(\mathbb {B}^T\mathbb {B}\left( \mathcal {V}_h\otimes \mathcal {T}_h\right)\) according to the Euclidean scalar product, and depends strongly on the conditioning of the operator \(\mathbb {B}^T\mathbb {B}. \)

Petrov–Galerkin PGD

In this section, another possible PGD computational method is presented, based on a Petrov–Galerkin criterion, also called MinMax [22]. Such a formulation is frequently used for solving PDEs which contain terms with odd order, implying a loss of symmetry in the weak formulation, as transport-dominated problems.

This Petrov–Galerkin formulation uses the weak formulation (4) where the unknown u and test function w are in different finite dimensional subspaces. Under the variable separation assumption, the test function w is taken as another couple of space and time functions \((\tilde{\psi },\tilde{\lambda }) \in \mathcal {V}\times \mathcal {T}\) such that the new PGD couple \((\psi ,\lambda ) \in \mathcal {V}\times \mathcal {T}\) is the optimal one that verifies the Galerkin orthogonality:

$$\begin{aligned} B\!\left( u_m + \psi \lambda , \tilde{\psi }^*\tilde{\lambda } + \tilde{\psi }\tilde{\lambda }^*\right) = L\!\left( \tilde{\psi }^*\tilde{\lambda } + \tilde{\psi }\tilde{\lambda }^*\right) \quad \forall \tilde{\psi }^*\in \mathcal {V},\tilde{\lambda }^* \in \mathcal {T}\end{aligned}$$

(18)

This leads to the two following orthogonality criteria:

$$\begin{aligned} B\!\left( u_m + \psi \lambda , \tilde{\psi }^*\tilde{\lambda }\right) = L\!\left( \tilde{\psi }^*\tilde{\lambda }\right) \quad \forall \tilde{\psi }^*\in \mathcal {V} \end{aligned}$$

(19a)

$$\begin{aligned} B\!\left( u_m + \psi \lambda , \tilde{\psi }\tilde{\lambda }^*\right) = L\!\left( \tilde{\psi }\tilde{\lambda }^*\right) \quad \forall \tilde{\lambda }^*\in \mathcal {T} \end{aligned}$$

(19b)

As a matter of fact, additional equations must be added in order to define the new functions \((\tilde{\psi },\tilde{\lambda }) \in \mathcal {V}\times \mathcal {T}\). The following orthogonality criteria (20a), (20b) are used, where \(\left\langle \!\left\langle \cdot , \cdot \right\rangle \!\right\rangle\) is an inner product on \(\mathcal {V}\otimes \mathcal {T}\). To compare PGD computational methods one another, we select the energy inner product (7a)–(7c).

$$\begin{aligned} B\!\left( \psi ^*\lambda , \tilde{\psi }\tilde{\lambda }\right) = \left\langle \!\left\langle \psi ^*\lambda , \psi \lambda \right\rangle \!\right\rangle _E \quad \forall \psi ^*\in \mathcal {V} \end{aligned}$$

(20a)

$$\begin{aligned} B\!\left( \psi \lambda ^*, \tilde{\psi }\tilde{\lambda }\right) = \left\langle \!\left\langle \psi \lambda ^*, \psi \lambda \right\rangle \!\right\rangle _E \quad \forall \lambda ^*\in \mathcal {T} \end{aligned}$$

(20b)

As for previous algorithms, an approximation of \((\psi ,\tilde{\psi },\lambda ,\tilde{\lambda }) \in \mathcal {V}^2\times \mathcal {T}^2\) is computed to verify (18) and (20a), (20b) simultaneously. As such a problem is non-linear, a power iterations strategy is chosen, where the four lower dimension problems are solved iteratively one after the other (see Algorithm 4).

Remark 6

As for all Galerkin based methods, the error \(u_{ex}-u_m\) is orthogonal to an approximation space \(\mathcal {L} = \left\{ \tilde{\psi }_i\otimes \tilde{\lambda }_i\right\} _{i = 1 \ldots m}\) of the test space according to the \(B(\cdot ,\cdot )\) scalar product.

Minimisation of the CRE

To control the PGD computational process, specific error indicators have been built in recent years. They particularly assess the error due to truncation of the sum in the separated decomposition (1). A first robust approach for PGD verification, using the concept of CRE, was proposed in [27, 28] leading to guaranteed and relevant error evaluation. It specificity lies on the way to construct the required dual fields, as \({\varphi (u_m)} = \mu \frac{\mathrm \partial ^{}{u_m}}{\mathrm \partial {x}^{}}\) does not verify equilibrium (3). The solution leads to a variable separation of \({\varphi (u_m)}\) with a static formulation, done from and after the classical Galerkin PGD space problem. The obtained data combine with prescribed ones, as displacements, traction and body forces define the starting point to evaluate the reduction error.

In this error estimator, the required dual field is computed after the PGD procedure and then cannot influence the solution. Here we propose to compute and control the PGD procedure on the fly with the CRE estimator.

A partial minimization

The full minimization leads to invert a huge problem. In the general case, it is computationally expensive. We propose here a partial minimization where condition (21) is not taken into account in the minimization process but afterwards. That leads to a new and promising PGD computational strategy.

Minimizing (22) under constraint (25) leads to the minimization in \(\mathcal {V}\otimes \mathcal {T}\otimes \mathcal {S}\) of:

$$\begin{aligned} \left( \psi ,\lambda ,Z\right) = \mathop {{{\mathrm{\arg \min }}}}\limits _{\psi ,\lambda ,Z} \int \limits _{\mathcal {I}}^{}{\int \limits _{\Omega }^{}{\frac{1}{\mu }\left( Q + \tau _m - \mu \frac{\mathrm \partial ^{}{u_m}}{\mathrm \partial {x}^{}} + \frac{\mathrm d^{}{\lambda }}{\mathrm d{t}^{}}Z - \mu \frac{\mathrm d^{}{\psi }}{\mathrm d{x}^{}}\lambda \right) ^2}\,\text {d}{x}}\,\text {d}{t} \end{aligned}$$

(29)

Minimization (29) leads to:

• According to \(\lambda\):

  $$\begin{aligned} 0 &= \int \limits _{\mathcal {I}}^{}{\int \limits _{\Omega }^{}{\mu \left( Q + \tau _m +\frac{\mathrm \partial ^{}{\lambda }}{\mathrm \partial {t}^{}}Z - \frac{\mathrm \partial ^{}{u_m}}{\mathrm \partial {x}^{}} - \frac{\mathrm d^{}{\psi }}{\mathrm d{x}^{}}\lambda \right) \left( Z\frac{\mathrm d^{}{\lambda ^*}}{\mathrm d{t}^{}} - \mu \frac{\mathrm d^{}{\psi }}{\mathrm d{x}^{}}\lambda ^*\right) }\,\text {d}{x}}\,\text {d}{t} \qquad \forall \lambda ^* \in \mathcal {T}\end{aligned}$$

  (30)
• According to \(\psi\). The field Z is fixed to the previous value computed at the previous iteration of the power iterations algorithm in first place:

  $$\begin{aligned} 0 = \int \limits _{\mathcal {I}}^{}{\int \limits _{\Omega }^{}{\left( \frac{\mathrm \partial ^{}{u_m}}{\mathrm \partial {x}^{}} + \frac{\mathrm d^{}{\psi }}{\mathrm d{x}^{}}\lambda - Q + \tau _m +\frac{\mathrm \partial ^{}{\lambda }}{\mathrm \partial {t}^{}}Z\right) \lambda \frac{\mathrm d^{}{\psi ^*}}{\mathrm d{x}^{}}}\,\text {d}{x}}\,\text {d}{t} \qquad \forall \psi ^* \in \mathcal {V}\end{aligned}$$

  (31)

• One can then determine Z to verify (25).

Remark 7

The CRE quantity \(\varphi _{m}-\mu \frac{\mathrm \partial ^{}{u_m}}{\mathrm \partial {x}^{}} = \tau _{m}+Q-\mu \frac{\mathrm \partial ^{}{u_m}}{\mathrm \partial {x}^{}}\) is orthogonal to \(\mathcal {V}\otimes \mathcal {T}\) and \(\mathcal {S}\otimes \mathcal {T}\) approximation spaces according to the energy scalar product.

Remark 8

For stationary problems, condition (24) does not apply. The idea is to seek \(\tau \in \mathcal {S}_0\) of the form:

$$\begin{aligned} \tau \simeq \tau _m = \sum _{i=1}^m \tilde{Z}_i(x,t) \end{aligned}$$

where each \(\tilde{Z}_i \in \mathcal {S}_0 = \left\{ w \in L^2(\Omega ,\mathcal {I}) \mid \forall z \in L^2(\Omega ,\mathcal {I}), \int \limits _{\mathcal {I}}^{}{\int \limits _{\Omega }^{}{w \frac{\mathrm \partial ^{}{z}}{\mathrm \partial {x}^{}}}\,\text {d}{x}}\,\text {d}{t} = 0\right\}\). Since potential and complementary energies are decoupled, each \(\tilde{Z}_i\) is decoupled from \(\psi _i\).

A non-separable lifting algorithm

Error estimation

Influence of the update phase

First we propose to compare the classical PGD computational methods presented in “Classical Proper Generalized Decomposition methods”, with or without the update step (see “Improvement of the decomposition”). Convergence curves are given in Figure 2 for a specific value of velocity set to \(v=L/2T\) for both reference and residual error estimators for classical PGD computational methods.

In Figure 2a, the first 10 modes of Galerkin/Petrov–Galerkin computational methods are close to the optimal decomposition obtained by SVD. Although extra modes degrade that convergence, they tend to follow the same path. Very poor convergence is observed for the classical Minimal Residual PGD computational method according to the reference error \(\overline{e_{ref}}\). For Galerkin/Petrov–Galerkin PGD computational methods, the update step improves significantly the quality of the PGD approximation without fitting the optimal one from SVD. They seem to result in the same PGD decomposition as the error is identical. To get an error of 1%, those two methods need 20 modes when the SVD requires 18 modes.

In general, the reference solution is unknown and convergence plots in Figure 2a cannot be obtained. Reduction error is therefore estimated through the residual error estimator \(\overline{e_{res}}\) as represented in Figure 2b. An estimation about 40 times larger than the reference one for all computational methods is observed. The residual error \(\overline{e_{res}}\) predicts that the best PGD decomposition comes from the Minimal Residual PGD computational method as it minimizes the residual, even if the reference reduction error says otherwise. The update step for this specific method does not improve the PGD decomposition. On the contrary, the update step highly improves Galerkin/Petrov Galerkin PGD computational methods to the same convergence curve, with a PGD decomposition better than the Minimal Residual PGD one. According to the residual error estimator, the SVD decomposition is not the optimal one, since we build it according to the energy norm rather than Euclidean norm.

As the purpose of reduction model is to reduce the computational cost of a brute force approximated solution, Figure 3 shows the computational cost of each PGD computational method where the time is averaged by \(t_{SVD}\), the time required to compute a brute force solution and the separated form through SVD.

In Figure 3 the reference reduction error is plotted as a function of the averaged time to reach it. For the same error level, the consideration of the update step for Galerkin/Petrov–Galerkin PGD computational method requires more computational time than the classical method. However, when using the update step fewer PGD modes are required than without update, thus enhancing storage footprint of the solution and post-processing manipulation.

The update step improves drastically the PGD solution and, in the author opinion, should always be used. In the following, the PGD computational methods will only refer to the PGD one with time function update step, unless otherwise specified.

Minimal CRE performances

In “Minimisation of the Constitutive Relation Error” we proposed a new PGD computational method based on the minimization of CRE, with two variants. In Figure 4, the convergence of each Minimal CRE PGD variant is represented with or without the update step.

We observe in Figure 4a, that the first 10 modes of Full/Partial Minimal CRE PGD computational methods are close to the optimal decomposition obtained by SVD. However extra modes degrade that convergence as the full minimization is better than the partial one. One should observe that when the update step is used, full and partial minimizations have the same convergence rate. As a matter of fact, the partial minimization with update gives similar results as the full minimization with update, for a lower complexity (Figure 5). Despite the absence of computational time gain in 1D, we hope that the partial minimization will present a better computational time in higher dimension.

In general, the reference solution is unknown and convergence plots in Figure 4a cannot be obtained. Since the new proposed Minimal CRE PGD computational methods have their own reduction error estimation (see “Minimisation of the Constitutive Relation Error”) evaluated through the CRE indicator, the reduction error is therefore estimated through the CRE estimator \(\overline{e_{CRE}}\) as represented in Figure 4b. As seen in Figure 4a, the update step improves the convergence of the computational methods close to the reference one. As a matter of fact, they are effective (accurate and guaranteed) as shown in Figure 6.

As a result, in the general case, one can only evaluate some error estimators depending on the computational methods. Then the CRE indicator obtained by minimizing the CRE leads to an improved and accurate reduction error estimation. Other PGD computational methods errors underestimate the reduction error. Unlike the Minimal CRE PGD solution, they require to compute extra modes to target a specific reference error.

The two CRE minimization strategies show great performances, robustness from minimization and give direct effective error estimator.

Limits of the PGD

In “A non-separable lifting algorithm”, we proposed a PGD computational method lifted with a non-separable known solution. Figure 7 represents the enrichment solution \(u_{\infty }\) solution over an infinite domain and the correction PGD solution \(u_{m}\) computed with a Galerkin PGD strategy with time functions update step. 3 modes are enough as the PGD solution matches the reference solution \(u_{ref}\) below a relative reference error of \(10^{-3}\). It is easy to observe that the correction term verifies the separation of variables from Figure 7, as the solution is smooth and continuous.

In Figure 8, the number of PGD modes required to obtain a reference error \(\overline{e_{ref}}\) below \(10^{-2}\) is studied depending on the loading velocity. While SVD and PGD computational methods require much more modes to catch the influence of the speed when the velocity increases, the lifting method only requires 2 PGD modes to get an accurate solution below a relative reference error of \(10^{-2}\). Such properties come from the lifting solution \(u_{\infty }\) that catches the influence of the speed. According to (6), the 35th singular value of \(u_{\infty }\) is below \(10^{-2}\) for \(v = \frac{L}{2T}\). It should be noted that when the velocity of the loading is greater than one, the flame leaves the domain. As a results, a plateau appears after that value.

Therefore, we rather propose to study the influence of the instationary term in the equilibrium equation (2b). To do so, we study the number of PGD modes required to meet a target error of \(10^{-2}\) according to the diffusion coefficient \({c/\mu }\), as shown in Figure 9. While classical PGD methods have difficulties to catch the right solution, minimizing the CRE leads to a better solution close to the optimal SVD one. As previously noted, the lifting technique requires much less modes to catch the solution, and does not depend of the influence of the diffusion term. Even if SVD or PGD computational methods require less modes to reach such an error, the lifting technique has the advantage not to compute those modes and therefore does not depend on the loading velocity.

From Figures 8 and 9, we study the influence of the transient part of the solution. Such a part does not verify the assumption of separated variables representation. When the diffusion term is neglectable, time does not influence the solution and a separated representation is implicit. On the other hand, time cannot be longer neglected, and extra separated modes are required.

In this section, the assumption of variable separation shows its limits for instationary problems that do not verify such an assumption. Initializing PGD strategies with a known unseparable solution circumvents that limit. Such more thoughtful strategy, presented as a lifting PGD technique, shows that the PGD method is not faulted.