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，用于求解参数依赖型双曲偏微分方程（PDEs），该工作延续自Marx等人（2020）的前期研究。该方法依托于非线性方程极弱解的定义，即满足Borel测度空间中线性方程的参数型熵值测度值（MV）解。我们将无限维线性问题近似为一个凸的、有限维的半定规划问题层级结构，即Lasserre层级结构。由此可获得与参数型熵值MV解相关联的占据测度矩序列的近似解，并证明该序列具有收敛性。最终，可基于该近似矩序列执行多种后处理操作。特别地，通过优化与近似测度关联的Christoffel-Darboux核（一种能够捕捉大部分不规则函数的强大逼近工具），可重构解的函数图像。同时，针对不确定性量化问题，可估计多种关注量，例如可直接估计解的光滑泛函期望值。我们通过数值实验评估了该方法的性能，实验对象为具有参数化初始条件或参数化通量函数的无粘Burgers方程。