Petrov-Galerkin formulations with optimal test functions allow for the stabilization of finite element simulations. In particular, given a discrete trial space, the optimal test space induces a numerical scheme delivering the best approximation in terms of a problem-dependent energy norm. This ideal approach has two shortcomings: first, we need to explicitly know the set of optimal test functions; and second, the optimal test functions may have large supports inducing expensive dense linear systems. Nevertheless, parametric families of PDEs are an example where it is worth investing some (offline) computational effort to obtain stabilized linear systems that can be solved efficiently, for a given set of parameters, in an online stage. Therefore, as a remedy for the first shortcoming, we explicitly compute (offline) a function mapping any PDE-parameter, to the matrix of coefficients of optimal test functions (in a basis expansion) associated with that PDE-parameter. Next, as a remedy for the second shortcoming, we use the low-rank approximation to hierarchically compress the (non-square) matrix of coefficients of optimal test functions. In order to accelerate this process, we train a neural network to learn a critical bottleneck of the compression algorithm (for a given set of PDE-parameters). When solving online the resulting (compressed) Petrov-Galerkin formulation, we employ a GMRES iterative solver with inexpensive matrix-vector multiplications thanks to the low-rank features of the compressed matrix. We perform experiments showing that the full online procedure as fast as the original (unstable) Galerkin approach. In other words, we get the stabilization with hierarchical matrices and neural networks practically for free. We illustrate our findings by means of 2D Eriksson-Johnson and Hemholtz model problems.
翻译:Petrov-Galerkin 配方, 具有最佳测试功能, 可以稳定定義元素的模拟。 特别是, 在一个离散的试验空间中, 最佳测试空间引出一个数字机制, 以基于问题的能源规范提供最佳近似。 这个理想的方法有两个缺点: 首先, 我们需要明确知道一套最佳测试功能; 第二, 最佳测试功能可能具有巨大的支持作用, 导致昂贵的密集线性系统。 然而, PDE 的参数组群是一个值得投资一些( 离线) 计算努力的例子, 以获得稳定线性线性系统, 这些系统可以在在线阶段为一组参数有效解决。 因此, 作为第一个短数的补救措施, 我们明确( 离线) 绘制任何 PDE 参数, 与PDE 参数相关的最佳测试函数矩阵矩阵矩阵( 基础扩展) 。 其次, 我们用低端的近位直线直线性直线性直线性直线性计算方法, 来( 以( 离线性) 优化测试功能的固定矩阵矩阵矩阵矩阵 。 为了快速加速运行这个进程, 我们用一个稳定的内基质的内压的内压网络 。 ( 将一个网络 向导 学习一个新的内压 演示中,, 我们用一个基压式的内基压式的网络, 向一个基压 演示一个基压式的网络, 向一个螺路路路路路路路路路路路路路路路数据压,, 。