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We develop a general framework for finding approximately-optimal preconditioners for solving linear systems. Leveraging this framework we obtain improved runtimes for fundamental preconditioning and linear system solving problems including the following. \begin{itemize} \item \textbf{Diagonal preconditioning.} We give an algorithm which, given positive definite $$\mathbf{K} \in \mathbb{R}^{d \times d}$$ with $$\mathrm{nnz}(\mathbf{K})$$ nonzero entries, computes an $$\epsilon$$-optimal diagonal preconditioner in time $$\widetilde{O}(\mathrm{nnz}(\mathbf{K}) \cdot \mathrm{poly}(\kappa^\star,\epsilon^{-1}))$$, where $$\kappa^\star$$ is the optimal condition number of the rescaled matrix. \item \textbf{Structured linear systems.} We give an algorithm which, given $$\mathbf{M} \in \mathbb{R}^{d \times d}$$ that is either the pseudoinverse of a graph Laplacian matrix or a constant spectral approximation of one, solves linear systems in $$\mathbf{M}$$ in $$\widetilde{O}(d^2)$$ time. \end{itemize} Our diagonal preconditioning results improve state-of-the-art runtimes of $$\Omega(d^{3.5})$$ attained by general-purpose semidefinite programming, and our solvers improve state-of-the-art runtimes of $$\Omega(d^{\omega})$$ where $$\omega > 2.3$$ is the current matrix multiplication constant. We attain our results via new algorithms for a class of semidefinite programs (SDPs) we call \emph{matrix-dictionary approximation SDPs}, which we leverage to solve an associated problem we call \emph{matrix-dictionary recovery}.
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Effectiveness and robustness revisited for a preconditioning technique based on structured incomplete factorization
Summary In this work, we provide new analysis for a preconditioning technique called structured incomplete factorization (SIF) for symmetric positive definite matrices. In this technique, a scaling and compression strategy is applied to construct SIF preconditioners, where off‐diagonal blocks of the original matrix are first scaled and then approximated by low‐rank forms. Some spectral behaviors after applying the preconditioner are shown. The effectiveness is confirmed with the aid of a type of two‐dimensional and three‐dimensional discretized model problems. We further show that previous studies on the robustness are too conservative. In fact, the practical multilevel version of the preconditioner has a robustness enhancement effect, and is unconditionally robust (or breakdown free) for the model problems regardless of the compression accuracy for the scaled off‐diagonal blocks. The studies give new insights into the SIF preconditioning technique and confirm that it is an effective and reliable way for designing structured preconditioners. The studies also provide useful tools for analyzing other structured preconditioners. Various spectral analysis results can be used to characterize other structured algorithms and study more general problems.
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- Award ID(s):
- 1819166
- PAR ID:
- 10375901
- Publisher / Repository:
- Wiley Blackwell (John Wiley & Sons)
- Date Published:
- Journal Name:
- Numerical Linear Algebra with Applications
- Volume:
- 27
- Issue:
- 3
- ISSN:
- 1070-5325
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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