In this work we propose a low rank approximation of high fidelity finite element simulations by utilizing weights corresponding to areas of high stress levels for an abdominal aortic aneurysm, i.e. a deformed blood vessel. We focus on the van Mises stress, which corresponds to the rupture risk of the aorta. This is modeled as a Gaussian Markov random field and we define our approximation as a basis of vectors that solve a series of optimization problems. Each of these problems describes the minimization of an expected weighted quadratic loss. The weights, which encapsulate the importance of each grid point of the finite elements, can be chosen freely - either data driven or by incorporating domain knowledge. Along with a more general discussion of mathematical properties we provide an effective numerical heuristic to compute the basis under general conditions. We explicitly explore two such bases on the surface of a high fidelity finite element grid and show their efficiency for compression. We further utilize the approach to predict the van Mises stress in areas of interest using low and high fidelity simulations. Due to the high dimension of the data we have to take extra care to keep the problem numerically feasible. This is also a major concern of this work.

翻译：本文提出一种低秩逼近方法，通过对腹主动脉瘤（即变形血管）高应力区域赋予权重，实现对高保真有限元模拟的低秩逼近。我们重点关注与主动脉破裂风险相关的冯·米塞斯应力。该问题建模为高斯马尔可夫随机场，并将逼近定义为求解一系列优化问题所得的向量基。每个优化问题描述对期望加权二次损失的最小化。权重可自由选择（基于数据驱动或结合领域知识），用于表征有限元网格各节点的重要性。在泛化数学性质讨论的基础上，我们提出在一般条件下有效计算该基的数值启发式方法。我们明确探索了高保真有限元网格表面上的两类基，并验证其压缩效率。进一步利用该方法，通过高低保真模拟预测感兴趣区域的冯·米塞斯应力。由于数据维度较高，需特别关注数值可行性问题，这也是本文的研究重点之一。