Proper Generalized Decomposition computational methods on a benchmark problem: introducing a new strategy based on Constitutive Relation Error minimization
 PierreEric Allier^{1}Email author,
 Ludovic Chamoin^{1} and
 Pierre Ladevèze^{1}
https://doi.org/10.1186/s4032301500384
© Allier et al. 2015
Received: 30 January 2015
Accepted: 29 June 2015
Published: 16 July 2015
Abstract
First, the effectivity of classical Proper Generalized Decomposition (PGD) computational methods is analyzed on a one dimensional transient diffusion benchmark problem, with a moving load. Classical PGD methods refer to Galerkin, Petrov–Galerkin and Minimum Residual formulations. A new and promising PGD computational method based on the Constitutive Relation Error concept is then proposed and provides an improved, immediate and robust reduction error estimation. All those methods are compared to a reference Singular Value Decomposition reduced solution using the energy norm. Eventually, the variable separation assumption itself (here time and space) is analyzed with respect to the loading velocity.
Keywords
Notations
All continuous fields or spaces are in classical format u, while Finite Element approximations fields \(u_h\) benefit of an h index. Discretized terms, usually represented by vectors of nodal values, are bolded \(\varvec{u}\) and matrices are in uppercase and doublestruck \(\mathbb {U}\).
Background
Nowadays, numerical simulations constitute a common tool in science and engineering activities. It is especially used for prediction and decision making, or simply for a better understanding of physical phenomena. However, in order to give an accurate representation of the real world, a large set of parameters and nonlinearities may need to be introduced in the mathematical models involved in the simulation, leading to important and often overwhelming computational effort. This results in a huge number of degrees of freedom, a drawback for such complex models, so that they cannot be tackled with classical brute force methods. Therefore, alternative numerical approaches are required, such as model reduction methods. These methods exploit the fact that the response of complex models can often be approximated with a reasonable accuracy by the response of a surrogate model, seen as the projection of the initial one on a lower dimensional functional basis [1–3]. Model reduction methods distinguish themselves by the way of defining and constructing the reduced basis. Among them are the Proper orthogonal decomposition (POD) [4, 5] and the Reduced Basis Method (RB) [6, 7]. While the first one seeks to extract the most relevant information from an already partial solution, the second one selects the most relevant calculation to perform in order to enrich the basis, thanks to an error indicator.
Even though PGD is usually very effective, a major question is to derive verification tools for controlling the calculation process. Basic results on a priori error estimation for separation of variable representation can be found in [11], and a first work providing strict bounds on global error in the PGD context is [19], in which specific error indicators are also given.
The first goal of this paper is to compare the effectivity of the classical PGD computational methods which are all built on a progressive algorithm [20], rather than solving the m basis functions at once. A particular attention is paid to the numerical strategy developed around PGD methods since the beginning of the 90’s [11]: for a timespace problem, a new PGD mode is computed using very few subiterations and at the end the full PGD time function basis is updated at once. In the literature, several variants of PGD have been proposed for the progressive construction of (1), respectively based on Galerkin formulation, Minimal Residual formulation [21] or Petrov–Galerkin formulation [22]. Performances of all these methods are compared on a benchmark problem proposed by S. Idelsohn, a one dimensional transient thermal problem with a moving thermal load. Secondly, we propose a new computational technique based on a PGD error indicator introduced in [23], minimizing the Constitutive Relation Error (CRE), with very promising properties. Finally, the effectivity of the variable separation hypothesis is analyzed with respect to the velocity of the moving thermal loading. In this context, a lifting PGD computational method is proposed to push the limits of that hypothesis.
The paper outline is as follows: in “The benchmark problem” we present the one dimensional transient thermal benchmark problem. In “Optimal reduced solution”, an optimal reduced solution, computed from a brute force finite element solution, is proposed through the Singular Value Decomposition (SVD). Classical definitions of PGD methods applied to the benchmark problem are reviewed in “Classical Proper Generalized Decomposition methods”. We then introduce in “Minimisation of the Constitutive Relation Error” a non classical definition of PGD, called CREPGD. In “A nonseparable lifting algorithm”, the lifting technique is addressed to study limits of variable separation. In “Results and discussion”, the different methods presented in this article are compared on the benchmark problem.
Methods
The benchmark problem
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.

the thermal constraints:$$\begin{aligned} u(0,t) = u(L,t) = 0 \quad \forall t \in \mathcal {I}\end{aligned}$$(2a)

the equilibrium equation:$$\begin{aligned} c \frac{\mathrm \partial ^{}{u}}{\mathrm \partial {t}^{}}\!(x,t) +\frac{\mathrm \partial ^{}{\varphi }}{\mathrm \partial {x}^{}}\!(x,t) = \delta (x  vt) \quad \forall (x,t) \in \Omega \times \mathcal {I}\end{aligned}$$(2b)

the constitutive relation:$$\begin{aligned} \varphi (x,t) =  \mu \frac{\mathrm \partial ^{}{u}}{\mathrm \partial {x}^{}}\!(x,t) \quad \forall (x,t) \in \Omega \times \mathcal {I}\end{aligned}$$(2c)

the initial condition:$$\begin{aligned} u(x,0) = 0 \quad \forall x \in \Omega \end{aligned}$$(2d)
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).
Remark 1
The reduced basis functions \(\psi _i \in \mathcal {V}_h\) (resp. \(\lambda _i \in \mathcal {T}_h\)) are orthogonal with respect to the inner product \(\left\langle \cdot , \cdot \right\rangle _{\mathcal {V}_h}\) (resp. \(\left\langle \cdot , \cdot \right\rangle _{\mathcal {T}_h}\)).
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

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 nonlinear problem. As this problem can be interpreted as a pseudoeigenproblem [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
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 subiterations \(k_{max} = 4\), letting the next PGD mode to correct the previous one if necessary.
Improvement of the decomposition
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.

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
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 transportdominated problems.
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 nonlinear, a power iterations strategy is chosen, where the four lower dimension problems are solved iteratively one after the other (see Algorithm 4).
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 full minimization

According to \(\lambda\):$$\begin{aligned} 0 &= \int \limits _{\mathcal {I}}^{}{\!\!\int \limits _{\Omega }^{}{\frac{1}{\mu }\left( Q + \tau _m  \mu \frac{\mathrm \partial ^{}{u_m}}{\mathrm \partial {x}^{}} + \frac{\mathrm d^{}{\lambda }}{\mathrm d{t}^{}} Z  \mu \lambda \frac{\mathrm d^{}{\psi }}{\mathrm d{x}^{}}\right) \!\left( \frac{\mathrm d^{}{\lambda ^*}}{\mathrm d{t}^{}}Z  \mu \lambda ^*\frac{\mathrm d^{}{\psi }}{\mathrm d{x}^{}} \right) }\,\text {d}{x}}\,\text {d}{t} \qquad \forall \lambda ^* \in \mathcal {T}\end{aligned}$$(28a)

According to \(\psi\):$$\begin{aligned} 0 &= 2\int \limits _{\mathcal {I}}^{}{\!\!\int \limits _{\Omega }^{}{\left( \mu \frac{\mathrm \partial ^{}{u_m}}{\mathrm \partial {x}^{}}  Q  \tau _m  \frac{\mathrm d^{}{\lambda }}{\mathrm d{t}^{}} Z + \mu \lambda \frac{\mathrm d^{}{\psi }}{\mathrm d{x}^{}}\right) \lambda \frac{\mathrm d^{}{\psi ^*}}{\mathrm d{x}^{}}}\,\text {d}{x}}\,\text {d}{t} + \int \limits _{\Omega }^{}{c \psi ^* {w} }\,\text {d}{x} \quad \forall \psi ^* \in \mathcal {V}\end{aligned}$$(28b)

According to Z:where \(\mathcal {S} = \left\{ w \in L^2(\Omega ) \mid \forall {g} \in L^2(\Omega ), \int \limits _{\Omega }^{}{{g} \frac{\mathrm d^{}{{w}}}{\mathrm d{x}^{}}}\,\text {d}{x} = 0\right\}\).$$\begin{aligned} 0 &= 2\int \limits _{\mathcal {I}}^{}{\!\!\int \limits _{\Omega }^{}{\frac{1}{\mu }\left( Q + \tau _m  \mu \frac{\mathrm \partial ^{}{u_m}}{\mathrm \partial {x}^{}} + \frac{\mathrm d^{}{\lambda }}{\mathrm d{t}^{}} Z  \mu \lambda \frac{\mathrm d^{}{\psi }}{\mathrm d{x}^{}}\right) \frac{\mathrm d^{}{\lambda }}{\mathrm d{t}^{}}Z^*}\,\text {d}{x}}\,\text {d}{t} + \int \limits _{\Omega }^{}{Z^* \frac{\mathrm \partial ^{}{{w}}}{\mathrm \partial {x}^{}}}\,\text {d}{x} \quad \forall Z^* \in \mathcal {S} \end{aligned}$$(28c)

According to w:$$\begin{aligned} 0 = \int \limits _{\Omega }^{}{Z \frac{\mathrm \partial ^{}{{w}^*}}{\mathrm \partial {x}^{}} + c \psi {w}^*}\,\text {d}{x} \quad \forall {w}^* \in \mathcal {V}\end{aligned}.$$(28d)
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.

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
A nonseparable lifting algorithm

an enrichment function, determined analytically in an infinite domain, which does not verify the separated decomposition;

an additional function, that aims at verifying the boundary conditions, and which verifies variables separation.
Results and discussion
On the benchmark problem presented in “The benchmark problem”, we compare the behavior of the different PGD strategies detailed in this paper. The comparison focuses on: (i) quality of approximation by studying the convergence of the Greedy algorithm; (ii) computational cost. Finally, the influence of time effects is studied through the loading velocity parameter, to show the main limitations of PGD.
The reference solution u is the classical Galerkin approximation \(u_h\), solution of (3), obtained through a brute force FEM. To avoid influence of discretization error, very refined space and time meshes are chosen.
After defining some error indicators, performances of the different PGD strategies are conducted on the benchmark problem with \(L={3.14}{m}\), \(T={1}{s}\), \(c={0.1}{J m^{3} K^{1}}\), \(\mu ={1}{W m^{1} K^{1}}\), \(v=\frac{L}{2T}\) to form a reasonable test case. Later in “Limits of the Proper Generalized Decomposition”, the influence of these coefficient values will be studied to illustrate limits of the PGD method.
Error estimation
 Reference reduction error :

Since the reference solution is known, an exact reduction error can be defined as the distance between the reference \(u_h\) and the reduced solution \(u_m\) according to the energy norm. As the mesh is highly refined, this reduction error is the exact one:$$\begin{aligned} e_{ref} = \Vert u_h  u_m\Vert _E, \quad \overline{e_{ref}} = \frac{e_{ref}}{\Vert u_h\Vert _E} \end{aligned}$$
 Residual error estimator :

In general, the reference solution \(u_h\) is not at hand. Using the equilibrium defects of the approximate solution \(u_m\), an approximated measure of the real error can be performed without knowledge of the reference solution.$$\begin{aligned} e_{res} = \Vert \mathcal {R}(u_m)\Vert _{L^2}, \quad \overline{e_{res}} = \frac{e_{res}}{\Vert \delta \Vert _{L^2}} \end{aligned}$$
 Constitutive Relation Error estimator :

As explained in “Minimisation of the Constitutive Relation Error”, minimizing the CRE leads to an immediate error estimator, evaluated without knowledge of the reference solution, as:$$\begin{aligned} e_{CRE} = \Vert \tau _m  \mu \frac{\mathrm \partial ^{}{u_m}}{\mathrm \partial {x}^{}}\Vert _E, \quad \overline{e_{CRE}} = \frac{e_{CRE}}{\Vert \tau _m + \mu \frac{\mathrm \partial ^{}{u_m}}{\mathrm \partial {x}^{}}\Vert _E} \end{aligned}$$
Influence of the update phase
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.
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 postprocessing 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
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 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.
Conclusion
We reviewed different possible PGD computational methods for the a priori construction of reduced order timedependent thermal model (ROM) based on the separated representation of the related solution. Different constructions have been analyzed, based on Galerkin, Minimal Residual and Petrov–Galerkin formulations of the evolution problem. They were compared on a one dimensional benchmark problem.
The numerical example has illustrated the importance of selecting a relevant and accurate reduction error estimator, as it is required to know when to stop the progressive algorithm. As the reference error estimator is generally out of reach, classical PGD computational methods usually lie on residual error estimator. As a matter of fact, this estimator is not effective nor accurate.
We proposed an innovative PGD construction based on minimizing the CRE. Its main advantage is to provide a robust, accurate and relevant reduction error indicator while maintaining advantages of PGD. Due to the use of CRE, discretization error is within easy reach and allows adaptive construction of PGD decomposition.
The update step highly improves convergence properties of all PGD computational methods. Putting aside the Minimal Residual PGD, all PGD computational methods coupled with this update step converged to the same PGD decomposition. It has the same convergence properties as an optimal reduced solution obtained by SVD.
Finally, limitations of PGD computational methods have been illustrated by lifting Galerkin PGD computational method with a nonseparable solution. Convergence of PGD computational methods tends to decrease when the separation of variables assumption cannot respect the problem to solve. We illustrated it for a time dependent problem, but parameters dependent problems may have the same behavior.
Endnote
^{a}Here, \(\mathbb {V}^{\dagger }\) denotes the adjoint matrix of \(\mathbb {V}\).
Declarations
Authors’ contributions
All authors have read and approved the final manuscript.
Compliance with ethical guidelines
Competing interests The authors declare that they have no competing interests.
Open AccessThis 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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