Approximate Schur‐Block ILU Preconditioners for Regularized Solution of Discrete Ill‐Posed Problems

Abstract

High order iterative methods with a recurrence formula for approximate matrix inversion are proposed such that the matrix multiplications and additions in the calculation of matrix polynomials for the hyperpower methods of orders of convergence p=4k+3, where k≥1 is integer, are reduced through factorizations and nested loops in which the iterations are defined using a recurrence formula. Therefore, the computational cost is lowered from κ=4k+3 to κ=k+4 matrix multiplications per step. An algorithm is proposed to obtain regularized solution of ill-posed discrete problems with noisy data by constructing approximate Schur-Block Incomplete LU (Schur-BILU) preconditioner and by preconditioning the one step stationary iterative method. From the proposed methods of approximate matrix inversion, the methods of orders p=7,11,15,19 are applied for approximating the Schur complement matrices. This algorithm is applied to solve two problems of Fredholm integral equation of first kind. The first example is the harmonic continuation problem and the second example is Phillip’s problem. Furthermore, experimental study on some nonsymmetric linear systems of coefficient matrices with strong indefinite symmetric components from Harwell-Boeing collection is also given. Numerical analysis for the regularized solutions of the considered problems is given and numerical comparisons with methods from the literature are provided through tables and figures.

Reusing Preconditioners in Projection Based Model Order Reduction Algorithms

“…Variations of ILU that exploit certain matrix constructs can also be developed. For example, ILU based upon Schur's complement [40]. Further, ILQ, SSOR and ADI are other kinds of preconditioning that fall under the implicit category [39].…”

“…Another example is where the approximate inverse of the coefficient matrix is used to compute an approximation to matrix's Schur's complement. This is then used to build an ILU preconditioner [40]. Now, we give the details of SPAI.…”

“…(a) Besides the currently used basic SPAI preconditioner, we will investigate the use of high-order convergent approximate inverse preconditioners [41], [42] as well as hybrid versions, which use a combination of factorization and approximate inverse techniques [40], [43].…”

Incomplete block‐matrix factorization of M‐matrices using two‐step iterative method for matrix inversion and preconditioning

“…17 Therein, the matrices were partitioned into 2 × 2 block form using graph partitioning, and various preconditioners were constructed using approximations to the Schur complement for right preconditioning of the generalized minimum residual method (GMRES). Most recently, Buranay and Iyikal 18 proposed iteratively constructed approximate Schur-block ILU preconditioners for the regularized solution of discrete ill-posed problems. The authors gave iterative methods of convergence order p = 7, 11, 15, 19 for the approximation of the Schur complement, and the constructed preconditioners were used for the one-step stationary iterative (OSSI) method.…”

A predictor-corrector iterative method for solving linear least squares problems and perturbation error analysis

“…Class 1 methods converge with order of convergence p = 3 * 2 k + 1 and Class 2 methods converge with order p = 5 * 2 k -1, requiring 2k + 4 Mms and 3k + 4 Mms, respectively. Most recently, high order iterative methods with a recurrence formula for approximate matrix inversion have been proposed in [19] with orders of convergence p = 4k + 3, where k ≥ 1 is an integer requiring k + 4 matrix multiplications per iteration. The methods of orders p = 7, 11, 15, 19 are used to construct approximate Schur-block incomplete LU preconditioners (Schur-BILU) for regularized solution of discrete ill-posed problems.…”