Incomplete Cholesky and its modified variant provide the largest work reduction among black-box preconditioners for CG, but fail on roughly 20% of problems, and AMD matrix ordering substantially outperforms RCM.
A Simple, Combinatorial Algorithm for Solving SDD Systems in Nearly-Linear Time
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abstract
In this paper, we present a simple combinatorial algorithm that solves symmetric diagonally dominant (SDD) linear systems in nearly-linear time. It uses very little of the machinery that previously appeared to be necessary for a such an algorithm. It does not require recursive preconditioning, spectral sparsification, or even the Chebyshev Method or Conjugate Gradient. After constructing a "nice" spanning tree of a graph associated with the linear system, the entire algorithm consists of the repeated application of a simple (non-recursive) update rule, which it implements using a lightweight data structure. The algorithm is numerically stable and can be implemented without the increased bit-precision required by previous solvers. As such, the algorithm has the fastest known running time under the standard unit-cost RAM model. We hope that the simplicity of the algorithm and the insights yielded by its analysis will be useful in both theory and practice.
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2025 1verdicts
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An Empirical Study of Conjugate Gradient Preconditioners for Solving Symmetric Positive Definite Systems of Linear Equations
Incomplete Cholesky and its modified variant provide the largest work reduction among black-box preconditioners for CG, but fail on roughly 20% of problems, and AMD matrix ordering substantially outperforms RCM.