Constructing memory-minimizing schedules for multifrontal methods

Abstract

We are interested in the memory usage of multifrontal methods. Starting from the algorithms introduced by Liu, we propose new schedules to allocate and process tasks that improve memory usage. This generalizes two existing factorization and memory-allocation schedules by allowing a more flexible task allocation together with a specific tree traversal. We present optimal algorithms for this new class of schedules, and demonstrate experimentally their benefit for some real-world matrices from sparse matrix collections where either the active memory or the total memory is minimized.

Combinatorial Problems in Solving Linear Systems

“…Liu [136] minimises the size of the active memory by rearranging the children of each node (hence creating an equivalent ordering) in order to minimise the peak active memory for processing the whole tree. In a series of papers, Agullo et al [2,3] and Guermouche and L'Excellent [102] elaborate on the issues addressed by Liu [136]. In these papers, algorithmic models that separate the concerns of I/O volume and the peak memory size are developed, and a new reorganisation of the computations within the context of an out-of-core multifrontal method are presented.…”

An out-of-core sparse Cholesky solver

“…Finally, compute the resulting t i and take the split point to be the p i that gives the smallest t i . Guermouche and L'Excellent [2006] proved they obtain the optimal t i . Fig.…”

“…However, numerical experiments have shown that this can also perform poorly because a chain of frontal matrices (at the active tree levels) must be stored. This led Guermouche and L'Excellent [2006] to propose computing, for each node, the optimal point at which to allocate the frontal matrix and start the assembly.…”

“…If it is allocated and the assembly started after p i children have been processed, Guermouche and L'Excellent [2006] showed that the total storage (including the active fronts) needed to process node i is…”

“…The third term of (6.2) should really be omitted if splitting is not advantageous at node i ( p i = n i ), but it can remain since it is less than the second term in this case. The change from (6.1) to (6.2) or (6.3) does not invalidate the algorithm of Guermouche and L'Excellent [2006] for finding the split point that minimizes the stack memory. The code to order the children and find the split point is outlined in Figure 3.…”