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Register a new userGreedy Block Gauss-Seidel Methods for Solving Large Linear Least Squares Problem
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Hanyu Li Dr.
Publication date
2020
fields
Informatics Engineering
and research's language is
English
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With a greedy strategy to construct control index set of coordinates firstly and then choosing the corresponding column submatrix in each iteration, we present a greedy block Gauss-Seidel (GBGS) method for solving large linear least squares problem. Theoretical analysis demonstrates that the convergence factor of the GBGS method can be much smaller than that of the greedy randomized coordinate descent (GRCD) method proposed recently in the literature. On the basis of the GBGS method, we further present a pseudoinverse-free greedy block Gauss-Seidel method, which doesnt need to calculate the Moore-Penrose pseudoinverse of the column submatrix in each iteration any more and hence can be achieved greater acceleration. Moreover, this method can also be used for distributed implementations. Numerical experiments show that, for the same accuracy, our methods can far outperform the GRCD method in terms of the iteration number and computing time.
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We present a novel greedy Gauss-Seidel method for solving large linear least squares problem. This method improves the greedy randomized coordinate descent (GRCD) method proposed recently by Bai and Wu [Bai ZZ, and Wu WT. On greedy randomized coordinate descent methods for solving large linear least-squares problems. Numer Linear Algebra Appl. 2019;26(4):1--15], which in turn improves the popular randomized Gauss-Seidel method. Convergence analysis of the new method is provided. Numerical experiments show that, for the same accuracy, our method outperforms the GRCD method in term of the computing time.
We consider the problem of efficiently solving large-scale linear least squares problems that have one or more linear constraints that must be satisfied exactly. Whilst some classical approaches are theoretically well founded, they can face difficulties when the matrix of constraints contains dense rows or if an algorithmic transformation used in the solution process results in a modified problem that is much denser than the original one. To address this, we propose modifications and new ideas, with an emphasis on requiring the constraints are satisfied with a small residual. We examine combining the null-space method with our recently developed algorithm for computing a null space basis matrix for a ``wide matrix. We further show that a direct elimination approach enhanced by careful pivoting can be effective in transforming the problem to an unconstrained sparse-dense least squares problem that can be solved with existing direct or iterative methods. We also present a number of solution variants that employ an augmented system formulation, which can be attractive when solving a sequence of related problems. Numerical experiments using problems coming from practical applications are used throughout to demonstrate the effectiveness of the different approaches.
The famous greedy randomized Kaczmarz (GRK) method uses the greedy selection rule on maximum distance to determine a subset of the indices of working rows. In this paper, with the greedy selection rule on maximum residual, we propose the greedy randomized Motzkin-Kaczmarz (GRMK) method for linear systems. The block version of the new method is also presented. We analyze the convergence of the two methods and provide the corresponding convergence factors. Extensive numerical experiments show that the GRMK method has almost the same performance as the GRK method for dense matrices and the former performs better in computing time for some sparse matrices, and the blo
Consider using the right-preconditioned generalized minimal residual (AB-GMRES) method, which is an efficient method for solving underdetermined least squares problems. Morikuni (Ph.D. thesis, 2013) showed that for some inconsistent and ill-conditioned problems, the iterates of the AB-GMRES method may diverge. This is mainly because the Hessenberg matrix in the GMRES method becomes very ill-conditioned so that the backward substitution of the resulting triangular system becomes numerically unstable. We propose a stabilized GMRES based on solving the normal equations corresponding to the above triangular system using the standard Cholesky decomposition. This has the effect of shifting upwards the tiny singular values of the Hessenberg matrix which lead to an inaccurate solution. Thus, the process becomes numerically stable and the system becomes consistent, rendering better convergence and a more accurate solution. Numerical experiments show that the proposed method is robust and efficient for solving inconsistent and ill-conditioned underdetermined least squares problems. The method can be considered as a way of making the GMRES stable for highly ill-conditioned inconsistent problems.
This paper is devoted to condition numbers of the total least squares problem with linear equality constraint (TLSE). With novel limit techniques, closed formulae for normwise, mixed and componentwise condition numbers of the TLSE problem are derived. Compact expressions and upper bounds for these condition numbers are also given to avoid the costly Kronecker product-based operations. The results unify the ones for the TLS problem. For TLSE problems with equilibratory input data, numerical experiments illustrate that normwise condition number-based estimate is sharp to evaluate the forward error of the solution, while for sparse and badly scaled matrices, mixed and componentwise condition numbers-based estimations are much tighter.
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