Filtered stochastic Galerkin methods for hyperbolic equations
Filtered stochastic Galerkin methods for hyperbolic equations
复制标题
DOI：
10.1016/j.jcp.2019.109073
复制
发表时间：
2018-08
期刊：
影响因子：
--
通讯作者：
J. Kusch;R. McClarren;M. Frank
中科院分区：
文献类型：
--
作者：
J. Kusch;R. McClarren;M. Frank
非线性双曲问题的不确定性量化在冲击波附近成为一个具有挑战性的任务。标准的侵入方法，如随机伽辽金（SG），导致振荡的解决方案，并可能导致非双曲矩系统。插入式多项式矩（IPM）方法保证双曲性，但数值代价较高。在本文中，我们过滤的广义多项式混沌（gPC）系数的SG近似，它允许一个数值上便宜的振荡减少。导出的滤波器基于Lasso回归，其将高阶的小gPC系数设置为零。我们自适应地自动选择滤波器的强度，以获得零值的最高阶矩。对Burgers方程和Euler方程进行了滤波SG方法的检验。结果表明，减少振荡的冲击，这导致一个改进的近似的期望值和方差相比，SG和IPM。
Uncertainty Quantification for nonlinear hyperbolic problems becomes a challenging task in the vicinity of shocks. Standard intrusive methods, such as Stochastic Galerkin (SG), lead to oscillatory solutions and can result in non-hyperbolic moment systems. The intrusive polynomial moment (IPM) method guarantees hyperbolicity but comes at higher numerical costs. In this paper, we filter the generalized polynomial chaos (gPC) coefficients of the SG approximation, which allows a numerically cheap reduction of oscillations. The derived filter is based on Lasso regression which sets small gPC coefficients of high order to zero. We adaptively and automatically choose the filter strength to obtain a zero-valued highest order moment. The filtered SG method is tested for Burgers' and the Euler equations. Results show a reduction of oscillations at shocks, which leads to an improved approximation of expectation values and the variance compared to SG and IPM.