论文标题
压缩等几何分析
Compressive Isogeometric Analysis
论文作者
论文摘要
在使用中度到高多项式程度的B-Splines求解偏微分方程（PDE）时，这项工作的动机是由于难以组装盖金基质的难度。为了减轻这个问题，我们提出了一种名为Cossiga的新方法（压缩等几何分析），该方法将IgA原理与托管结合在一起，这是一种基于压缩感应的最近引入的PDES稀疏恢复方法。 Cossiga仅组装合适的Iga Petrov-Galerkin离散化的一小部分，并且只要PDE溶液充分稀疏或可压缩，即当它的大多数系数为零或可忽略时，则有效。通过使用B-Splines的多级词典而不是基础，可以促进解决方案的稀疏性。得益于稀疏性以及仅组装完整离散矩阵的一小部分的事实，该提出的技术有可能导致大量的计算节省。我们通过广泛的数值研究表明了cossiga对2D和3D泊松方程求解的有效性。
This work is motivated by the difficulty in assembling the Galerkin matrix when solving Partial Differential Equations (PDEs) with Isogeometric Analysis (IGA) using B-splines of moderate-to-high polynomial degree. To mitigate this problem, we propose a novel methodology named CossIGA (COmpreSSive IsoGeometric Analysis), which combines the IGA principle with CORSING, a recently introduced sparse recovery approach for PDEs based on compressive sensing. CossIGA assembles only a small portion of a suitable IGA Petrov-Galerkin discretization and is effective whenever the PDE solution is sufficiently sparse or compressible, i.e., when most of its coefficients are zero or negligible. The sparsity of the solution is promoted by employing a multilevel dictionary of B-splines as opposed to a basis. Thanks to sparsity and the fact that only a fraction of the full discretization matrix is assembled, the proposed technique has the potential to lead to significant computational savings. We show the effectiveness of CossIGA for the solution of the 2D and 3D Poisson equation over nontrivial geometries by means of an extensive numerical investigation.
