A moment approach for entropy solutions of parameter-dependent hyperbolic conservation laws
Clément Cardoen, Swann Marx, Anthony Nouy, Nicolas Seguin
TL;DR
The paper develops a moment-SOS framework for parameter-dependent hyperbolic conservation laws by adopting entropy measure-valued (MV) solutions, which linearize the PDE constraints when expressed on occupation measures. It casts the MV problem as a generalized moment problem (GMP) and solves it via Lasserre's hierarchy of semidefinite programs, enabling offline computation of moments and efficient online post-processing. Key contributions include a robust mechanism to reconstruct solution graphs using the Christoffel-Darboux kernel and to extract statistical quantities of interest through moment-completion techniques, demonstrated on the inviscid Burgers equation with parametrised data. The approach provides a mesh-free, convergent method to handle discontinuities and uncertainty, offering practical tools for uncertainty quantification and sensitivity analysis in hyperbolic systems.
Abstract
We propose a numerical method to solve parameter-dependent hyperbolic partial differential equations (PDEs) with a moment approach, based on a previous work from Marx et al. (2020). This approach relies on a very weak notion of solution of nonlinear equations, namely parametric entropy measure-valued (MV) solutions, satisfying linear equations in the space of Borel measures. The infinite-dimensional linear problem is approximated by a hierarchy of convex, finite-dimensional, semidefinite programming problems, called Lasserre's hierarchy. This gives us a sequence of approximations of the moments of the occupation measure associated with the parametric entropy MV solution, which is proved to converge. In the end, several post-treatments can be performed from this approximate moments sequence. In particular, the graph of the solution can be reconstructed from an optimization of the Christoffel-Darboux kernel associated with the approximate measure, that is a powerful approximation tool able to capture a large class of irregular functions. Also, for uncertainty quantification problems, several quantities of interest can be estimated, sometimes directly such as the expectation of smooth functionals of the solutions. The performance of our approach is evaluated through numerical experiments on the inviscid Burgers equation with parametrised initial conditions or parametrised flux function.