NHR PerfLab Seminar: Towards exascale simulations using the submatrix method and approximate computing

We present the submatrix method and a novel linear-scaling electronic-structure method in conjunction with approximate computing, as well as the implementation of the technique in CP2K. Even though initially proposed for inverse p-th roots [1], it has recently been recognized that the submatrix method represents a general method to approximate arbitrary matrix functions such as the matrix-sign function of large sparse matrices. The Matrix-sign function is the essential workhorse of linear-scaling electronic-structure theory, and we present an intuitive chemical justification for the accuracy of the submatrix method. We will discuss the efficient implementation of the submatrix method into CP2K with a special focus on limiting communication between compute nodes. The resulting compute kernel is the sign function of a relatively small but dense matrix. Our optimized implementation with a simple diagonalization-based evaluation of the sign function of the submatrices outperforms the Newton-Schulz Sign iteration in initial results [2], especially for larger cutoffs of matrix elements. This observation shows that the submatrix method will be a valuable tool in the context of approximate computing.

[1] https://doi.org/10.1145/3218176.3218231

[2] https://arxiv.org/abs/2003.03868