Efficient Method for Moore-Penrose Inverse Problems Involving Symmetric Structures Based on Group Theory

Abstract

The Moore-Penrose inverse has many applications in civil engineering, such as structural control, nonlinear buckling, and form-finding. However, solving the generalized inverse requires ample computational resources, especially for large-sized matrices. An efficient method based on group theory for the Moore-Penrose inverse problems for symmetric structures is proposed, which can deal with not only well-conditioned but also rank deficient matrices. First, the QR decomposition algorithm is chosen to evaluate the generalized inverse of any sparse and rank deficient matrix. In comparison with other well established algorithms, the QR method has superiority in computation efficiency and accuracy. Then, a group-theoretic approach to computing the Moore-Penrose inverse for problems involving symmetric structures is described. Based on the inherent symmetry and the irreducible representations, the orthogonal transformation matrices are deduced to express the inverse problem in a symmetry-adapted coordinate system. The original problem is transferred into computing the generalized inverse of many independent submatrices. Numerical experiments on three different types of structures with cyclic or dihedral symmetry are carried out. It is concluded from the numerical results and comparisons with two conventional methods that the proposed technique is efficient and accurate.