Condition number complexity of an elementary algorithm for computing a reliable solution of a conic linear system

BACKGROUND: The limited-angle reconstruction problem is of both theoretical and practical importance. Due to the severe ill-posedness of the problem, it is very challenging to get a valid reconstructed result from the known small limited-angle projection data. The theoretical ill-posedness leads the normal equation A T Ax = A T b of the linear system derived by discretizing the Radon transform to be severely ill-posed, which is quantified as the large condition number of A T A. OBJECTIVE: To develop and test a new valid algorithm for improving the limited-angle image reconstruction with the known appropriately small angle range from [ 0 , π 3 ] ∼ [ 0 , π 2 ] . METHODS: We propose a reweighted method of improving the condition number of A T Ax = A T b and the corresponding preconditioned Landweber iteration scheme. The weight means multiplying A T Ax = A T b by a matrix related to A T A, and the weighting process is repeated multiple times. In the experiment, the condition number of the coefficient matrix in the reweighted linear system decreases monotonically to 1 as the weighting times approaches infinity. RESULTS: The numerical experiments showed that the proposed algorithm is significantly superior to other iterative algorithms (Landweber, Cimmino, NWL-a and AEDS) and can reconstruct a valid image from the known appropriately small angle range. CONCLUSIONS: The proposed algorithm is effective for the limited-angle reconstruction problem with the known appropriately small angle range.

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A Simple Estimate of the Condition Number of a Linear System

College Mathematics Journal ◽

10.1080/07468342.1995.11973657 ◽

1995 ◽

Vol 26 (1) ◽

pp. 2-5 ◽

Cited By ~ 5

Author(s):

Heinrich W. Guggenheimer ◽

Alan S. Edelman ◽

Charles R. Johnson

Keyword(s):

Linear System ◽

Condition Number ◽

Simple Estimate

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Analysis of the Structured Perturbation for the BCSCB Linear System

Abstract and Applied Analysis ◽

10.1155/2015/471362 ◽

2015 ◽

Vol 2015 ◽

pp. 1-8

Author(s):

Xia Tang ◽

Zhaolin Jiang

Keyword(s):

Linear System ◽

Relative Error ◽

Condition Number ◽

Spectral Decomposition ◽

Circulant Matrix ◽

Networked Systems ◽

Perturbation Bound ◽

Numerical Example ◽

Backward Perturbation ◽

Skew Circulant

Circulant and block circulant type matrices are important tools in solving networked systems. In this paper, based on the style spectral decomposition of the basic circulant matrix and the basic skew circulant matrix, the block style spectral decomposition of the BCSCB matrix is obtained. And then, the structure perturbation is analysed, which includes the condition number and relative error of the BCSCB linear system. Then the optimal backward perturbation bound of the BCSCB linear system is discussed. Simultaneously, the algorithm for the optimal backward perturbation bound is given. Finally, a numerical example is provided to verify the effectiveness of the algorithm.

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A Deluxe FETI-DP Preconditioner for a Composite Finite Element and DG Method

Computational Methods in Applied Mathematics ◽

10.1515/cmam-2015-0025 ◽

2015 ◽

Vol 15 (4) ◽

pp. 465-482

Author(s):

Maksymilian Dryja ◽

Juan Galvis ◽

Marcus Sarkis

Keyword(s):

Finite Element ◽

Linear System ◽

Condition Number ◽

Numerical Experiments ◽

Mesh Size ◽

Discontinuous Coefficients ◽

Two Dimensional ◽

Dg Method ◽

Numer Anal ◽

Appl Math

AbstractIn this paper, we present and analyze a FETI-DP solver with deluxe scaling for a Nitsche-type discretization [Comput. Methods Appl. Math. 3 (2003), 76–85], [SIAM J. Numer. Anal. 49 (2011), 1761–1787] based on a discontinuous Galerkin (DG) method for elliptic two-dimensional problems with discontinuous coefficients and non-matching meshes only across subdomains. We establish a condition number estimate for the preconditioned linear system which is scalable with respect to the number of subdomains, is quasi-optimal polylogarithmic with respect to subdomain mesh size, and is independent of coefficient discontinuities and ratio of mesh sizes across subdomain interfaces. Numerical experiments support the theory and show that the deluxe scaling improves significantly the performance over classical scaling.

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A Simple Estimate of the Condition Number of a Linear System

College Mathematics Journal ◽

10.2307/2687283 ◽

1995 ◽

Vol 26 (1) ◽

pp. 2 ◽

Cited By ~ 17

Author(s):

Heinrich W. Guggenheimer ◽

Alan S. Edelman ◽

Charles R. Johnson

Keyword(s):

Linear System ◽

Condition Number ◽

Simple Estimate

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Application of an iterative Golub-Kahan algorithm to structural mechanics problems with multi-point constraints

Advanced Modeling and Simulation in Engineering Sciences ◽

10.1186/s40323-020-00181-2 ◽

2020 ◽

Vol 7 (1) ◽

Author(s):

Carola Kruse ◽

Vincent Darrigrand ◽

Nicolas Tardieu ◽

Mario Arioli ◽

Ulrich Rüde

Keyword(s):

Finite Element ◽

Linear System ◽

Iterative Method ◽

Condition Number ◽

Degrees Of Freedom ◽

Mesh Size ◽

Element Model ◽

Structural Mechanics ◽

Iterative Solvers ◽

Test Problems

AbstractKinematic relationships between degrees of freedom, also named multi-point constraints, are frequently used in structural mechanics. In this paper, the Craig variant of the Golub-Kahan bidiagonalization algorithm is used as an iterative method to solve the arising linear system with a saddle point structure. The condition number of the preconditioned operator is shown to be close to unity and independent of the mesh size. This property is proved theoretically and illustrated on a sequence of test problems of increasing complexity, including concrete structures enforced with pretension cables and the coupled finite element model of a reactor containment building. The Golub-Kahan algorithm converges in only a small number of steps for all considered test problems and discretization sizes. Furthermore, it is robust in practical cases that are otherwise considered to be difficult for iterative solvers.

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