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Register a new userRCHOL: Randomized Cholesky Factorization for Solving SDD Linear Systems
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Added by
Chao Chen
Publication date
2020
fields
Informatics Engineering
and research's language is
English
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We introduce a randomized algorithm, namely RCHOL, to construct an approximate Cholesky factorization for a given Laplacian matrix (a.k.a., graph Laplacian). From a graph perspective, the exact Cholesky factorization introduces a clique in the underlying graph after eliminating a row/column. By randomization, RCHOL only retains a sparse subset of the edges in the clique using a random sampling developed by Spielman and Kyng. We prove RCHOL is breakdown-free and apply it to solving large sparse linear systems with symmetric diagonally dominant matrices. In addition, we parallelize RCHOL based on the nested dissection ordering for shared-memory machines. We report numerical experiments that demonstrate the robustness and the scalability of RCHOL. For example, our parallel code scaled up to 64 threads on a single node for solving the 3D Poisson equation, discretized with the 7-point stencil on a $1024times 1024 times 1024$ grid, a problem that has one billion unknowns.
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We study the use of Krylov subspace recycling for the solution of a sequence of slowly-changing families of linear systems, where each family consists of shifted linear systems that differ in the coefficient matrix only by multiples of the identity. Our aim is to explore the simultaneous solution of each family of shifted systems within the framework of subspace recycling, using one augmented subspace to extract candidate solutions for all the shifted systems. The ideal method would use the same augmented subspace for all systems and have fixed storage requirements, independent of the number of shifted systems per family. We show that a method satisfying both requirements cannot exist in this framework. As an alternative, we introduce two schemes. One constructs a separate deflation space for each shifted system but solves each family of shifted systems simultaneously. The other builds only one recycled subspace and constructs approximate corrections to the solutions of the shifted systems at each cycle of the iterative linear solver while only minimizing the base system residual. At convergence of the base system solution, we apply the method recursively to the remaining unconverged systems. We present numerical examples involving systems arising in lattice quantum chromodynamics.
We present a parallel hierarchical solver for general sparse linear systems on distributed-memory machines. For large-scale problems, this fully algebraic algorithm is faster and more memory-efficient than sparse direct solvers because it exploits the low-rank structure of fill-in blocks. Depending on the accuracy of low-rank approximations, the hierarchical solver can be used either as a direct solver or as a preconditioner. The parallel algorithm is based on data decomposition and requires only local communication for updating boundary data on every processor. Moreover, the computation-to-communication ratio of the parallel algorithm is approximately the volume-to-surface-area ratio of the subdomain owned by every processor. We present various numerical results to demonstrate the versatility and scalability of the parallel algorithm.
The linear equations that arise in interior methods for constrained optimization are sparse symmetric indefinite and become extremely ill-conditioned as the interior method converges. These linear systems present a challenge for existing solver frameworks based on sparse LU or LDL^T decompositions. We benchmark five well known direct linear solver packages using matrices extracted from power grid optimization problems. The achieved solution accuracy varies greatly among the packages. None of the tested packages delivers significant GPU acceleration for our test cases.
Some new rigorous perturbation bounds for the generalized Cholesky factorization with normwise or componentwise perturbations in the given matrix are obtained, where the componentwise perturbation has the form of backward rounding error for the generalized Cholesky factorization algorithm. These bounds can be much tighter than some existing ones while the conditions for them to hold are simple and moderate.
We propose to compute a sparse approximate inverse Cholesky factor $L$ of a dense covariance matrix $Theta$ by minimizing the Kullback-Leibler divergence between the Gaussian distributions $mathcal{N}(0, Theta)$ and $mathcal{N}(0, L^{-top} L^{-1})$, subject to a sparsity constraint. Surprisingly, this problem has a closed-form solution that can be computed efficiently, recovering the popular Vecchia approximation in spatial statistics. Based on recent results on the approximate sparsity of inverse Cholesky factors of $Theta$ obtained from pairwise evaluation of Greens functions of elliptic boundary-value problems at points ${x_{i}}_{1 leq i leq N} subset mathbb{R}^{d}$, we propose an elimination ordering and sparsity pattern that allows us to compute $epsilon$-approximate inverse Cholesky factors of such $Theta$ in computational complexity $mathcal{O}(N log(N/epsilon)^d)$ in space and $mathcal{O}(N log(N/epsilon)^{2d})$ in time. To the best of our knowledge, this is the best asymptotic complexity for this class of problems. Furthermore, our method is embarrassingly parallel, automatically exploits low-dimensional structure in the data, and can perform Gaussian-process regression in linear (in $N$) space complexity. Motivated by the optimality properties of our methods, we propose methods for applying it to the joint covariance of training and prediction points in Gaussian-process regression, greatly improving stability and computational cost. Finally, we show how to apply our method to the important setting of Gaussian processes with additive noise, sacrificing neither accuracy nor computational complexity.
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