Robust model reduction by \(L^{1}\)norm minimization and approximation via dictionaries: application to nonlinear hyperbolic problems
 Rémi Abgrall^{1},
 David Amsallem^{2} and
 Roxana Crisovan^{3}Email author
DOI: 10.1186/s4032301500553
© The Author(s) 2016
Received: 15 October 2015
Accepted: 8 December 2015
Published: 5 January 2016
Abstract
We propose a novel model reduction approach for the approximation of non linear hyperbolic equations in the scalar and the system cases. The approach relies on an offline computation of a dictionary of solutions together with an online \(L^1\) norm minimization of the residual. It is shown why this is a natural framework for hyperbolic problems and tested on nonlinear problems such as Burgers’ equation and the onedimensional Euler equations involving shocks and discontinuities. Efficient algorithms are presented for the computation of the \(L^1\)norm minimizer, both in the cases of linear and nonlinear residuals. Results indicate that the method has the potential of being accurate when involving only very few modes, generating physically acceptable, oscillationfree, solutions.
Keywords
Model reduction Dictionaries \(L^{1}\)norm residual minimizationBackground
Many engineering applications require the ability to simulate the behavior of a physical system in realtime. This requirement holds in particular when a full parametric exploration of the behavior of the system is sought. In aerodynamics, such an exploration can be done to compute the flow around an aircraft for varying boundary conditions or to design its shape to maximize lift and minimize drag. Uncertainty quantification also requires a large number of simulations with varying parameters in order to propagate chaos by means of a MonteCarlo method or calibrating input parameters by a Markov chain technique. A third important application is flow control.
When such a large number of simulations is required, the cost of one simulation is critical to the application at hand. This cost can be lowered by using sophisticated computer science techniques such as parallelization but such techniques are usually not enough to allow full parametric exploration, especially when computational resources are limited.

Burgers’ equation for which \(U=u\) is scalar:

Its unsteady version,with periodic boundary conditions$$\begin{aligned} \dfrac{\partial u}{\partial t}+\dfrac{\partial }{\partial x}\left( \frac{1}{2}u^2\right) =0, \quad u(x,0)=u_0(x) \end{aligned}$$

It steady version with weak Dirichlet boundary conditions


The onedimensional compressible Euler equations for whichand the perfect gas equation of state holds:$$\begin{aligned} U=(\rho , \rho u, E), F(U) = \left( \rho u, \rho u^2+p, u(E+p)\right) \end{aligned}$$\(\rho \) denotes the density, u the velocity, p the pressure and E the energy.$$\begin{aligned} p=(\gamma 1)\left( E\frac{1}{2}\rho u^2\right) . \end{aligned}$$

An example of a steady flow through a nozzle.
Instead of allowing any value of the solution degrees of freedom \(\mathbf {u}\), model reduction however restricts the solution to be contained in a subspace of the underlying highdimensional space. This subspace is determined by an optimized reduced basis that is determined in a training phase. Thus, a large number of degrees of freedom (say millions) are represented by only a few number of coefficients in the representation of the full solution in terms of the reduced basis vectors, leading to important computational savings. Two important questions arise at this point: (1) how can an optimal reduced basis be constructed? and (2) how can the evolution of the reduced coefficients be computed in a stable fashion?
Concerning compressible fluids, there is another difficulty. In problem (4), one needs a norm. In the case of incompressible flows, a natural norm is related to the kinetic energy. For compressible materials, however, one needs to take into account the density, velocity and the energy, i.e. the thermodynamics. A simple \(L^2\)norm cannot be used because one cannot combine in a quadratic manner these variables, for dimensional reasons. Only a nondimensionalization of the variables can alleviate the dimensionality issue [11, 12].
The natural equivalent of the \(L^2\)norm is however related to the entropy, which is not quadratic: if a minimization problem can be set up, its solution is non trivial. These arguments were raised in [10], and an energybased norm was developed in [13] for linearized compressible flows.
To circumvent those issues, an approach based on a dictionary of solutions [14] is developed in this work as an alternative to using a truncated reduced basis based on POD. The elements of this dictionary are solutions \(\mathbf {u}(t_l;\varvec{\mu }_j)\) computed for varying values of time \(t_l\) and parameter \(\varvec{\mu }_j\in \mathbb {R}^m\). Selecting appropriate parameter samples \(\varvec{\mu }_j\in \mathcal {D} \subset \mathbb {R}^m\) is a crucial step that can affect the accuracy of the reducedorder model in the parameter domain. Greedy sampling procedures have been developed when error estimates are known [7, 9, 15–17]. In this work, we do not elaborate much on this, we are more focused on showing that such a method can actually work. The strategy to look for the “best” \(\varvec{\mu }\) in this context will be the topic of further research.
In addition to choosing an appropriate dictionary \(\mathcal {D}\), selecting an approach for computing a reduced solution based on that dictionary is also crucial. For selfadjoint systems, Galerkin projection is a natural approach but it there is no motivation for using Galerkin projection for nonlinear compressible flows. Instead, strategies based on the minimization of the residual arising from the reduced approximation have been successfully developed for compressible flows in [1, 2, 11]. These approaches rely on a minimization of the residual in the \(L^2\) sense. In the present work, this minimization problem is extended to the more general minimization using a \(L^q\)norm, with emphasis on \(q=1\). For nonlinear systems, an additional step, hyperreduction, is required to ensure an efficient solution of the reduced system [11, 18]. Hyperreduction is not considered in this work but will be the subject of followup work.
Methods
This section is organized as follows. Motivations for using the \(L^1\) norm in the case of hyperbolic systems are presented first. We show that \(q=1\) is very closely linked to the concept of weak solutions of hyperbolic problems. Then, the proposed model reduction approach is developed in both the steady and unsteady cases. Finally, the proposed procedure is applied to the model reduction of several steady and unsteady systems and conclusions are given in the end.
Motivation for the \(L^1\)norm
In solving minimization problems, it is quite usual to minimize some residual with respect to the \(L^q\) norm for suitable q. The choice \(q=2\) is very common because it amounts to minimize in some least square sense and many efficient algorithms are available. In the case of hyperbolic problems, as we are concerned with here, this is still a convenient choice (after proper adimensionalization as mentioned above), but it might not be the most natural one. For example in [19, 20] it is shown at least experimentally, that the numerical solution has an excellent non oscillatory behavior, without doing explicitly anything but to minimize the \(L^1\) norm of the PDE residual. In fact, this observation was our original motivation for choosing the \(L^1\) norm, since we are interested in keeping the non oscillatory nature of the solution. In this section, we further justify the choice of the \(L^1\) norm applied to the residual, and show that it is closely related to the weak formulation of the problem. The discussion is here formal.
Thanks to this definition, we see that if we define the spacetime flux \(\mathcal {F}=(U,F)\), U is a weak solution if and only if the total variation of \(\mathcal {F}\) vanishes, \( TV\big ( \mathcal {F}) =0.\)
When \(\mathcal {I}\) is not equal to the set of degrees of freedom, then something new happens. We expect precisely to exploit this idea, or ideas related to this.
In the remainder of this paper, this idea is exploited in the case of model reduction, for which \(\mathcal {I}\) is not equal to the set of grid points and TV semi norm slightly modified in order to guaranty that a unique solution to the minimization problem exists, as well as the minimization problem is as easy as possible to solve.
Formulation
Highdimensional model
Model reduction by residual minimization over a dictionary
Steady problems
The parameter vector \(\varvec{\mu }\in \mathcal {P}\subset \mathbb {R}^m\) can, for instance, parametrize the boundary conditions associated with the steadystate problem. The parametric domain of interest \(\mathcal {P}\) is assumed here to be a bounded set of \(\mathbb {R}^m\).
The solution manifold \(\mathcal {M} = \left\{ \mathbf {u}(\varvec{\mu })~\text {s.t}~ \varvec{\mu }\in \mathcal {P}\subset \mathbb {R}^m\right\} \) is assumed to be of small dimension. This manifold \(\mathcal {M}\) belongs to \(L^\infty ({\mathbb R}^d)\cap BV({\mathbb R}^d)\), and thus can be locally described by some mapping \(\theta : \mathcal {P}\mapsto L^\infty ({\mathbb R}^d)\cap BV({\mathbb R}^d)\). To approximate this mapping, we consider a family of r parameters in \(\mathcal {P}\), \(\{\varvec{\mu }_\ell \}_{\ell =1}^r\), and compute the associated solutions \(\left\{ \mathbf {u}(\varvec{\mu }_\ell )\right\} _{\ell =1}^r\) of (8).
In order to minimize J when \(\mathbf {r}\) is a linear function of \(\varvec{\beta }\), the Linear Programming (LP) approach is considered, involving the solution of an optimization problem with \(2m+r\) variables and 3m constraints.
When \(\mathbf {r}\) is a nonlinear function of \(\varvec{\beta }\), a GaussNewtonlike procedure can be used in combination with the LP approach. Unicity of the solution can be guaranteed by setting the regularization term \(\eta >0\). That’s why we are not doing the linear example.
Remark

Decreasing the dimensionality of the solution space from N to r is not enough to gain computational speedup when the system to be solved is nonlinear. An additional level of approximation, hyperreduction, is necessary.

A careful selection of the sample parameter samples \(\left\{ \varvec{\mu }_\ell \right\} _{\ell =1}^r\) is necessary in order to generate a reducedorder model that is accurate in the entire parameter domain \(\mathcal {P}\). Greedy sampling techniques, associated with a posteriori error estimates, have been successfully used to construct reduced models that are robust and accurate in a parameter domain \(\mathcal {P}\). These techniques are not considered in this paper but will also be the focus of future work.
Unsteady problems
For simplicity, in the remainder of this section, we assume that only the initial condition \(\mathbf {u}^0(\varvec{\mu })\) depends on a parameter vector \(\varvec{\mu }\in \mathcal {P}\subset \mathbb {R}^m\). Again, the family of solutions \(\mathbf {u}(\varvec{\mu })\) of the Cauchy problem (8) is then conjectured to belong to a low dimensional manifold \(\mathcal {M}\) when the initial condition is parametrized in (8b).
To approximate this mapping, we consider a family of r parameters in \(\mathcal {P}\), \(\{\varvec{\mu }_\ell \}_{\ell =1}^r\), and compute the associated solutions of (8) for respective initial conditions \(\mathbf {u}^0(\varvec{\mu }_\ell )\), \(\ell =1, \ldots , r\).
 1.Initialization: determine the reduced coefficients \(\{\alpha ^0_\ell (\varvec{\mu })\}_{\ell =1}^r\) as:for a given choice of functional \(J(\mathbf {u},\varvec{\beta })\).$$\begin{aligned} \varvec{\alpha }^0(\varvec{\mu }):=\left( \alpha _1^0(\varvec{\mu }), \ldots , \alpha _r^0(\varvec{\mu })\right) = \mathop {{{\mathrm{\arg \!\min }}}}\limits _{\varvec{\beta }\,=\, \left( \beta _1,\ldots ,\beta _r\right) }J\left( \sum _{\ell =1}^r \beta _\ell \mathbf {u}^0({\varvec{\mu }_\ell }),\varvec{\beta }\right) , \end{aligned}$$
 2.Assume that \(\varvec{\alpha }^n(\varvec{\mu }) = (\alpha _1^n(\varvec{\mu }), \ldots , \alpha _r^n(\varvec{\mu }))\) is known, determine \(\varvec{\alpha }^{n+1} = (\alpha _1^{n+1}, \ldots , \alpha _r^{n+1})\) such that:where$$\begin{aligned}&{\varvec{\alpha }}^{n+1}(\varvec{\mu })= \mathop {{{\mathrm{\arg \!\min }}}}\limits _{\varvec{\beta }\,=\,(\beta _1,\ldots ,\beta _r)}J\Bigg ( \sum _{\ell =1}^r \beta _\ell \mathbf {u}^{n+1}({\varvec{\mu }_\ell })\mathbf {w}^n(\varvec{\mu })\dfrac{\Delta t}{\Delta x} \bigg (\mathbf {f}_{1/2}(\mathbf {w}^n)\mathbf {f}_{1/2}(\mathbf {w}^n)\bigg ),\varvec{\beta }\Bigg ) \end{aligned}$$$$\begin{aligned} \mathbf {w}^n(\varvec{\mu })=\sum _{\ell =1}^r \alpha ^n_\ell (\varvec{\mu }) \mathbf {u}^{n}({\varvec{\mu }_\ell }). \end{aligned}$$
A few immediate remarks can be made.
Remark

In the case of a linear flux, Problem (1) is linear. If \(\mathcal {S}_t\) is the mapping between the initial condition \(u_0\) and the solution at time t, we have \(\mathcal {S}_t(u+v)=\mathcal {S}_t(u)+\mathcal {S}_t(v)\). This means the exact solution of the Cauchy problem with \(U_0=\sum _\ell \alpha _\ell ^0 U_0(\varvec{\mu }_\ell )\) is \(\mathcal {S}_t(U_0)=\sum _\ell \alpha _\ell ^0 \mathcal {S}_t(U({\varvec{\mu }_\ell ,0}))\). In the case of a linear scheme, minimizing the functional J should result in \(\varvec{\alpha }^n=\varvec{\alpha }^0\) for any \(n\ge 0\).

In the case of an explicit background scheme, the choice of the numerical flux, how high order is reached, and the choice of time stepping has no influence on the overall procedure: any subtime step would be treated similarly. In this paper, we have chosen a first order method with Euler time stepping in the case of unsteady problem.

In the case of an implicit scheme, a Newtonlike procedure can be applied to minimize the functional as in [11]. At each time step, the procedure is then identical as in the steady case described above.
Results and discussion
Model reduction of unsteady problems
Unsteady Burgers’ equation
After the shock, the \(L^1\)normtype solutions are all close to each other and the shock is rather well reproduced with, however, an artifact that develops for longer times, as seen at \(t=\pi \). Nevertheless, the \(L^1\)normtype solutions are within the bounds of the “exact” solution, and no large oscillation develops.
In a second set of numerical experiments, we consider the influence of the sampling parameter set included in the dictionary \(\mathcal {D}\). We consider two dictionaries \(\mathcal {D}_1=\{0.4,0.45, 0.55, 0.6\}\) and \(\mathcal {D}_0=\{0,0.2,0.4,0.45, 0.55, 0.6,1.0\}\), for the same target value of \(\mu ^\star =0.5\). These choices amounts to selecting samples close to the target value 0.5 while varying elements of the dictionary that are not close to 0.5 (see Fig. 4).
We see that refining the dictionary has a positive influence as the target solution is much closer to the dictionary elements. This is confirmed by additional experiments where the samples of \(\mu \) used to generate the dictionary where more numerous and closer to 0.5 (not reported here). The \(L^1\)normtype solutions are however unaffected by the presence of these “outliers” in the dictionary.
Euler equations
This problem is parametrized by the initial conditions \(U_0(x;\mu )\). To define the parametrized initial conditions of the problem, the Lax and Sod cases are first introduced as follows.

Either we reconstruct together the discretized density vectors \(\varvec{\rho }\), momentum \(\mathbf {m}=\varvec{\rho }\mathbf {u}\) and energy \(\mathbf {E}\), i.e. the state variable at time \(t_n\) using only one coefficient vector \(\varvec{\alpha }^n=(\alpha ^n_1,\ldots ,\alpha ^n_r)\)Here the \(\{\alpha _j^n\}_{j=1}^r\) are obtained by minimizing J on the density components of the state because the density enable to detect fans, contact discontinuities and shocks, contrarily to pressure and velocity which are constant across contact waves. Doing so we expect to control better the numerical oscillations, if any, than with the other physical variables. Similar arguments could be applied with the other conserved variables as well.$$\begin{aligned} \mathbf {u}^n=\begin{pmatrix} \varvec{\rho }^{n}\\ \mathbf {m}^n\\ \mathbf {E}^n\end{pmatrix} \approx \sum _{j=1}^r \alpha _j^n \mathbf {u}^n(\mu _j). \end{aligned}$$(13)

Alternatively, we reconstruct each conserved variable separatelywhere the minimization procedures are done independently on each conserved variable.$$\begin{aligned} \varvec{\rho }^n\approx \sum _{j=1}^r \alpha _{j}^n \varvec{\rho }^n(\mu _j), \quad \mathbf {m}^n\approx \sum _{j=1}^r \alpha _{j}^n \mathbf {m}^n(\mu _j), \quad \mathbf {E}^n\approx \sum _{j=1}^r \alpha _{j}^n \mathbf {E}^n(\mu _j). \end{aligned}$$(14)
From both figures, we can see that the overall structure of the solutions is correct. Nevertheless, there are differences that can be highlighted. From Fig. 6, we can observe that the density predictions, besides an undershoot at the shock, are well reproduced. However, we cannot recover correct values of the initial velocity (see left boundary), because there is no reason to believe that the coefficient \(\varvec{\alpha }\), evaluated from the density only, will also be correct for the momentum. A careful observation of the pressure plot also reveals the same behavior which is not satisfactory. For the same reason, if any other single variable is used for a global approximation of each conservative variables, there no reason why better qualitative results could be obtained.
This problem does not occur with the second strategy for the reconstruction (14): the correct initial values are recovered. We have some slight problems on the velocity, between the contact and the shock.
All this being said, the solution using three distinct coefficients obtained independently is of significantly much better quality than the one using only one expansion.
Model reduction of steady problems
Nozzle flow
Conclusions
A novel model reduction that relies on a dictionary approach is developed and tested on several steady and unsteady hyperbolic problems. All of the solutions of the problem tested are parametrized and have regions of their spatial domain with discontinuities, leading to solutions with very distinct behaviors, such as different wave speeds and shock locations, making them challenging to reduce using classical projectionbased model reduction techniques. To address this challenge, the proposed approach is based on a dictionary of solutions is coupled with a functional minimization. The analysis and numerical experiments conducted in this work show that the proposed approach is robust (at least for onedimensional problems) and performs the best when the functional is of \(L^1\)normtype. As an extension to this work, other related minimization techniques which are less CPU intensive will be considered.
Current work includes a multidimensional fluid case, an error estimate, the storage of the dictionary and an application of the hyperreduction to the dictionary framework.
Abbreviations
 HDM:

High dimensional model
 PDE:

Partial differential equations
 POD:

Proper orthogonal decomposition
 LP:

Linear programming
 TV:

Total variation
 BV:

Bounded variation
Declarations
Acknowledgements
The first author has been funded in part by the MECASIF Project (2013–2017) funded by the French “Fonds Unique Interministériel” and SNF Grant # 200021_153604 of the Swiss National Foundation. The second author would like to acknowledge partial support by the Army Research Laboratory through the Army High Performance Computing Research Center under Cooperative Agreement W911NF0720027, and partial support by the Office of Naval Research under Grant No. N000141110707. The third author has been supported in part by the SNF Grant # 200021_153604 of the Swiss National Foundation. This document does not necessarily reflect the position of these institutions, and no official endorsement should be inferred. Rémi Abgrall would also like to thank Y. Maday, LJLL, Université Pierre and Marie Curie for several very insightful conversations.
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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