BiCGCR2: A new extension of conjugate residual method for solving non-Hermitian linear systems

In the present paper, we introduce a new extension of the conjugate residual (CR) for solving non-Hermitian linear systems with the aim of developing an alternative basic solver to the established biconjugate gradient (BiCG), biconjugate residual (BiCR) and biconjugate A-orthogonal residual (BiCOR) methods. The proposed Krylov subspace method, referred to as the BiCGCR2 method, is based on short-term vector recurrences and is mathematically equivalent to both BiCR and BiCOR. We demonstrate by extensive numerical experiments that the proposed iterative solver has often better convergence performance than BiCG, BiCR and BiCOR. Hence, it may be exploited for the development of new variants of non-optimal Krylov subspace methods

ABS-Based Direct Method for Solving Complex Systems of Linear Equations

Efficient solution of linear systems of equations is one of the central topics of numerical computation. Linear systems with complex coefficients arise from various physics and quantum chemistry problems. In this paper, we propose a novel ABS-based algorithm, which is able to solve complex systems of linear equations. Theoretical analysis is given to highlight the basic features of our new algorithm. Four variants of our algorithm were also implemented and intensively tested on randomly generated full and sparse matrices and real-life problems. The results of numerical experiments reveal that our ABS-based algorithm is able to compute the solution with high accuracy. The performance of our algorithm was compared with a commercially available software, Matlab’s mldivide (\) algorithm. Our algorithm outperformed the Matlab algorithm in most cases in terms of computational accuracy. These results expand the practical usefulness of our algorithm

Fast Microwave Tomography Algorithm for Breast Cancer Imaging

Microwave tomography has shown promise for breast cancer imaging. The microwaves are harmless to body tissues, which makes microwave tomography a safe adjuvant screening to mammography. Although many clinical studies have shown the effectiveness of regular screening for the detection of breast cancer, the anatomy of the breast and its critical tissues challenge the identification and diagnosis of tumors in this region. Detection of tumors in the breast is more challenging in heterogeneously dense and extremely dense breasts, and microwave tomography has the potential to be effective in such cases. The sensitivity of microwaves to various breast tissues and the comfort and safety of the screening method have made microwave tomography an attractive imaging technique. Despite the need for an alternative screening technique, microwave tomography has not yet been introduced as a screening modality in regular health care, and is still subject to research. The main obstacles are imperfect hardware systems and inefficient imaging algorithms. The immense computational costs for the image reconstruction algorithm present a crucial challenge. 2D imaging algorithms are proposed to reduce the amount of hardware resources required and the imaging time. Although 2D microwave tomography algorithms are computationally less expensive, few imaging groups have been successful in integrating the acquired 3D data into the 2D tomography algorithms for clinical applications. The microwave tomography algorithms include two main computation problems: the forward problem and the inverse problem. The first part of this thesis focuses on a new fast forward solver, the 2D discrete dipole approximation (DDA), which is formulated and modeled. The effect of frequency, sampling number, target size, and contrast on the accuracy of the solver are studied. Additionally, the 2D DDA time efficiency and computation time as a single forward solver are investigated.\ua0 The second part of this thesis focuses on the inverse problem. This portion of the algorithm is based on a log-magnitude and phase transformation optimization problem and is formulated as the Gauss-Newton iterative algorithm. The synthetic data from a finite-element-based solver (COMSOL Multiphysics) and the experimental data acquired from the breast imaging system at Chalmers University of Technology are used to evaluate the DDA-based image reconstruction algorithm. The investigations of modeling and computational complexity show that the 2D DDA is a fast and accurate forward solver that can be embedded in tomography algorithms to produce images in seconds. The successful development and implementation in this thesis of 2D tomographic breast imaging with acceptable accuracy and high computational cost efficiency has provided significant savings in time and in-use memory and is a dramatic improvement over previous implementations