A Flexible-Scalar Splitting Iterative Method for Linear Inverse Problems with Complex Symmetric Matrix
DOI:
https://doi.org/10.53560/PPASA(62-4)697Keywords:
Inverse Problem, Complex Symmetric Matrix, Splitting, Iterative Method, Flexible-scalarAbstract
This paper introduces a flexible scalar-splitting (f-SCSP) iterative scheme and examines its convergence properties. The approach also yields a straightforward matrix-splitting preconditioner for the original linear system. To confirm the theoretical results and evaluate practical performance, comprehensive numerical examinations are performed on various test cases. The findings indicate that the proposed method is practical, reliable, and more efficient than existing techniques for handling demanding classes of complex symmetric linear systems.
References
S.R. Arridge. Optical tomography in medical imaging. Inverse Problems 15(2): R41-R93 (1999).
D. Bertaccini. Efficient preconditioning for sequences of parametric complex symmetric linear systems. Electronic Transactions on Numerical 18: 49-64 (2004).
A. Feriani, F. Perotti, and V. Simoncini. Iterative system solvers for the frequency analysis of linear mechanical systems. Computational Methods in Applied Mechanics and Engineering 190: 1719-1739 (2000).
A. Frommer, T. Lippert, B. Medeke, and K. Schilling (Eds.). Numerical challenges in lattice quantum chromodynamics. Proceedings, Joint Interdisciplinary Workshop, Wuppertal, Germany (2000).
B. Poirier. Efficient preconditioning scheme for block partitioned matrices with structured sparsity. Numerical Linear Algebra with Applications 7: 715-726 (2000).
W.V. Dijk and F.M. Toyama. Accurate numerical solutions of the time-dependent Schrödinger equation. Physical Review E 75: 036707 (2007).
Z. Ahmed, Z.A. Kalhoro, A.W. Shaikh, M.S.R. Baloch, and O.A. Rajput. An improved iterative scheme using successive over-relaxation for solution of linear system of equations. Proceedings of the Pakistan Academy of Sciences: Part A. Physical and computational Sciences 59(3): 35-43 (2022).
M. Kanwal, Z. Ahmed, and S. Jamali. Development of generalized refinement strategies in composite stationary iterative solvers for linear systems. Southern Journal of Research 5(02(01)): 1-13 (2025).
O. Axelsson and A. Kucherov. Real valued iterative methods for solving complex symmetric linear systems. Numerical Linear Algebra with Applications 7: 197-218 (2000).
M. Benzi and D. Bertaccini. Block preconditioning of real valued iterative algorithms for complex linear systems. IMA Journal of Numerical Analysis 28: 598-618 (2007).
Z.Z. Bai. On preconditioned iteration methods for complex linear systems. Journal of Engineering Mathematics 93: 41-60 (2014).
Z.Z. Bai, M. Benzi, and F. Chen. Modified HSS iteration methods for a class of complex symmetric linear systems. Computing 87: 93-111 (2010).
Z.Z. Bai, M. Benzi, and F. Chen. On preconditioned MHSS iteration methods for complex symmetric linear systems. Numerical Algorithms 56: 297-317 (2011).
T. Wang, Q. Zheng, and L. Lu. A new iteration method for a class of complex symmetric linear systems. Journal of Computational and Applied Mathematics 325: 188-197 (2017).
D.K. Salkuyeh, D. Hezari, and V. Edalatpour. Generalized successive overrelaxation iterative method for a class of complex symmetric linear system of equations. International Journal of Computer Mathematics 92: 802-815 (2015).
D. Hezari, V. Edalatpour, and D.K. Salkuyeh. Preconditioned GSOR iterative method for a class of complex symmetric system of linear equations. Numerical Linear Algebra with Applications 22: 761-776 (2015).
O. Axelsson and D.K. Salkuyeh. A new version of a preconditioning method for certain two-by-two block matrices with square blocks. BIT Numerical Mathematics 59: 321-342 (2019).
X. Xie and H. Li. On preconditioned Euler-extrapolated single-step Hermitian and skew-Hermitian splitting method for complex symmetric linear systems. Japan Journal of Industrial and Applied Mathematics 8: 503-518 (2020).
Y. Xiang and N.M. Zhang. On the preconditioned conjugate gradient method for complex symmetric systems. Applied Mathematics Letters 120:107250 (2021).
D.K. Salkuyeh. A Preconditioner for Complex Symmetric System of Linear Equations with Indefinite Hermitian Part. Bulletin of the Iranian Mathematical Society 51: 25-25 (2025).
P.P. Zhao, S.D. Li, and R.P. Wen. Single-step HSS methods for a class of complex symmetric linear systems. Communications in Applied Mathematics and Computation 31: 200-212 (2017).
R.P. Wen, F.J. Ren, and Y.Q. Gao. On convergence of splitting iteration methods for the complex symmetric linear systems. Mathematica Applicanda 29: 173-182 (2016).
R.P. Wen, S.D. Li, and F.J. Ren. A new splitting and preconditioner for iteratively solving a class of complex symmetric linear systems. Mathematica Applicanda 27: 65-72 (2014).
H.A. van-der-Vorst and J.B.M. Melissen. A Petrov–Galerkin type method for solving Ax = b, where A is symmetric complex. IEEE Transactions on Magnetics 26: 706-708 (1990).
R.W. Freund. Conjugate gradient-type methods for linear systems with complex symmetric coefficient matrices. SIAM Journal on Scientific and Statistical Computing 13: 425-448 (1992).
A. Bunse Gerstner and R. Stöver. On a conjugate gradient–type method for solving complex symmetric linear systems. Linear Algebra and Its Applications 287: 105-123 (1999).
M. Clemens, T. Weiland, and U. Van-Rienen. Comparison of Krylov type methods for complex linear systems applied to high voltage problems. IEEE Transactions on Magnetics 34: 3335-3338 (1998).
D. Hezari, D.K. Salkuyeh, and V. Edalatpour. A new iterative method for solving a class of complex symmetric system of linear equations. Numerical Algorithms 73: 927-955 (2016).
D.K. Salkuyeh. Two-step scale-splitting method for solving complex symmetric system of linear equations. Arxiv Preprint Arxiv: 1705.02468 (2017).
D.K. Salkuyeh and T.S. Siahkolaei. Two-parameter TSCSP method for solving complex symmetric system of linear equations. Calcolo 55: 8 (2018).
Z. Zheng, F.L. Huang, and Y.C. Peng. Double step scale splitting iteration method for a class of complex symmetric linear systems. Applied Mathematics Letters 73: 91-97 (2017).
B. Li, J. Cui, Z. Huang, and X. Xie. A dual-parameter double-step splitting iteration method for solving complex symmetric linear equations. Applications of Mathematics 69: 311-337 (2024).
B. Li, J. Cui, Z. Huang, and X. Xie. Two Quasi-combining Real and Imaginary Parts Iteration Methods for Solving Complex Symmetric System of Linear Equations. Communications on Applied Mathematics and Computation (2024). https://doi.org/10.1007/s42967-024-00448-0.
