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

TL;DR

This paper introduces efficient hierarchical algorithms for generating orthogonal matrices and performing matrix-vector multiplications in biomolecular rigid body simulations, significantly reducing computational complexity.

Contribution

It presents a novel method to construct orthonormal bases and perform matrix-vector operations with optimal asymptotic complexity in biomolecular simulations.

Findings

Constructed orthogonal basis matrices using O(n log n) operations.

Developed hierarchical algorithms with O(n) complexity for matrix-vector multiplications.

Demonstrated performance and accuracy through numerical experiments.

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 $6 \times 3n$ transformation matrix $Z$ that maps the force on each bead to $p$'s translational and rotational forces (a $6\times 1$ vector), and $V$ the row space of $Z$, we show how to explicitly construct the $(3n-6) \times 3n$ matrix $\tilde{Q}$ consisting of $(3n-6)$ orthonormal basis vectors of $V^{\perp}$ (orthogonal complement of $V$) using only $\mathcal{O}(n \log n)$ operations and storage. For applications where only the matrix-vector multiplications $\tilde{Q}{\bf v}$ and $\tilde{Q}^T {\bf v}$ are needed, we introduce asymptotically optimal $\mathcal{O}(n)$ hierarchical algorithms without explicitly forming $\tilde{Q}$. Preliminary numerical results are presented to demonstrate the performance and accuracy of the numerical algorithms.