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：我们提出了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，该方法建立在Marx等人（2020）先前工作的基础上。该方法依赖于非线性方程的一种极弱解概念——参数熵测值解，该解在Borel测度空间中满足线性方程。通过称为Lasserre层次结构的一系列凸的、有限维的半定规划问题来逼近该无限维线性问题。由此我们得到了与参数熵测值解相关的占据测度矩的近似序列，并证明了该序列的收敛性。最终，可基于该近似矩序列进行多种后处理操作。特别地，可通过优化与近似测度相关的Christoffel-Darboux核来重构解的图像，该核是一种能够捕捉一大类不规则函数的强大逼近工具。此外，对于不确定性量化问题，可以估计多种感兴趣的物理量，有时可直接计算解的光滑泛函的期望值。我们通过数值实验评估了该方法在具有参数化初始条件或参数化通量函数的无粘性Burgers方程上的表现。