Multiscale modeling using goal-oriented adaptivity and numerical homogenization. Part II: Algorithms for the Moore–Penrose pseudoinverse

This paper is the second in this series to develop a numerical homogenization method for heterogeneous media and integrate it with goal-oriented finite element mesh adaptivity. The first paper developed the mathematical formulation of local homogenization. This was done using an appropriate averaging of the Moore–Penrose pseudoinverse of the element stiffness matrix. We also presented numerical verification of this approach and other numerical results. In the current paper, we present four algorithms to compute the Moore–Penrose pseudoinverse that also exploit the sparsity. These are based on QR factorization, a priori knowledge of the null-space, a regularization based characterization, and lastly an iterative algorithm based on proper splittings of matrices. We analyze some of these algorithms in detail for homogenizing hexahedral elements in the molecular base model of densification in Step and Flash Imprint Lithography.