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.

翻译：本文提出一种基于矩方法的数值方法，用于求解参数依赖的双曲型偏微分方程（PDEs）。该方法源自Marx等人（2020）的前期工作，其核心思想是利用非线性方程的一种极弱解概念——即参数熵测度值（MV）解，该解满足Borel测度空间中的线性方程。我们通过一系列凸的有限维半定规划问题（即Lasserre层级）来逼近这一无限维线性问题。由此获得与参数熵MV解相关联的占据测度矩的近似序列，并证明了该序列的收敛性。基于这个近似矩序列可进行多种后处理操作：特别地，可通过优化近似测度对应的Christoffel-Darboux核来重构解的图形，这种强大的逼近工具能够捕捉大量不规则函数；同时，针对不确定性量化问题，可估算多种关注量（例如解的光滑泛函期望），部分情况下可直接计算。通过带参数化初始条件或通量函数的无黏Burgers方程的数值实验，验证了本方法的有效性。