Parallel algorithms for block tridiagonal matrices with applications

The scaling of device sizes, along with an increased demand for detailed and accurate simulation, has resulted in extraordinary computational challenges. We ameliorate several of these challenges through the design of novel divide- and-conquer algorithms that are applicable to a broad class of simulation problems. Our techniques rely on the decomposition of the underlying system matrices, enabling efficient implementation through parallel computing techniques. The versatility and computational efficiency of our approach is demonstrated through two specific applications. First is the analysis of nano-scale devices based upon the Non-Equilibrium Green\u27s Function (NEGF) formalism which allows for the calculation of quantum effects through the solution of structured matrix equations. The second is the transient simulation of power meshes which involves solutions for structured linear systems of equations. The inherently parallel framework we have constructed allows for computing resources to be flexibly allocated toward either speeding up the solution of a problem of a given size, or solving problems of larger sizes in comparable time. As an illustration we stably generate the dynamics for silicon nanowires through the use of an atomistic model consisting of over one million atomic orbitals. This analysis has been viewed to be computationally daunting when considering a NEGF based analysis using distributed computing resources