论文标题
通过在线自适应减少网格的参数化运输占主导地位的问题过度还原
Hyper-reduction for parametrized transport dominated problems via online-adaptive reduced meshes
论文作者
论文摘要
我们提出了一种有效的残余最小化技术，用于对参数化双曲线偏微分方程的非线性模型阶降。我们的非线性近似空间是在移动的空间结构域上评估的快照的跨度，我们通过残留最小化计算减少近似。为了加快残差最小化，我们计算并最大程度地减少了网格（最好是小）子集（所谓的还原网格）上的残留物。由于我们近似空间的非线性，我们表明，与解决方案相似，残差也表现出运输型行为。为了说明这种行为，我们通过沿空间域“移动”以参数依赖性偏移来介绍减少网格的在线适应性。我们还提出了我们的方法的扩展，以使空间变换不同于转移。数值实验展示了我们方法的有效性以及由非自适应降低的网格引起的不准确性。
We propose an efficient residual minimization technique for the nonlinear model-order reduction of parameterized hyperbolic partial differential equations. Our nonlinear approximation space is a span of snapshots evaluated on a shifted spatial domain, and we compute our reduced approximation via residual minimization. To speed-up the residual minimization, we compute and minimize the residual on a (preferably small) subset of the mesh, the so-called reduced mesh. Due to the nonlinearity of our approximation space we show that, similar to the solution, the residual also exhibits transport-type behaviour. To account for this behaviour, we introduce online-adaptivity in the reduced mesh by "moving" it along the spatial domain with parameter dependent shifts. We also present an extension of our method to spatial transforms different from shifting. Numerical experiments showcase the effectiveness of our method and the inaccuracies resulting from a non-adaptive reduced mesh.
