This paper is concerned with the finite element discretization of the data driven approach according to arXiv:1510.04232 for the solution of PDEs with a material law arising from measurement data. To simplify the setting, we focus on a scalar diffusion problem instead of a problem in elasticity. It is proven that the data convergence analysis from arXiv:1708.02880 carries over to the finite element discretization as long as $H(\mathrm{div})$-conforming finite elements such as the Raviart-Thomas element are used. As a corollary, minimizers of the discretized problems converge in data in the sense of arXiv:1708.02880, as the mesh size tends to zero and the approximation of the local material data set gets more and more accurate. We moreover present several heuristics for the solution of the discretized data driven problems, which is equivalent to a quadratic semi-assignment problem and therefore NP-hard. We test these heuristics by means of two examples and it turns out that the "classical" alternating projection method according to arXiv:1510.04232 is superior w.r.t. the ratio of accuracy and computational time.
翻译:本文根据arXiv:151.0044232, 数据驱动方法的有限元素离散, 用于用测量数据产生的物质法解决 PDE 。 为了简化设置, 我们侧重于一个斜体扩散问题, 而不是弹性问题。 事实证明, ArXiv: 170.8/ 2880的数据趋同分析, 只要使用Ravirt- Thomas 元素等符合数据的有限元素, 即 Ravirt- Thoomas 元素, 就会延续到数据离散的有限元素离散化。 作为必然结果, 离散问题在 arxiv: 170.8/ 2880 意义上的数据中最小化的问题会集中到 。 由于mash 大小趋向为零, 本地材料数据集的近似近度越来越准确。 此外, 我们对离散数据驱动问题的解决方案提出了几种偏差, 这相当于四重半分配问题, 因此NP- 硬化 。 我们用两个例子来测试这些超值。 我们用两个例子来检验这些离子体问题, 。 。 从 aclasticalalcalal comlical complectation roduction roduction by by acluction roviewdal by rout.</s>