Robust Model Reduction Of Hyperbolic Problems by $L^1$ norm Minimization and Dictionary Approximation

We are interested in the model reduction techniques for hyperbolic problems, in particular in fluids. This paper, which is a continuation of  proposes a dictionary approach coupled with a L 1 minimisation approach. We develop the method and analyse it in simplified one dimensional cases. We show in this case that error bound with the full model can be obtained provided a suitable minimisation approach is chosen. The capability of the algorithm is then shown on non linear scalar problems, one dimensional unsteady fluid problems and two dimensional steady compressible problems. The paper ends with a discussion on the cost of the method. 1.  Model reduction is becoming an essential tool to enable applications requiring either real time predictions or the evaluation of a large number of partial differential equations (PDE) based computational models. The first category encompasses optimal control [28, 34] and model predictive control [26, 4] Routine analysis and parametrized studies  design optimization [33, 6] and the quantification of uncertainty [14] are applications pertaining to the second category, to name just a few. In all of these applications, the large dimensional ity associated with the discretized partial equations prevents their solution in real time. Model reduction reduces that cost by restricting the solution to a subspace of the solution space. This subspace is usually described by a small number of reduced basis vectors. In turn, a projection step reduces the dimensionality of the system of discrete equations considered, enabling their fast solution. While the model reduction of elliptic and parabolic PDEs has been the subject of numerous studies [30, 40] and its theory is well understood  reducing hyperbolic equations has proved to be much more challenging [9] More specifically, moving waves and discontinuities such as shocks require a large number of basis vectors to accurately approximate these features [18] This characterizes these problems as ones with large Kolmogorov n widths [10] To circumvent this issue, approaches based on local bases [5, 20, 7, 35] reduce the Kolmogorov n width by considering local subspaces. The locality can be characterized in parameters [5] time [20] or state space [7] In the present work, an approach based on dictionaries is considered [29, 13] More specifically, solutions corresponding to various time and parameter instances are collected and stored in such a dictionary. Each solution will then be considered as a reduced basis vector. In turn, localization in time and space can be easily enforced by only considering basis vectors corresponding to restricted sub domains of the time and parameters spaces. In addition to the reduction in number of basis vectors, this paper will demonstrate that a key advantage of a dictionary approach is a better approximation of states having sharp gradients and discontinuities. In particular, it will be demonstrated that avoiding basis truncation such as the one occurring in Proper Orthogonal Decomposition (POD) [38] or Non Negative Matrix Factorization  avoid Gibbs phenomenon. In addition to the choice of reduced basis, a key ingredient in projection based model reduction is the definition of the reduced system of equations. For symmetric systems such as those arising in elliptic and parabolic PDEs, Galerkin projection is the method of choice. For non symmetric systems, however, it has been shown that minimizing the L 2 norm of the residual is preferable for stability considerations [16, 17] In the present paper, model reduction based on