Abstract
The hierarchical interpolative factorization (HIF) offers an efficient way for solving or preconditioning elliptic partial differential equations. By exploiting locality and low-rank properties of the operators, the HIF achieves quasi-linear complexity for factorizing the discrete positive definite elliptic operator and linear complexity for solving the associated linear system. In this paper, the distributed-memory HIF (DHIF) is introduced as a parallel and distributed-memory implementation of the HIF. The DHIF organizes the processes in a hierarchical structure and keeps the communication as local as possible. The computation complexity is O(frac{Nlog N}{P}) and O(frac{N}{P}) for constructing and applying the DHIF, respectively, where N is the size of the problem and P is the number of processes. The communication complexity is O(sqrt{P}log ^3 P)alpha + O(frac{N^{2/3}}{sqrt{P}})beta where alpha is the latency and beta is the inverse bandwidth. Extensive numerical examples are performed on the NERSC Edison system with up to 8192 processes. The numerical results agree with the complexity analysis and demonstrate the efficiency and scalability of the DHIF.
Highlights
1 Background This paper proposes an efficient distributed-memory algorithm for solving elliptic partial differential equations (PDEs) of the form
This paper proposes the first distributed-memory hierarchical interpolative factorization (DHIF) for solving very large-scale problems
191.15 tsetup is the setup time which is identical to tf in previous examples for DHIF; tsolve is the total iterative solving time using GMRES; niter is the number of iterations in GMRES
Summary
This paper proposes an efficient distributed-memory algorithm for solving elliptic partial differential equations (PDEs) of the form,. Instead of directly applying the hierarchical matrix structure to the 3D problems, these methods apply it to the representation of the frontal matrices (i.e., the interactions between the lower-dimensional fronts). These methods are of linear or quasi-linear complexities in theory with much small prefactors. Based on the key observation that the number of skeleton points on each front scales linearly as the one-dimensional fronts, the HIF factorizes the matrix A (and A−1) as a product of sparse matrices that contains only O(N ) nonzero entries in total. The HIF shows significant saving in terms of computational resources required for 3D problems
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