We consider an operator equation with noisy data where the noise level is given. The case of noisy data is especially actual in ill-posed problems. We formulate a criterion for comparison of accuracy of two approximate solutions get by arbitrary (different) methods. This generalizes previous results about monotonicity of error of approximate solutions generated by the same method but using different parameters.
REFERENCES
1.
2.
P.P.B.
Eggermont
, V.N.
LaRiccia
, M.Z.
Nashed
, GEM Int. J. Geomath.
3
, pp. 11
–178
, (2012
).3.
H. W.
Engl
, M.
Hanke
, and A.
Neubauer
, Regularization of Inverse Problems
, Kluwer, Dordrecht
, 1996
.4.
A.
Ganina
, U.
Hämarik
and U.
Kangro
, Mathematical Modelling and Analysis
19
, pp. 299
–308
, (2014
).5.
U.
Hämarik
, On the discretization error in regularized projection methods with parameter choice by discrepancy principle, in Ill-posed problems in natural sciences
edited by A.N.
Tikhonov
(VSP
, Utrecht
1992
), pp. 24
–28
.6.
U.
Hämarik
, Monotonicity of error and choice of the stopping index in iterative regularization methods, in Differential and Integral Equations: Theory and Numerical Analysis
edited by A.
Pedas
(Estonian Mathematical Society
, Tartu, 1999
), pp. 15–30.7.
U.
Hämarik
, E.
Avi
and A.
Ganina
, Mathematical Modelling and Analysis
7
, pp. 241
–252
, (2002
).8.
U.
Hämarik
, U.
Kangro
, R.
Palm
, T.
Raus
and U.
Tautenhahn
, Inverse Probl. Sci. Eng.
22
, pp. 10
Ű30
, (2014
).9.
U.
Hämarik
and R.
Palm
, Mathematical Modelling and Analysis
12
, pp. 61
–70
, (2007
).10.
U.
Hämarik
, R.
Palm
, and T.
Raus
, Journal of Inverse and Ill-Posed Problems
15
, pp. 277
–294
, (2007
).11.
U.
Hämarik
, R.
Palm
, and T.
Raus
, Numer. Funct. Anal. Optim.
30
, pp. 924
–950
, (2009
).12.
U.
Hämarik
, R.
Palm
, and T.
Raus
, Mathematical Modelling and Analysis
15
, pp. 55
–68
, (2010
).13.
U.
Hämarik
, R.
Palm
, and T.
Raus
, Calcolo
48
, 47
–59
, (2011
).14.
U.
Hämarik
, R.
Palm
, and T.
Raus
, J. Comput. Appl. Math.
236
, pp. 2146
–2157
, (2012
).15.
16.
U.
Hämarik
and T.
Raus
, Journal of Inverse and Ill-Posed Problems
14
, pp. 251
–266
, (2006
).17.
U.
Hämarik
and T.
Raus
, Numer. Funct. Anal. Optim.
30
, pp. 951
–970
, (2009
).18.
U.
Hämarik
and U.
Tautenhahn
, BIT
41
, pp. 1029
–1038
, (2001
).19.
M.
Hanke
, Numer. Funct. Anal. Optim.
35
, pp. 1500
–1510
, (2014
).20.
M.
Hanke
, T.
Raus
, SIAM J. Sci. Comput.
4
, pp. 956
–972
, (1996
).21.
B.
Jin
, D.
Lorenz
, SIAM J. Numer. Anal.
48
, 48
, pp. 1208
–1229
, (2010).22.
Q.
Jin
, P.
Mathe
, SIAM/ASA J. Uncertain. Quantif.
, 1
, pp. 386
–407
, (2013
).23.
24.
P.
Mathe
and S. V.
Pereverzev
, Inverse Problems
19
, pp. 789
–803
, (2003
).25.
P.
Mathe
and U.
Tautenhahn
, Inverse Problems
27
, 035016
, (2011
).26.
P.
Mathe
and U.
Tautenhahn
, J. Inverse Ill-Posed Probl.
19
, 859
–879
, (2011
).27.
28.
S.
Pereverzev
, E.
Schock
, SIAM J. Numer. Anal.
43
, pp. 2060
–2076
, (2005
).29.
T.
Raus
and U.
Hämarik
, Mathematical Modelling and Analysis
14
, pp. 99
–108
, (2009
).30.
T.
Raus
and U.
Hämarik
, Mathematical Modelling and Analysis
14
, pp. 187
–198
, (2009
).31.
T.
Reginska
, SIAM J. Sci. Comput.
17
, pp. 740
–749
, (1996
).32.
U.
Tautenhahn
and U.
Hämarik
, Inverse Problems
15
, pp. 1487
–1505
, (1999
).33.
G. M.
Vainikko
and A. Yu.
Veretennikov
, Iteration Procedures in Ill-Posed Problems
, Nauka
, Moscow
, 1986
. In Russian
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