Hierarchical Orthogonal Matrix Generation and Matrix-Vector Multiplications in Rigid Body Simulations

Abstract

In this paper, we apply the hierarchical modeling technique and study some numerical linear algebra problems arising from the Brownian dynamics simulations of biomolecular systems where molecules are modeled as ensembles of rigid bodies. Given a rigid body p consisting of n beads, the 6×3n transformation matrix Z that maps the force on each bead to p’s translational and rotational forces (a 6 × 1 vector), and V the row space of Z, we show how to explicitly construct the (3n – 6) × 3n matrix trueQ˜ consisting of (3n – 6) orthonormal basis vectors of V⊥ (orthogonal complement of V) using only O(nlogn) operations and storage. For applications where only the matrix-vector multiplications Q˜V and trueQ˜TV are needed, we introduce asymptotically optimal O(n) hierarchical algorithms without explicitly forming trueQ˜. Preliminary numerical results are presented to demonstrate the performance and accuracy of the numerical algorithms.