Scalable parallelization for the solution of phonon Boltzmann Transport Equation

The Boltzmann Transport Equation (BTE) for phonons is often used to predict thermal transport at submicron scales in semiconductors. The BTE is a seven-dimensional nonlinear integro-differential equation, resulting in difficulty in its solution even after linearization under the single relaxation time approximation. Furthermore, parallelization and load balancing are challenging, given the high dimensionality and variability of the linear systems' conditioning. This work presents a 'synthetic' scalable parallelization method for solving the BTE on large-scale systems. The method includes cell-based parallelization, combined band+cell-based parallelization, and batching technique. The essential computational ingredient of cell-based parallelization is a sparse matrix-vector product (SpMV) that can be integrated with an existing linear algebra library like PETSc. The combined approach enhances the cell-based method by further parallelizing the band dimension to take advantage of low inter-band communication costs. For the batched approach, we developed a batched SpMV that enables multiple linear systems to be solved simultaneously, merging many MPI messages to reduce communication costs, thus maintaining scalability when the grain size becomes very small. We present numerical experiments to demonstrate our method's excellent speedups and scalability up to 16384 cores for a problem with 12.6 billion unknowns.