A class of generic factored and multi-level recursive approximate inverse techniques for solving general sparse systems

Abstract

Purpose – The purpose of this paper is to propose novel factored approximate sparse inverse schemes and multi-level methods for the solution of large sparse linear systems. Design/methodology/approach – The main motive for the derivation of the various generic preconditioning schemes lies to the efficiency and effectiveness of factored preconditioning schemes in conjunction with Krylov subspace iterative methods as well as multi-level techniques for solving various model problems. Factored approximate inverses, namely, Generic Factored Approximate Sparse Inverse, require less fill-in and are computed faster due to the reduced number of nonzero elements. A modified column wise approach, namely, Modified Generic Factored Approximate Sparse Inverse, is also proposed to further enhance performance. The multi-level approximate inverse scheme, namely, Multi-level Algebraic Recursive Generic Approximate Inverse Solver, utilizes a multi-level hierarchy formed using Block Independent Set reordering scheme and an approximation of the Schur complement that results in the solution of reduced order linear systems thus enhancing performance and convergence behavior. Moreover, a theoretical estimate for the quality of the multi-level approximate inverse is also provided. Findings – Application of the proposed schemes to various model problems is discussed and numerical results are given concerning the convergence behavior and the convergence factors. The results are comparatively better than results by other researchers for some of the model problems. Research limitations/implications – Further enhancements are investigated for the proposed factored approximate inverse schemes as well as the multi-level techniques to improve quality of the schemes. Furthermore, the proposed schemes rely on the definition of multiple parameters that for some problems require thorough testing, thus adaptive techniques to define the values of the various parameters are currently under research. Moreover, parallel schemes will be investigated. Originality/value – The proposed approximate inverse preconditioning schemes as well as multi-level schemes are efficient computational methods that are valuable for computer scientists and for scientists and engineers in engineering computations.

Higher Order Finite Difference Scheme for solving 3D Black-Scholes equation based on Generic Factored Approximate Sparse Inverse Preconditioning using Reordering Schemes

“…For the time integration the implicit Backward Difference Formulas (BDF) in conjunction with implicit Runge-Kutta methods was used. The resulting linear system of algebraic equations was solved at each time step by the Preconditioned BiConjugate Gradient Stabilized (PBiCG-STAB) method, [16], in conjunction with the Modified Generic Factored Approximate Sparse Inverse (MGenFAspI) scheme, [5,6], using reordering schemes, [2]. The number of nonzero elements in the sparsity patterns required to compute the factors of the MGenFAspI matrix are rapidly growing as the levels of fill (lfill) parameter increases.…”

“…This modified approach is performed column-wise by utilizing a restricted solution algorithm to compute each column of the approximate inverse factors. The modified approach minimizes the searches for elements and enhances performance of the method, [5].…”

“…The system (28) is solved only for the nonzero elements given by the approximate inverse sparsity pattern algorithm, [3,5]. This modified approach is performed column-wise by utilizing a restricted solution algorithm to compute each column of the approximate inverse factors.…”

“…In order to improve the convergence behavior of the Preconditioned BiConjugate Gradient Stabilized (PBiCG-STAB) method, the Modified Generic Factored Approximate Sparse Inverse (MGenFAspI) matrix technique, [5], was considered to be used as a preconditioner. The proposed scheme constructs an inverse of the form M=GH where G is the upper triangular factor and H is the lower triangular factor of the approximate inverse M. Let us consider the Incomplete LU factorization of the coefficient matrix A:…”

“…The sparsity patterns of the factors G, H are computed according to the approximate inverse sparsity pattern algorithm, [3], applied to the patterns of L and U factors. The computation of the Modified Generic Factored Approximate Sparse Inverse (MGenFAspI) is computed by solving the following systems, [5]:…”