Weighted high dimensional data reduction of finite element features: an application on high pressure of an abdominal aortic aneurysm

In this work we propose a low rank approximation of areal, particularly three dimensional, data utilizing additional weights. This way we enable effective compression if additional information indicates that parts of the data are of higher interest than others. The guiding example are high fidelity finite element simulations of an abdominal aortic aneurysm, i.e. a deformed blood vessel. The additional weights encapsulate the areas of high stress, which we assume indicates the rupture risk of the aorta. The stress values on the grid are 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. We provide an effective numerical heuristic to compute the basis under general conditions, which relies on the sparsity of the precision matrix to ensure acceptable computing time even for large grids. We explicitly explore two such bases on the surface of a high fidelity finite element grid and show their efficiency for compression. Finally, we utilize the approach as part of a larger model to predict the van Mises stress in areas of interest using low and high fidelity simulations.