Stochastic Preconditioning for Diagonally Dominant Matrices

Abstract

Abstract. This paper presents a new stochastic preconditioning approach for large sparse matrices. For the class of matrices that are row-wise and column-wise irreducibly diagonally dominant, we prove that an incomplete LDL T factorization in a symmetric case or an incomplete LDU factorization in an asymmetric case can be obtained from random walks, and used as a preconditioner. It is argued that our factor matrices have better quality, i.e., better accuracy-size tradeoffs, than preconditioners produced by existing incomplete factorization methods. Therefore a resulting preconditioned Krylov-subspace iterative solver requires less computation than traditional methods to solve a set of linear equations with the same error tolerance. The advantage increases for larger and denser matrices. These claims are verified by numerical tests, and we provide techniques that can potentially extend the theory to non-diagonally-dominant matrices.

Incremental power network analysis using backward random walks

“…We study an interesting correspondence between the backward random walks of Section II.A with LU decomposition of the LHS matrix of Eq. (1), G. This relation can be used to find a quick and moderately accurate LU factorization of G which can be used for a variety of applications, e.g., as a preconditioner for an iterative method similar to the work of [16]. The work of [16] showed the relation between the UL factors of the LHS and the forward random walks, and this is its counterpart for backward walks.…”

“…(1), G. This relation can be used to find a quick and moderately accurate LU factorization of G which can be used for a variety of applications, e.g., as a preconditioner for an iterative method similar to the work of [16]. The work of [16] showed the relation between the UL factors of the LHS and the forward random walks, and this is its counterpart for backward walks. This relationship is not specifically used by the incremental solver but is pointed out for completeness.…”

“…Similarly the forward random walk game is constructed based on probability matrix of Eq. (11) which is computed by normalizing to rows of matrix G [16]. Fig.…”

“…Hence, if a rough but fast estimation of G −1 is desired, backward random walks is a suitable candidate. The number of walks for a desired accuracy can be determined dynamically in a manner similar to that derived in [16]. It can be shown that for a given relative error of δ, the number of walks needed for the relative error of random walk method to be less than or equal to δ, with confidence of α (e.g., 99%), is given by:…”

A hierarchical matrix inversion algorithm for vectorless power grid verification

“…In addition, it is proposed in [4] that the preconditioned conjugate gradient (PCG) method [5], [6] can be employed to compute c l with a proper choice of preconditioner. The stochastic preconditioning technique proposed in [10] is applied in [4] to generate the preconditioner using random walks [11].…”