A double optimal descent algorithm for iteratively solving ill-posed linear inverse problems

References

• Stewart G. Introduction to matrix computations. New York (NY): Academic Press; 1973.
   Google Scholar
• Kunisch K, Zou J. Iterative choices of regularization parameters in linear inverse problems. Inverse Probl. 1998;14:1247–1264.
   Web of Science ®Google Scholar
• Wang Y, Xiao T. Fast realization algorithms for determining regularization parameters in linear inverse problems. Inverse Probl. 2001;17:281–291.
   Web of Science ®Google Scholar
• Xie J, Zou J. An improved model function method for choosing regularization parameters in linear inverse problems. Inverse Probl. 2002;18:631–643.
   Web of Science ®Google Scholar
• Resmerita E. Regularization of ill-posed problems in Banach spaces: convergence rates. Inverse Probl. 2005;21:1303–1314.
   Web of Science ®Google Scholar
• Chehab JP, Laminie J. Differential equations and solution of linear systems. Numer. Algorithms. 2005;40:103–124.
   Web of Science ®Google Scholar
• Helmke U, Moore JB. Optimization and dynamical systems. Berlin: Springer; 1994.
   Google Scholar
• Liu CS. Optimally generalized regularization methods for solving linear inverse problems. CMC: Comput. Mater. Con. 2012;29:103–127.
   Web of Science ®Google Scholar
• Liu CS. Scaled vector regularization method to solve ill-posed linear problems. Appl. Math. Comput. 2012;218:10602–10616.
   Web of Science ®Google Scholar
• Liu CS. A dynamical Tikhonov regularization for solving ill-posed linear algebraic systems. Acta Appl. Math. 2012;123:285–307.
   Web of Science ®Google Scholar
• Liu CS. An optimal preconditioner with an alternate relaxation parameter used to solve ill-posed linear problems. CMES: Comput. Model. Eng. Sci. 2013;92:241–269.
   Web of Science ®Google Scholar
• Liu CS. A globally optimal iterative algorithm to solve an ill-posed linear system. CMES: Comput. Model. Eng. Sci. 2012;84:383–403.
   Web of Science ®Google Scholar
• Liu CS. An optimal tri-vector iterative algorithm for solving ill-posed linear inverse problems. Inverse Probl. Sci. Eng. 2013;21:650–681.
   Web of Science ®Google Scholar
• Liu CS. A globally optimal tri-vector method to solve an ill-posed linear system. J. Comput. Appl. Math. 2014;260:18–35.
   Web of Science ®Google Scholar
• Barzilai J, Borwein JM. Two point step size gradient methods. IMA J. Numer. Anal. 1988;8:141–148.
   Web of Science ®Google Scholar
• Akaike H. On a successive transformation of probability distribution and its application to the analysis of the optimum gradient method. Ann. Inst. Stat. Math. Tokyo. 1959;11:1–16.
   Web of Science ®Google Scholar
• Saad Y, Schultz MH. GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems. SIAM J. Sci. Stat. Comput. 1986;7:856–869.
   Web of Science ®Google Scholar
• Liu CS, Atluri SN. An iterative method using an optimal descent vector, for solving an ill-conditioned system Bx = b, better and faster than the conjugate gradient method. CMES: Comput. Model. Eng. Sci. 2011;80:275–298.
   Web of Science ®Google Scholar
• Liu CS. A doubly optimized solution of linear equations system expressed in an affine Krylov subspace. J. Comput. Appl. Math. 2014;260:375–394.
   Web of Science ®Google Scholar
• Liu CS. Discussing a more fundamental concept than the minimal residual method for solving linear system in a Krylov subspace. J. Math. Res. 2013;5:58–70.
   Google Scholar
• Liu CS. An optimally generalized steepest-descent algorithm for solving ill-posed linear systems. J. Appl. Math. 2013; ID 154358, 15 p.
   PubMed Web of Science ®Google Scholar
• Jacoby SLS, Kowalik JS, Pizzo JT. Iterative methods for nonlinear optimization problems. New Jersey (NJ): Prentice-Hall; 1972.
   Google Scholar
• Ostrowski AM. Solution of equations in Euclidean and Banach spaces. 3rd ed. New York (NY): Academic Press; 1973.
   Google Scholar
• Liu CS. An optimal multi-vector iterative algorithm in a Krylov subspace for solving the ill-posed linear inverse problems. CMC: Comput. Mater. Con. 2013;33:175–198.
   Web of Science ®Google Scholar
• Dongarra J, Sullivan F. Guest editors’ introduction to the top 10 algorithms. Comput. Sci. Eng. 2000;2:22–23.
   Web of Science ®Google Scholar
• Simoncini V, Szyld DB. Recent computational developments in Krylov subspace methods for linear systems. Numer. Linear Algebra Appl. 2007;14:1–59.
   Web of Science ®Google Scholar
• Saad Y. Krylov subspace methods for solving large unsymmetric linear systems. Math. Comput. 1981;37:105–126.
   Web of Science ®Google Scholar
• Freund RW, Nachtigal NM. QMR: a quasi-minimal residual method for non-Hermitian linear systems. Numer. Math. 1991;60:315–339.
   Web of Science ®Google Scholar
• van Den Eshof J, Sleijpen GLG. Inexact Krylov subspace methods for linear systems. SIAM J. Matrix Anal. Appl. 2004;26:125–153.
   Web of Science ®Google Scholar
• Hestenes MR, Stiefel EL. Methods of conjugate gradients for solving linear systems. J. Res. Nat. Bur. Stand. 1952;49:409–436.
   Google Scholar
• Lanczos C. Solution of systems of linear equations by minimized iterations. J. Res. Nat. Bur. Stand. 1952;49:33–53.
   Google Scholar
• Paige CC, Saunders MA. Solution of sparse indefinite systems of linear equations. SIAM J. Numer. Anal. 1975;12:617–629.
   Web of Science ®Google Scholar
• Fletcher R. Conjugate gradient methods for indefinite systems. Lecture notes in Math. Vol. 506. Berlin: Springer-Verlag; 1976. p. 73–89.
   Google Scholar
• Sonneveld P. CGS: a fast Lanczos-type solver for nonsymmetric linear systems. SIAM J. Sci. Stat. Comput. 1989;10:36–52.
   Web of Science ®Google Scholar
• van der Vorst HA. Bi-CGSTAB: a fast and smoothly converging variant of Bi-CG for the solution of nonsymmetric linear systems. SIAM J. Sci. Stat. Comput. 1992;13:631–644.
   Web of Science ®Google Scholar
• Saad Y, van der Vorst HA. Iterative solution of linear systems in the 20th century. J. Comput. Appl. Math. 2000;123:1–33.
   Web of Science ®Google Scholar
• Saad Y. Iterative methods for sparse linear systems. 2nd ed. Pennsylvania (PA): SIAM; 2003.
   Google Scholar
• van der Vorst HA. Iterative Krylov methods for large linear systems. New York (NY): Cambridge University Press; 2003.
   Google Scholar
• Liu CS. The concept of best vector used to solve ill-posed linear inverse problems. CMES: Comput. Model. Eng. Sci. 2012;83:499–525.
   Web of Science ®Google Scholar
• Calvetti C, Lewis B, Reichel L. GMRES-type methods for inconsistent systems. Linear Algebra Appl. 2000;316:157–169.
   Web of Science ®Google Scholar
• Kuroiwa N, Nodera T. A note on the GMRES method for linear discrete ill-posed problems. Adv. Appl. Math. Mech. 2009;1:816–829.
   Web of Science ®Google Scholar
• Matinfar M, Zareamoghaddam H, Eslami M, Saeidy M. GMRES implementations and residual smoothing techniques for solving ill-posed linear systems. Comput. Math. Appl. 2012;63:1–13.
   Web of Science ®Google Scholar
• Morikuni K, Reichel L, Hayami K. FGMRES for linear discrete ill-posed problems. Appl. Numer. Math. 2014;75:175–187.
   Web of Science ®Google Scholar
• Yin JF, Hayami K. Preconditioned GMRES methods with incomplete Givens orthogonalization method for large sparse least-squares problems. J. Comput. Appl. Math. 2009;226:177–186.
   Web of Science ®Google Scholar
• Horn RA, Johnson CR. Matrix analysis. New York (NY): Cambridge University Press; 1985.
   Google Scholar
• Liu CS, Hong HK, Atluri SN. Novel algorithms based on the conjugate gradient method for inverting ill-conditioned matrices, and a new regularization method to solve ill-posed linear systems. CMES: Comput. Model. Eng. Sci. 2010;60:279–308.
   Web of Science ®Google Scholar
• Liu CS. A vector regularization method to solve linear inverse problems. Inverse Probl. Sci. Eng. 2013. dx.doi.org/10.1080/17415977.2013.823415.
   PubMedGoogle Scholar
• Hon YC, Li M. A discrepancy principle for the source points location in using the MFS for solving the BHCP. Int. J. Comput. Methods. 2009;6:181–197.
   Web of Science ®Google Scholar
• Liu CS. The method of fundamental solutions for solving the backward heat conduction problem with conditioning by a new post-conditioner. Numer. Heat Transfer B: Fundam. 2011;60:57–72.
   Web of Science ®Google Scholar
• Liu CS. A two-side equilibration method to reduce the condition number of an ill-posed linear system. CMES: Comput. Model. Eng. Sci. 2013;91:17–42.
   Web of Science ®Google Scholar