Structure-Aware Calculation of Many-Electron Wave Function Overlaps on Multicore Processors

Abstract

We introduce a new algorithm that exploits the relationship between the determinants of a sequence of matrices that appear in the calculation of many-electron wave function overlaps, yielding a considerable reduction of the theoretical cost. The resulting enhanced algorithm is embarrassingly parallel and our comparison against the (embarrassingly parallel version of) original algorithm, on a computer node with 40 physical cores, shows acceleration factors which are close to 7 for the largest problems, consistent with the theoretical dierence.

Efficient update of determinants for many-electron wave function overlaps

“…In addition, the computational cost of the algorithm is dominated by the operation(s) performed in the innermost loop. In [21], the authors introduced a variant of the naive algorithm that exploited these two observations in order to update an initial factorization, computed at the beginning of each iteration of the loop indexed by r 2 , and obtain from that all the determinants of the two innermost loops with a reduced cost.…”

“…which corresponds to the matrix that is obtained by eliminating columns c 1 , c 2 from U. The key to the cost savings for the algorithm described in [21] lies in exploiting the "close-to-upper triangular" structure of U −||c 1 ,c 2 when reducing this matrix to upper triangular form. Concretely, consider the partitioning…”

“…where u B corresponds to the bottom row of U . The base algorithm that exploits the relationship between the matrices factorized in the inner two loops proposed in [21]…”

“…In [21], the reduction of the appropriate blocks of U 2 , U 3 to upper triangular form was done via Gauss transforms inside the innermost loop (indexed by c 2 ). Taking into account the costs in Table 1, the global cost of the algorithm in [21] is given by…”

“…In Section 2. we briefly review the Mathematical-Physics problem. Next, in Section 3, we introduce a straight-forward algorithm for computing the determinants of the minors and the basic columnwise re-utilization algorithm [14,21]. In Section 4, we describe in detail several algorithmic (and implementation) optimization strategies that can be applied to the columnwise reutilization algorithms.…”