A robust hierarchical solver for ill-conditioned systems with applications to ice sheet modeling

Abstract

A hierarchical solver is proposed for solving sparse ill-conditioned linear systems in parallel. The solver is based on a modification of the LoRaSp method, but employs a deferred-compression technique, which provably reduces the approximation error and significantly improves efficiency. Moreover, the deferred-compression technique introduces minimal overhead and does not affect parallelism. As a result, the new solver achieves linear computational complexity under mild assumptions and excellent parallel scalability. To demonstrate the performance of the new solver, we focus on applying it to solve sparse linear systems arising from ice sheet modeling. The strong anisotropic phenomena associated with the thin structure of ice sheets creates serious challenges for existing solvers. To address the anisotropy, we additionally developed a customized partitioning scheme for the solver, which captures the strong-coupling direction accurately. In general, the partitioning can be computed algebraically with existing software packages, and thus the new solver is generalizable for solving other sparse linear systems. Our results show that ice sheet problems of about 300 million degrees of freedom have been solved in just a few minutes using a thousand processors.

Sparse hierarchical solvers with guaranteed convergence

“…As a result, the relative error tolerance needs to be reduced in order to enforce a given absolute tolerance. This point is further discussed and addressed, for example, in the work of Chen et al…”

Effectiveness and robustness revisited for a preconditioning technique based on structured incomplete factorization

“…Here, we show that both types of schemes are similarly effective and also give a more intuitive explanation of the effectiveness by extending Equation (10). 3is k. For the Cholesky SIF factorL in Equation (6), the Equation (9) holds with the nonzero singular values ofĈ in Equation (10) given by…”

“…We anticipate that the analysis here can serve as a starting point for studying and designing SIF preconditioners for more practical discretized problems. Readers who are interested in numerical evidences for practical sparse problems are referred to References 7,10,11,14,15. Remark 2. The computational complexity of the SIF scheme for sparse matrices can be understood as follows.…”

“…These multilevel test results roughly follow similar patterns in the estimates for the prototype case after Corollary 3. Numerical tests for more practical problems can be found in References 7,10,11,14,15. Note that the effectiveness results in Reference 7 has some strict robustness requirements in order to guarantee that the multilevel SIF scheme does not break down or the approximation to A remains positive definite. In the following section, we show that such requirements are too conservative and may be relaxed.…”

“…Practical numerical tests have shown superior convergence results as compared with standard structured preconditioning based on direct off-diagonal compression. 7 The scaling and compression strategy is later also followed by a series of other work [9][10][11][12] for designing structured preconditioners for both dense and sparse matrices. Related ideas also appear in some work for preconditioning sparse matrices.…”