Hierarchical Orthogonal Matrix Generation and Matrix-Vector Multiplications in Rigid Body Simulations
This work addresses computational bottlenecks in Brownian dynamics simulations of biomolecular systems modeled as rigid bodies, offering asymptotically faster algorithms for a specific numerical linear algebra problem.
The paper presents a hierarchical method to construct an orthonormal basis for the orthogonal complement of the row space of a transformation matrix in rigid body simulations, achieving O(n log n) operations and storage, and introduces O(n) algorithms for matrix-vector multiplications without explicit matrix formation.
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.