Compressive isogeometric analysis

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.