An iterative method to compute minimum norm solutions of ill-posed problems in Hilbert spaces

We study an algorithm to compute minimum norm solution of ill-posed problems in Hilbert spaces and investigate its regularizing properties with discrepancy principle stopping rule. This algorithm results from straightly applying the LSQR method to the main problem before discretizing. In fact, the p...

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Published in:	Afrika mathematica Vol. 30; no. 5-6; pp. 797 - 816
Main Authors:	Jozi, Meisam, Karimi, Saeed, Salkuyeh, Davod Khojasteh
Format:	Journal Article
Language:	English
Published:	Berlin/Heidelberg Springer Berlin Heidelberg 01.09.2019 Springer Nature B.V
Subjects:	Algorithms Applications of Mathematics Hilbert space History of Mathematical Sciences Ill posed problems Integral equations Iterative methods Mathematics Mathematics and Statistics Mathematics Education Regularization Regularization methods Regularization method 45A05 Minimum norm 45P05 47B38 45Q05 First kind equations 45N05 47B34 Ill-posed problem algorithm
ISSN:	1012-9405, 2190-7668
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Abstract	We study an algorithm to compute minimum norm solution of ill-posed problems in Hilbert spaces and investigate its regularizing properties with discrepancy principle stopping rule. This algorithm results from straightly applying the LSQR method to the main problem before discretizing. In fact, the proposed algorithm obtains a sequence of approximate solutions of the original problem. In order to test the new algorithm, it is implemented to solve system of linear integral equations of the first kind and some examples are given. Moreover, we compare the presented algorithm with the Tikhonov regularization method to compute the least norm solution when there are more than one solution.
AbstractList	We study an algorithm to compute minimum norm solution of ill-posed problems in Hilbert spaces and investigate its regularizing properties with discrepancy principle stopping rule. This algorithm results from straightly applying the LSQR method to the main problem before discretizing. In fact, the proposed algorithm obtains a sequence of approximate solutions of the original problem. In order to test the new algorithm, it is implemented to solve system of linear integral equations of the first kind and some examples are given. Moreover, we compare the presented algorithm with the Tikhonov regularization method to compute the least norm solution when there are more than one solution.
Author	Karimi, Saeed Jozi, Meisam Salkuyeh, Davod Khojasteh
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Math.19771713414547491210.1007/BF01932285 – reference: MorozovVAMethods for Solving Incorrectly Posed Problems1984BerlinSpringer10.1007/978-1-4612-5280-1 – reference: BazánFSVCunhaMCCBorgesLSExtension of GKB-FP algorithm to large-scale general-form Tikhonov regularizationNumer. Linear Algebra Appl.201421316339320557910.1002/nla.18741340.65071 – reference: GolubGHVan LoanCFMatrix Computation1996BaltimoreJohns Hopkins University Press0865.65009 – reference: KammererWNashedMZIterative methods for best approximate solutions of integral equations of the first and second kindsJ. Math. Anal. Appl.19724054757332067710.1016/0022-247X(72)90002-90246.45015 – reference: KirschAAn Introduction to the Mathematical Theory of Inverse Problems2011New YorkSpringer10.1007/978-1-4419-8474-61213.35004 – reference: KressRLinear Integral Equations2014BerlinSpringer10.1007/978-1-4614-9593-21328.45001 – reference: HämarikUKaltenbacherBKangroUResmeritaERegularization by discretization in Banach spacesInverse Probl20163212834706471382.65160 – reference: TikhonovANSolution of incorrectly formulated problems and the regularization methodSoviet Math. Dokl19635103510380141.11001 – reference: HansenPCRank-Deficient and Discrete Ill-Posed Problems: Numerical Aspects of Linear Inversion1998PhiladelphiaSIAM10.1137/1.9780898719697 – reference: van der SluisAvan der VorstHAThe rate of convergence of conjugate gradientsNumer. Math.19864854356083961610.1007/BF013894500596.65015 – reference: HansenPCDiscrete Inverse Problems: Insight and Algorithms2010PhiladelphiaSIAM10.1137/1.97808987188361197.65054 – reference: EnglHWNeubauerAAn improved version of Marti’s method for solving ill-posed linear integral equationsMath. Comput.1985454054168049320578.65135 – reference: TikhonovANRegularization of incorrectly posed problemsSoviet Math. Dokl19634162416270183.11601 – reference: VarahJMPitfalls in the numerical solution of linear ill-posed problemsSIAM J. Sci. Stat. Comput.1983416417669717110.1137/09040120533.65082 – reference: BauerFLukasMAComparing parameter choice methods for regularization of ill-posed problemsMath. Comput. Simul.20118117951841279972910.1016/j.matcom.2011.01.0161220.65063 – reference: HankeMAccelerated Landweber iterations for the solution of ill-posed equationsNumer. Math.199160341373113719810.1007/BF013857270745.65038 – reference: BazánFSVBorgesLSGKB-FP: an algorithm for large-scale discrete ill-posed problemsBIT Numer. Math.201050481507271982510.1007/s10543-010-0275-31207.65039 – reference: MartiJTAn algorithm for computing minimum norm solutions of Fredholm integral equations of the first kindSIAM J. Numer. Anal.1978151071107651268310.1137/07150710399.65093 – reference: NashedMZWahbaGConvergence rates of approximate least squares solution of linear integral and operator equations of the first kindMath. Comput.197428698046189510.1090/S0025-5718-1974-0461895-10273.45012 – reference: WazwazAMlinear and Nonlinear Integral Equations Methods and Applications2011ChicagoSpringer10.1007/978-3-642-21449-31227.45002 – reference: AtkinsonKEThe Numerical Solution of Integral Equations of the Second Kind1997New YorkCambridge University Press10.1017/CBO97805116263400899.65077 – reference: ReichelLRodriguezGOld and new parameter choice rules for discrete ill-posed problemsNumer. Algorithm2013636587304040310.1007/s11075-012-9612-81267.65045 – reference: BorgesLSBazánFSVCunhaMCCAutomatic stopping rule for iterative methods in discrete ill-posed problemsComput. Appl. Math.20153411751197339753210.1007/s40314-014-0174-31337.65034 – reference: HuangYJiaZSome results on the regularization of LSQR for large-scale discrete ill-posed problemsSci. 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Title	An iterative method to compute minimum norm solutions of ill-posed problems in Hilbert spaces
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