Parallel numerical solution to large-scale eigenvalue problem in master equation of protein folding kinetics

A master equation characterizes the time-evolution of trajectories, the transition of states in protein folding kinetics. Numerical solution of the master equation requires calculating eigenvalues for the corresponding large scale eigenvalue problem. In this paper, we present a parallel computing technique to compute the eigenvalues of the matrix with an N-dimensional vector of the instantaneous probability of the N conformations. Parallelization of the implicitly restarted Arnoldi method is successfully implemented on a PC-based Linux cluster. The parallelization scheme used in this work mainly partitions the operations of the matrix. For the Arnoldi factorization, we replicate the upper Hessenberg matrix H_m for each processor, and distribute the set of Arnoldi vectors V_m among processors. Each processor performs its own operations. This algorithm is implemented on a PC-based Linux cluster with message passing interface (MPI) libraries. Our preliminary numerical experiment performing on the 32-nodes PC-based Linux cluster has shown that the maximum difference among CPUs is within 10%. A 23 times speedup and 72% parallel efficiency are also attained for the tested cases. This approach enables us to explore large scale dynamics of protein folding.