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Operator-theoretic and regularization approaches to ill-posed problems. (English) Zbl 1464.47006

With this survey paper, the authors, both experts in the field of inverse and ill-posed problems, present an excellent overview of the history and new developments in the context of operator-theoretic and regularization approaches to ill-posed problems. Although the paper is published as a journal article, it can be considered as a textbook on regularization theory and practice with focus on its functional analytic background. It is hoped that numerous mathematicians, young scientists, and practitioners who are dealing with inverse problems and with the phenomenon of ill-posedness take note of this outstanding work. The readers will also find the comprehensive bibliography with more than 250 references very useful. The paper is highly recommended to all who want to understand the philosophy of resolution with respect to inverse problems which are expressed by linear and nonlinear operator equations formulated in abstract Banach, Hilbert, or metric spaces. Moreover, the interconnections between the formulation of the problems in infinite-dimensional spaces with proper ill-posedness and its discrete analogues with ill-conditioning phenomena as immediate consequences of the underlying ill-posedness are outlined. For any ill-posed applied inverse problem in natural sciences, engineering, or finance, the practitioner has to decide whether first to regularize the operator equation and then discretize it or, alternatively, first to discretize the problem and then regularize the finite-dimensional system is the better and more appropriate approach. This, however, requires a deep understanding of the operator-theoretic basics, and the paper under review can help to achieve this knowledge.
The second author is the doyen of operator theory with respect to ill-posed problems. From the reviewer’s point of view, some chapters of the article can be seen in some sense as a mathematical legacy of Nashed’s achievements over almost 50 years. One can find more than 25 of his publications, partially written with coauthors, in the references. The article also reminds the reader of some valuable concepts in inverse problem theory developed by Nashed, for example, Nashed’s ill-posedness definition and classification, first published in 1987, and distinguishing ill-posedness of type I and type II. On the other hand, the first author is the doyen of Geomathematics in Germany. He has founded the International Journal of Geomathematics and managed the Handbook of Geomathematics. Many interesting and challenging applications of regularization theory come from the field of geosciences and remote sensing by satellites.
The comprehensive article consists of four chapters. In Chapter 1, the authors announce in a short introduction their philosophy with respect to ill-posed problems and regularization, and they list there journals, the history of textbooks, and important articles of this field. Chapter 2, entitled “Solvability of ill-posed operator equations”, gives a short overview of linear inverse problems formulated in infinite-dimensional spaces, their discretization and regularization. The generalized inverse, a main topic of Nashed’s research, plays a relevant role there. Classical regularization methods like Tikhonov regularization, truncated singular value regularization, and asymptotic regularization are mentioned. Moreover, the standard techniques for analyzing linear ill-posed problems with compact forward operators by singular values and Fourier expansions are introduced.
Chapter 3, entitled “Operator methodologies of resolution”, discusses the philosophy and methods for obtaining stable approximate solutions to ill-posed problems. The authors distinguish between “solution” of an operator equation and “resolution” for an inverse problem. With respect to the instability effect coming from ill-posedness and chances for overcoming it or at least its reduction by exploiting additional a priori information, the computational standpoint is relevant for the authors. In this chapter and in the subsequent Chapter 4, entitled “Reconstruction methods and regularizing filters”, they focus on three aspects of the problem area: (a) strategy of resolution and reconstruction, (b) regularization-approximation schema, (c) regularization algorithms. In Chapter 3, the concept of the authors concerning regularization with occurring error bounds are outlined. Moreover, the use of compactness and a priori bounds is discussed. Tikhonov’s regularization method, the role of source conditions and convergence rates are also briefly under consideration. The chapter is concluded by the characterization of regularizers. This, however, requires a deep analysis of linear operators in Banach and Hilbert spaces, the role of their null-spaces and ranges, and of associated generalized inverses.
Reconstruction methods based on regularizing filter factors form the content of the final Chapter 4, encompassing more than 60 pages. Assuming that the forward operator of the inverse problem is a bounded and compact linear operator mapping between infinite-dimensional Hilbert spaces with non-closed range, preferably linear regularization procedures are analyzed in eight sections of this chapter. In an introductory prologue of 16 pages, the basics of the theory of linear regularization methods are outlined, meeting optimality concepts, worst case errors, regularizing filters, the choice of regularization parameters, source conditions, and convergence rates. The Tikhonov-Phillips regularization is discussed in detail as an important example for exercising all these theory components. In Section 4.1, generalized inverses in reproducing kernel Hilbert spaces are presented and used in order to construct appropriate regularization techniques. For approaching the computational aspects, discretization is necessary and Section 4.2 follows this requirement by studying projection methods. On the other hand, Section 4.3 turns to multiscale methods as regularization schemes. Again with respect to computational aspects and as alternative to variational regularization, iterative regularization methods are in the focus of Section 4.4. It is unavoidable in regularization theory to take into account also stochastic aspects with respect to noisy data or a randomization of the solution in a Bayesian setting. Section 4.5 briefly meets such concepts. Mollifier methods are considered in Section 4.6, and the Backus-Gilbert method is discussed in Section 4.7. The commendable Section 4.8 on numerical dilemmas and methodologies completes Chapter 4.

MSC:

47A52 Linear operators and ill-posed problems, regularization
65J20 Numerical solutions of ill-posed problems in abstract spaces; regularization
65N21 Numerical methods for inverse problems for boundary value problems involving PDEs
86A22 Inverse problems in geophysics
35R30 Inverse problems for PDEs
47-02 Research exposition (monographs, survey articles) pertaining to operator theory
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[1] Acar, R; Vogel, CR, Analysis of bounded variation penalty methods for ill-posed problems, Inverse Prob., 10, 1217-1229, (1994) · Zbl 0809.35151
[2] Adams, R.A.: Sobolev Spaces. Academic Press, New York (1975) · Zbl 0314.46030
[3] Alber, YI, The solution of nonlinear equations with monotone operators in Banach spaces, Sib. Math. J., 16, 1-8, (1975)
[4] Alber, YI, Iterative regularization in Banach spaces, Soviet Math. (Iz. VUZ), 30, 1-8, (1986)
[5] Alber, YI, The regularization method for variational inequalities with nonsmooth unbounded operators in Banach spaces, Appl. Math. Lett., 6, 63-68, (1993) · Zbl 0797.49022
[6] Alber, YI, Generalized projection operators in Banach spaces: properties and applications, Funct. Differ. Equ. Proc. Isr. Semin., 1, 1-21, (1994) · Zbl 0882.47046
[7] Alber, YI; Kartsatos, A (ed.), Metric and generalized projection operators in Banach spaces: properties and applications, 15-50, (1996), New York · Zbl 0883.47083
[8] Alber, YI; Notik, A, Perturbed unstable variational inequalities with unbounded operator on approximately given sets, Set Valued Anal., 1, 393-402, (1993) · Zbl 0815.49010
[9] Alber, YI; Reich, S, An iterative method for solving a class of nonlinear operator equations in Banach spaces, Panamerian Math. J., 4, 39-54, (1994) · Zbl 0851.47043
[10] Alber, YI; Rjazanceva, I, Variational inequalities with discontinuous monotone mappings, Sov. Math. Dokl., 25, 206-210, (1982) · Zbl 0532.46026
[11] Albert, A.: Regressions and the Moore-Penrose Pseudoinverse. Academic Press, New York (1972) · Zbl 0253.62030
[12] Angell, T.S., Nashed, M.Z.: Operator-theroetic and computational aspects of ill-posed problems in antenna theory. In: Proceedings of Symposia in Pure Mathematics Theory of Networks and Syst. pp. 499-511, Delft University of Technology, The Netherlands (1979)
[13] Anger, G, A characterization of inverse gravimetric source problem through extremal measures, Rev. Geophys. Space Phys., 19, 299-306, (1981) · Zbl 0437.35068
[14] Anger, G.: Inverse Problems in Differential Equations. Akademie-Verlag, Berlin (1990) · Zbl 0707.35157
[15] Aronszajn, N, Theory of reproducing kernels, Trans. Am. Math. Soc., 68, 337-404, (1950) · Zbl 0037.20701
[16] Aubert, G., Kornprobst, P.: Mathematical Problems in Image Processing, Applied Mathematical Sciences, vol. 147, 2nd edn. Springer, New York (2006) · Zbl 1110.35001
[17] Baart, M.L.: Methods for Solution of Fredholm Integral Equations of the First Kind. CSIR, Pretoria, Technical Report, vol. 104 (1979) · Zbl 0437.65096
[18] Baer, R.: Linear Algebra and Projective Geometry. Academic Press, New York (1952) · Zbl 0049.38103
[19] Backus, GE; Gilbert, F, Numerical applications of a formalism for geophysical inverse problems, Geophys. J.R. Astron. Soc., 13, 247-276, (1967)
[20] Bakusinskii, AB, A general method for constructing regularizing algorithms for a linear incorrect equation in Hilbert space, U.S.S.R. Comput. Math. Meth. Phys., 7, 279-284, (1967)
[21] Bakusinskii, AB, On the principle of iterative regularization, U.S.S.R. Comput. Math. Meth. Phys., 19, 256-260, (1979)
[22] Barzaghi, R; Sansò, F, Remarks on the inverse gravimetric problem, Boll. Geod. Scienze Affini, 45, 203-216, (1986) · Zbl 0904.62138
[23] Baumeister, J.: Stable Solution of Inverse Problems. Vieweg, Braunschweig (1987) · Zbl 0623.35008
[24] Ben-Israel, A., Greville, T.N.E.: Generalized Inverses: Theory and Applications. Wiley-Interscience, New York (1974) · Zbl 0305.15001
[25] Bertero, M., de Mol, C., Viano, G.A.: Linear inverse problems with discrete data. Inverse Prob. 4: 573-594 (1985/88) · Zbl 0653.65042
[26] Bertero, M; Brianzi, P; Pike, ER; Rebolia, L, Linear regularizing algorithms for positive solutions of linear inverse problems, Proc. R. Soc. Lond. A, 415, 257-275, (1988) · Zbl 0636.65145
[27] Bertero, M; DeMol, C; Viano, GA; Baltes, HP (ed.), The stability of inverse problems, (1979), Berlin
[28] Binder, A; Engl, HW; Groetsch, CW; Neubauer, A; Scherzer, O, Weakly closed nonlinear operators and parameter identification in parabolic equations by tiknonov regularization, Appl. Anal., 55, 215-234, (1994) · Zbl 0835.65078
[29] Bissantz, H; Hohage, T; Munk, A; Ruymgaart, F, Convergence rates of general regularization methods for statistical inverse probelms and applications, SIAM J. Numer. Anal., 45, 2610-2626, (2007) · Zbl 1234.62062
[30] Bjerhammar, A, Rectangular reciprocal matrices, with special reference to geodetic calculations, Bull. Géod., 25, 188-220, (1951) · Zbl 0043.12203
[31] Bjerhammar, A.: Theory of Errors and Generalized Matrix Inverses. Elsevier Scientific Publ. Co., Amsterdam (1973) · Zbl 0267.65002
[32] Boullion, T.L., Odell, P.L.: Generalized Inverse Matrices. Wiley-Interscience, New York (1971) · Zbl 0223.15002
[33] Bruck, RE, A strongly convergent iterative solution of \(0∈ Ux\) for a maximal montone operator \(U\) in Hilbert space, J. Math. Anal. Appl., 48, 114-126, (1974) · Zbl 0288.47048
[34] Burger, M; Kaltenbacher, B, Regularizing Newton-Kaczmarz methods for nonlinear ill-posed problems, SIAM J. Numer. Anal., 44, 1775-1797, (2006) · Zbl 1112.65049
[35] Cavalier, L; Golubev, GK, Risk hull method and regularization by projections of illposed inverse problems, Ann. Stat., 34, 1653-1677, (2006) · Zbl 1246.62082
[36] Cavalier, L; Golubev, GK; Picard, D; Tsybakov, AB, Oracle inequalities for inverse problems, Ann. Stat., 30, 843-874, (2002) · Zbl 1029.62032
[37] Craven, BD; Nashed, MZ, Generalized implicit function theorems when the derivative has no bounded inverse: theory, methods, and applications, Nonlinear Anal., 6, 375-387, (1982) · Zbl 0486.58004
[38] Davis, P.J.: Interpolation and Approximation. Blaisdell, New York (1963) · Zbl 0111.06003
[39] Defrise, M., de Mol, C.: A note on stopping rules for iterative regulaization methods and filtered SVD. In: Inverse Problems: An Interdisziplinary Study. pp. 261-268, Academic Press, San Diego (1987) · Zbl 0691.65040
[40] Desbat, L; Girard, D, The “minimum reconstruction error” choice of regularization parameters: some more efficient methods and their application of deconvolution problems, SIAM J. Sci. Comptu., 16, 187-1403, (1995) · Zbl 0840.65138
[41] Diaz, JB; Metcalf, FT, On interation procedures for equations of the first kind, \(Ax=y\), and picard’s criterion for the existence of a solution, Math. Comput., 24, 923-935, (1970) · Zbl 0217.43401
[42] Dicken, V; Maass, P, Wavelet-Galerkin methods for ill-posed problems, J. Inverse Ill-posed Probl., 4, 203-222, (1996) · Zbl 0867.65026
[43] Dimitiev, VI; Il’inski, AS; Svenshnikov, AG, The developments of mathematical methods for the study of direct and inverse problems in electrodynamics, Russ. Math. Surv., 31, 133-152, (1976)
[44] Dobson, DC; Scherzer, O, Analysis of regularized total variation penalty methods for denoising, Inverse Prob., 12, 601-617, (1996) · Zbl 0866.65041
[45] Donoho, DL, Nonlinear solution of linear inverse problems by wavelet-vaguelette decomposition, Appl. Comput. Harm. Anal., 2, 101-126, (1995) · Zbl 0826.65117
[46] Donoho, DL; Johnstone, IM, Minimax estimation via wavelet shrinkage, Ann. Stat., 26, 879-921, (1998) · Zbl 0935.62041
[47] Dunford, N., Schwarz, J.: Linear Operators, vol. II. Wiley-Interscience, New York (1963)
[48] Duris, CS, Optimal quadrature formulas using generalized inverses. part I. general theory and minimum variance formulas, Math. Comput., 25, 495-504, (1971) · Zbl 0221.65050
[49] Eggermont, PPB, Maximum entropy regularization for Fredholm integral equations of the first kind, SIAM J. Math. Anal., 24, 1557-1576, (1993) · Zbl 0791.65099
[50] Eggermont, P.P.B., LaRicca, V.N.: Maximum Penalized Likelihood Estimation. Volume II. Regression. Springer, New York (2009)
[51] Eggermont, PPB; LaRicca, VN; Nashed, MZ, On weakly bounded noise in ill-posed problems, Inverse Prob., 25, 115018-115032, (2009) · Zbl 1191.65056
[52] Eggermont, PPB; LaRicca, VN; Nashed, MZ, Moment discretization of ill-posed problems with discrete weakly bounded noise, Int. J. Geomath., 3, 155-178, (2012) · Zbl 1272.47016
[53] Eggermont, PN; LaRiccia, V; Nashed, MZ; Freeden, W (ed.); Nashed, MZ (ed.); Sonar, T (ed.), Noise models for ill-posed problems, No. 2, 1633-1658, (2015), New York
[54] Eicke, B, Iteration methods for convexly constrained ill-posed problems in Hilbert space, Numer. Funct. Anal. Optim, 13, 413-429, (1992) · Zbl 0769.65026
[55] Eisenhart, C.: Carl Friedrich Gauss, vol. VI, pp. 74-81. International Encyclopedia of Social Sciences, New York (1986)
[56] Ekeland, I., Temam, R.: Convex Analysis and Variational Problems. North Holland, Amsterdam (1976) · Zbl 0322.90046
[57] Elden, L, Algorithms for the regularization of ill-conditioned least squares problems, BIT, 17, 134-145, (1977) · Zbl 0362.65105
[58] Elden, L, An algorithm for the regularization of ill-conditioned banded least squares problems, SIAM J. Sci. Stat. Comput., 5, 237-254, (1984) · Zbl 0546.65014
[59] Engl, H, Discrepancy principles for Tikhonov regularization of ill-posed problems, leading to optimal convergence rates, J. Optim. Theory Appl., 52, 209-215, (1987) · Zbl 0586.65045
[60] Engl, H.: Integralgleichungen. Springer Lehrbuch Mathematik, Wien (1997) · Zbl 0898.45001
[61] Engl, HW; Nashed, MZ, Stochastic projectional schemes for random linear operator equations of the first and second kinds, Numer. Funct. Anal. Optim., 1, 451-473, (1979) · Zbl 0446.60048
[62] Engl, HW; Nashed, MZ, New extremal characterizations of generalized inverses of linear operators, J. Math. Anal. Appl., 82, 566-586, (1981) · Zbl 0492.47012
[63] Engl, H.W., Hanke, M., Neubauer, A.: Regularization of Inverse Problems. Kluwer, Dordrecht (1996) · Zbl 0859.65054
[64] Engl, HW; Kunisch, K; Neubauer, A, Convergence rates for Tikhonov regularisation of non-linear ill-posed problems, Inverse Prob., 5, 523-540, (1989) · Zbl 0695.65037
[65] Engl, H., Louis, A.K., Rundell, W. (eds.): Inverse Problems in Geophysical Applications. SIAM, Philadelphia (1997) · Zbl 0857.00035
[66] Flemming, J; Hofmann, B, A new approach to source conditions in regularization with general residual term, Numer. Funct. Anal. Optim., 31, 254-284, (2010) · Zbl 1195.47040
[67] Frankenberger, H; Hanke, M, Kernel polynomials for the solution of indefinite and ill-posed problems, Numer. Algorithms, 25, 197-212, (2002) · Zbl 0996.65057
[68] Franklin, JN, Well-posed stochastic extensions to ill-posed linear problems, J. Math. Anal. Appl., 31, 682-716, (1970) · Zbl 0198.20601
[69] Franklin, JN, On thikhonov’s method for ill-posed problems, Math. Comput., 28, 889-907, (1974) · Zbl 0297.65053
[70] Fredholm, I, Sur une classe d’equations functionelles, Acta Math., 27, 365-390, (1903) · JFM 34.0422.02
[71] Freeden, W, On approximation by harmonic splines, Manuscr. Geod., 6, 193-244, (1981) · Zbl 0475.41010
[72] Freeden, W.: Multiscale Modelling of Spaceborne Geodata. B.G. Teubner, Stuttgart, Leipzig (1999) · Zbl 0957.65123
[73] Freeden, W., Gutting, M.: Special Functions of Mathematical (Geo)Physics. Birkhäuser, Basel (2013) · Zbl 1268.33001
[74] Freeden, W; Maier, T, Spectral and multiscale signal-to-noise thresholding of spherical vector fields, Comput. Geosci., 7, 215-250, (2003) · Zbl 1026.42036
[75] Freeden, W., Michel, V.: Multiscale Potential Theory (with Applications to Geoscience). Birkhäuser, Boston (2004) · Zbl 1067.31001
[76] Freeden, W; Nutz, H, Satellite gravity gradiometry as tensorial inverse problem, Int. J. Geomath., 2, 177-218, (2012) · Zbl 1252.86005
[77] Freeden, W; Schneider, F, Regularization wavelets and multiresolution, Inverse Prob., 14, 493-515, (1998) · Zbl 0902.65019
[78] Freeden, W; Schreiner, M; Freeden, W (ed.); Nashed, MZ (ed.); Sonar, T (ed.), Satellite gravity gradiometry (SGG): from scalar to tensorial solution, (2015), New-York
[79] Freeden, W; Witte, B, A combined (spline-)interpolation and smoothing method for the determination of the gravitational potential from heterogeneous data, Bull. Géod., 56, 53-62, (1982)
[80] Freeden, W; Michel, V; Nutz, H, Satellite-to-satellite tracking and satellite gravity gradiometry (advanced techniques for high-resolution geopotential field determination), J. Eng. Math., 43, 19-56, (2002) · Zbl 1043.70012
[81] Freeden, W., Nashed, Z., Sonar, T. (Eds.): Handbook of Geomathematics, vols. 1,2, and 3, 2nd edn. Springer, New-York (2015)
[82] Freeden, W., Schneider, F., Schreiner, M.: Gradiometry—an inverse problem in modern satellite geodesy, In: Engl, H.W., Louis, A., Rundell, W. (eds.), GAMM-SIAM Symposium on Inverse Problems: Geophysical Applications, pp. 179-239 (1997) · Zbl 0866.65091
[83] Friedrich, K.: Allgemeine für die Rechenpraxis geeignete Lösung für die Aufgaben der kleinsten Absolutsumme und der günstigsten Gewichtsverteilung. Z. Vermess. 337-358 (1937)
[84] Gajewski, H., Gröger, K., Zacharias, K.: Nichtlineare Operatorgleichungen und Operatordifferentialglei-chungen. Akademie-Verlag, Berlin (1974) · Zbl 0289.47029
[85] Galerkin, BG, Expansions in stability problems for elastic rods and plates (in Russian), Vestn. Inzkenorov, 19, 897-908, (1915)
[86] Gauss, C.F.: Theoria Motus Corporum Coelestium in Sectionibus Conicis Solem Ambientium, Hamburg (1809), Werke 7. Translated into English by C.H, Davis (1963) · Zbl 1234.01016
[87] Gauss, C.F.: Theoria Combinationis Observationum Erroribus Minimis Obnoxiae, vol. 1. Teil, Göttingen (1821)
[88] Gebbauer, B; Scherzer, O, Impedance-acoustic tomography, SIAM J. Appl. Math., 69, 565-576, (2008) · Zbl 1159.92027
[89] Gerhards, C.: Spherical Multiscale Methods in Terms of Locally Supported Wavelts: Theory and Application to Geomagnetic Modeling. Ph.D.- Thesis, Geomathematics Group, University of Kaiserslautern (2011) · JFM 43.0425.04
[90] Gfrerer, H, An a posteriori parameter choice for ordinary and iterated Tikhonov regularization of ill-posed problems leading to optimal convergence rates, Math. Comput., 49, 523-542, (1987) · Zbl 0631.65056
[91] Gfrerer, H, Supplement to: an a posteriori parameter choice for ordinary and iterated Tikhonov regularization of ill-posed problems leading to optimal convergence rates, Math. Comput., 49, s5-s12, (1987) · Zbl 0631.65057
[92] Glockner, O.: On Numerical Aspects of Gravitational Field Modelling from SST and SGG by Harmonic Splines and Wavelets (with Application to CHAMP Data), Ph.D. Thesis, Geomathematics Group, University of Kaiserslautern (2001)
[93] Gholami, A; Siahkoohi, HR, Regularization of linear and nonlinear geophysical ill-posed problems with joint sparsity constraints, Geophys. J. Int., 180, 871-882, (2010)
[94] Goldstine, H.H.: A History of Numerical Analysis from the 16th Through the 19th Century. Springer, New York (1977) · Zbl 0402.01005
[95] Golub, G.H., Van Loan, C.F.: Marix Computations, 3rd edn. The John Hopkins University Press, Baltimore, MD (1996)
[96] Grafarend, E.W.: Six lectures on geodesy and global geodynamics. In: Moritz, H., Sünkel, H. (eds.) Proceedings of the Third International Summer School in the Mountains, pp. 531-685 (1982) · Zbl 0941.65056
[97] Grafarend, E.W., Awange, J.L.: Applications of Linear and Nonlinear Models. Springer, Berlin (2012) · Zbl 1054.86008
[98] Graves, J; Prenter, PM, On generalized iterative filters for ill-posed problems, Numer. Math., 30, 281-299, (1978) · Zbl 0425.65075
[99] Grenander, U.: Abstract Inference. Wiley, New York (1981) · Zbl 0505.62069
[100] Groetsch, C.W.: Generalized Inverses of Linear Operators. Marcel Dekker. Inc., New York (1977) · Zbl 0358.47001
[101] Groetsch, C.W.: The Theory of Tikhonov Regularization for Fredholm Equations of the First Kind. Pitman, London (1984) · Zbl 0545.65034
[102] Groetsch, C.W.: Inverse Problems in the Mathematical Science. Vieweg, Braunschweig (1993) · Zbl 0779.45001
[103] Groetsch, CW; Hämmerlin, G (ed.); Hoffmann, KH (ed.), Comments on morozov’s discrepancy principle, 97-104, (1983), Basel
[104] Groetsch, C.W., Scherzer, O.: Iterative stabilization and edge detection. In: Nashed, M.Z., Scherzer, O. (eds.) Contemporary Mathematics, vol. 313, pp. 129-141. American Mathematical Society, Providence, RI (2002) · Zbl 1028.65056
[105] Hadamard, J, Sur LES problémes aux dérivés partielles et leur signification physique, Princeton Univ. Bull., 13, 49-52, (1902)
[106] Hadamard, J.: Lectures on the Cauchy Problem in Linear Partial Differential Equations. Yale University Press, New Haven (1923) · JFM 49.0725.04
[107] Haddad, R.A., Parsons, T.W.: Digital Signal Processing: Theory, Applications and Hardware. Computer Science Press, Rockville (1991)
[108] Hanke, M.: Conjugate Gradient Type Methods for Ill-Posed Problems. Pitman Research Notes in Mathematics. Longman House, Harlow (1995) · Zbl 0830.65043
[109] Hanke, M; Hansen, PC, Regularization methods for large scale problems, Surv. Math. Ind., 3, 253-315, (1993) · Zbl 0805.65058
[110] Hanke, M; Scherzer, O, Inverse probelms light: numerical differentiation, Am. Math. Mon., 108, 512-521, (2001) · Zbl 1002.65029
[111] Hanke, M; Vogel, CR, Two-level preconditioners for regularized inverse problems, Numer. Math., 83, 385-402, (1999) · Zbl 0941.65056
[112] Hansen, PC, Analysis of discrete ill-posed problems by means of the L-curve, SIAM Rev., 34, 561-580, (1992) · Zbl 0770.65026
[113] Hanson, RJ, A numerical methods for solving Fredholm inegral equations of the first kind, SIAM J. Numer. Anal., 8, 616-662, (1971) · Zbl 0199.50803
[114] Hegland, M, Variable Hilbert scales and their interpolation inequalities with applications to Tikhonov regularization, Appl. Anal., 59, 207-223, (1995) · Zbl 0841.65039
[115] Helmert, F.: Die Ausgleichungsrechnung nach der Methode der kleinsten Quadrate. Teubner, Berlin (1907) · JFM 38.0276.01
[116] Heuser, H.: Funktionalanalysis. 4. Auflage, Teubner (1975) · Zbl 1252.86005
[117] Hilbert, D.: Grundzüge einer allgemeinen Theorie der linearen Integralgleichungen. Teubner, Leipzig (1912) · JFM 43.0423.01
[118] Hille, E, Introduction to the general theory of reproducing kernels, Rocky Mt. J. Math., 2, 321-368, (1972) · Zbl 0266.30009
[119] Hofmann, B.: Inverse Probleme. Teubner, Leipzig (1999) · Zbl 0942.45004
[120] Hofmann, B; Mathé, P; Weiszäcker, H, Regularisation in Hilbert space under unbounded operators and general source conditions, Inverse Prob., 25, 115-130, (2009)
[121] Hohage, T; Pricop, M, Nonlinear Tikhonov regularization in Hilbert scales for inverse boundary value problems with random noise, Inverse Probl. Imaging, 2, 271-290, (2008) · Zbl 1160.65023
[122] Hurwitz, WA, On the pseudo-resolvent to the kernel of an integral equation, Trans. Am. Math. Soc., 13, 405-418, (1912) · JFM 43.0425.04
[123] Ismail, M., Nashed, M.Z., Zayed, A., Ghaleb, A.: Mathematical Analysis, Wavelets and Signal Processing. Contemporary Mathematics, vol. 190. American Mathematical Society, Providence, RI (1995) · Zbl 0829.00023
[124] Ivanov, VK; Kudrinskii, VYu, Approximate solution of linear operator equations in Hilbert space by the method of least squares. I, Z. Vycisl. Mat. i Mat. Fiz, 6, 831-944, (1966)
[125] Jacobsen, M; Hansen, PC; Saunders, MA, Subspace preconditioned LSQR for discrete ill-posed problems, BIT Numer. Math., 43, 975-989, (2003) · Zbl 1046.65030
[126] Jorgensen, P; Tian, F, Graph Laplacians and discrete reproducing kernel Hilbert spaces from restrictions, Stochastic Analysis and Applications, 34, 722-747, (2016) · Zbl 1380.46020
[127] Kaipio, J., Somersalo, E.: Statistical and Computational Inverse Problems. Springer, Berlin (2005) · Zbl 1068.65022
[128] Kammerer, WJ; Nashed, MZ, The convergence of the conjugate gradient method for singular linear operator equations, SIAM J. Numer. Anal., 9, 165-181, (1972) · Zbl 0243.65026
[129] Kammerer, WJ; Nashed, MZ, Iterative methods for best approximate solutions of linear integral equation of the first and second kind, J. Math. Anal. Appl., 40, 547-573, (1972) · Zbl 0246.45015
[130] Kantorowitsch, L.W., Akilow, G.P.: Funktionalanalysis in Normierten Räumen. Akademie-Verlag, Berlin (1964) · Zbl 0359.46017
[131] Kato, T, Perturbation theory for nullity definciency and other quantities of linear operators, J. Anal. Math., 6, 271-322, (1958)
[132] Kirsch, A.: An Introduction to the Mathematical Theory of Inverse Problems, 2nd edn. Springer, Heidelberg (1996) · Zbl 0865.35004
[133] Kowar, R; Scherzer, O; Romanov, S (ed.); Kabanikhin, SI (ed.); Anikonov, YE (ed.); Bukhgeim, AL (ed.), Convergence analysis of a Landweber-Kaczmarz method for sovling nonlinear ill-posed problems, (2002), Zeist
[134] Kress, R.: Linear Integral Equations, 2nd edn. Springer, Berlin (1989) · Zbl 0671.45001
[135] Kreyszig, E.: Introductory Functional Analysis with Applications. Wiley, New York (1978) · Zbl 0368.46014
[136] Larson, D., Massopust, P., Nashed, M.Z., Nguyen, M.C., Papadakis, M., Zayed, A. (Eds.): Frames and Operator Theory in Analysis and Signal Processing. Contemporary Mathematics, vol. 451, American Mathematical Society, Providence, RI (2008) · Zbl 1132.42001
[137] Lavrentiev, M.M.: Some Improperly Posed Problems of Mathematicsl Physics, Izdat. Sibirsk. Otdel, Akad. Nauk. SSSR, Novosibirsk (1962), Englisch Transl., Springer Tracts in Natural Philosophy, Vol. 11, Springer-Verlag, Berlin (1967) · Zbl 0791.65099
[138] Lieusternik, L.A., Sobolev, V.J.: Elements of Functional Analysis. Ungar, New York (1961)
[139] Lin, Y; Brown, LD, Statistical properties of the method of regularization with periodic Gaussian reproducing kernel, Ann. Stat., 32, 1723-1743, (2004) · Zbl 1045.62026
[140] Liskovets, OA, Regularization of variational inqualities with pseudo-monotone operators on approximately given sets, Differ. Equ., 11, 1970-1977, (1989)
[141] Liu, F; Nashed, MZ, Tikhonov regularization of nonlinear ill-posed problems with closed operators in Hilbert scales, J. Inverse Ill-Posed Prob., 5, 363-376, (1997) · Zbl 0890.65056
[142] Locker, J; Prenter, PM, Regularization with differential operators, J. Math. Anal. Appl., 74, 504-529, (1980) · Zbl 0447.65023
[143] Louis, A.K.: Inverse und schlecht gestellte Probleme. Teubner, Stuttgart (1989) · Zbl 0667.65045
[144] Louis, AK; Maass, P, A mollifier method for linear equations of the first kind, Inverse Prob., 6, 427-440, (1989) · Zbl 0713.65040
[145] Louis, A.K., Maass, P., Rieder, A.: Wavelets: Theorie und Anwendungen. B. G. Teubner Studienbücher, Stuttgart (1998) · Zbl 0897.42018
[146] Mair, BA; Ruymgaart, FH, Statistical inverse estimation in Hilbert scales, SIAM J. Appl. Math., 56, 1424-1444, (1996) · Zbl 0864.62020
[147] Mathé, P, The lepskii principle revisited, Inverse Prob., 22, 111-115, (2006) · Zbl 1095.65045
[148] Mathé, P., Hofmann, B.: How general are general source conditions? Inverse Probl. 24. https://doi.org/10.1088/0266-5611/24/1/015009 (2008)
[149] Mathé, P; Pereverzev, SV, The discretized discrepancy principle under general source conditions, J. Complex., 22, 371-381, (2006) · Zbl 1095.65046
[150] Mathé, P; Pereverzev, SV, Regularization of some linear ill-posed problems with discretized random noisy data, Math. Comput., 75, 1913-1929, (2006) · Zbl 1103.62031
[151] Mathé, P; Pereverzev, SV, Geometry of linear ill-posed problems in variable Hilbert scales, Inverse Prob., 19, 789-803, (2003) · Zbl 1026.65040
[152] Marti, JT, An algorithm for computing minimum norm solutions of Fredholm integral equaions of the first kind, SIAM J. Numer. Anal., 15, 1071-1076, (1978) · Zbl 0399.65093
[153] Marti, JT, On the convergence of an algorithm computing minimum-norm solutions of ill-posed problems, Math. Comput., 34, 521-527, (1980) · Zbl 0439.65036
[154] Meissl, P.A.: A Study of Covariance Functions Related to the Earth’s Disturbing Potential. Department of Geodetic Science, vol. 151. The Ohio State University, Columbus, OH (1971)
[155] Meissl, PA, Hilbert spaces and their applications to geodetic least squares problems, Boll. Geod. Sci. Aff., 1, 181-210, (1976) · Zbl 0342.46010
[156] Michel, V.: A Multiscale Method for the Gravimetry Problem: Theoretical and Numerical Aspects of Harmonic and Anharmonic Modelling. Ph.D.-thesis, Geomathematics Group, University of Kaiserslautern, Shaker, Aachen (1999)
[157] Michel, V, Scale continuous, scale discretized and scale discrete harmonic wavelets for the outer and the inner space of a sphere and their application to an inverse problem in geomathematics, Appl. Comput. Harm. Anal. (ACHA), 12, 77-99, (2002) · Zbl 0995.42027
[158] Michel, V.: A Multiscale Approximation for Operator Equations in Separable Hilbert Spaces—Case Study: Reconstruction and Description of the Earth’s Interior. Habilitation Thesis, University of Kaiserslautern, Geomathematics Group, Shaker, Aachen (2002b)
[159] Miller, K, Least squares methods for ill-posed problems with a prescribed bounded, SIAM J. Math. Anal., 1, 52-74, (1970) · Zbl 0214.14804
[160] Moore, EH, On the reciprocal of the general algebraic matrix, Bull. Am. Math. Soc., 26, 394-395, (1920)
[161] Moore, EH, General analysis, Mem. Am. Math. Soc., 1, 197-209, (1935)
[162] Moritz, H.: Advanced Physical Geodesy. Herbert Wichmann Verlag, Karlsruhe (1980)
[163] Morozov, VA, On the solution of functional equations by the method of regularization, Sov. Math. Doklady, 7, 414-41, (1966) · Zbl 0187.12203
[164] Morozov, V.A.: Methods for Solving Incorrectly Posed Problems. (in Russian), Moscow University, (1975) (English transl. editor M.Z. Nashed), Springer, New York (1984) · Zbl 0851.47043
[165] Morozov, V.A.: Methods for Solving Incorrectly Posed Problems. Springer, New York (1984)
[166] Nagel, H-H, On the estimation of optical flow: relations between new approaches and some new results, Artif. Intell., 33, 299-324, (1987)
[167] Nashed, MZ, Steepest descent for singular linear opertor equations, SIAM J. Numer. Anal., 7, 358-362, (1970) · Zbl 0221.65097
[168] Nashed, MZ; Rall, LB (ed.), Generalized inverses, normal solvability and iteration for singular operator equations, 311-359, (1971), New York
[169] Nashed, MZ; Rall, LB (ed.), Differentiability and related properties of nonlinear operators: some aspects of the role of differentials in nonlinear functional analyis, 103-309, (1971), New York
[170] Nashed, M.Z.: Some Aspects of Regularization and Approximations of Solutions of Ill-Posed Operator Equations. In: Proceedings of the 1972 Army Numerical Analysis Conf., Edgewood, MD, pp. 163-181 (1972)
[171] Nashed, MZ; Colton, DL (ed.); Gilbert, RP (ed.), Approximate regularized solutions to improperly posed linear integral and operator equations, No. 430, 289-322, (1974), New York
[172] Nashed, M.Z. (ed.): Generalized Inverses and Applications. Academic Press, New York, San Francisco, London (1976a) · Zbl 0346.15001
[173] Nashed, M.Z.: Aspects of generalized inverses in analysis and regularization. In: Nashed, M.Z. (ed.) Generalized Inverses and Applications. pp. 193-244, Academic Press, New York (1976b)
[174] Nashed, M.Z.: Perturbations and approximation for generalized inverses and linear operators. In: Nashed, M.Z. (ed.) Generalized Inverses and Applications. pp. 325-396, Academic Press, New York, San Francisco, London (1976c)
[175] Nashed, MZ, On moment-discretization and least squares solutions of linear integration equations of the first kind, J. Math. Anal. Appl., 53, 359-366, (1976) · Zbl 0327.45022
[176] Nashed, M.Z.: Regularization and approximation of ill-posed problems in system theory. In: Meyer, G.G.L., Westgate, C.R. (eds.) Proceedings of the 1979 Conference on Information Sciences and Systems. pp. 568-575, The Johns Hopkins University, New York (1979)
[177] Nashed, M.Z.: New applications of generalized inverses in system and control theory. In: Thomas, J.B. (ed.) Proceedings of the 1980 Conferences on Information Sciences and Systems. pp. 353-358. Princeton. NJ, Princeton (1980)
[178] Nashed, MZ; Herman, GT (ed.); Natterer, F (ed.), Continuous and semicontinuous analogous of iterative method of cimmino and Kaczmarz with applications to the inverse Radon transform, 160-178, (1981), New York
[179] Nashed, MZ, Operator-theoretic and computational approaches to ill-posed problems with applications to antenna theory, IEEE Trans. Antennas Propag., 29, 220-231, (1981) · Zbl 0477.65092
[180] Nashed, MZ; Engl, H (ed.); Groetsch, CW (ed.), A new approach to classification and regularization of ill-posed operator equations, (1987), Boston
[181] Nashed, MZ, Inner, outer, and generalized inverses in Banach and Hilbert spaces, Numer. Funct. Anal. Optim., 9, 261-326, (1987) · Zbl 0633.47001
[182] Nashed, M.Z.: Inverse problems, moment problems and signal processing: un menage a trois, mathematics in science and technology. In: Siddiqi, A.H., Singh, R.C. Manchanda, P. (eds.) Mathematical Models, Methods, and Applications. pp. 1-19, World Scientific, New Jersey (2010)
[183] Nashed, MZ; Engl, HW; Bharucha-Reid, AT (ed.), Random generalized inverses and approximate solution of random operator equations, 149-210, (1979), New York
[184] Nashed, MZ; Lin, F; Kartsatos, A (ed.), On nonlinear ill-posed problems ii: monotone operator equaions and monotone variational inequalities, 223-240, (1996), New York · Zbl 0880.47041
[185] Nashed, MZ; Scherzer, O, Stable approximation of nondifferentiable optimization problems with variational inequalities, Contemp. Math., 204, 155-170, (1997) · Zbl 0868.49008
[186] Nashed, MZ; Scherzer, O, Stable approximation of a minimal surface problem with variational inequalities, Abst. Appl. Anal., 2, 137-161, (1997) · Zbl 0937.49020
[187] Nashed, M.Z., Scherzer, O. (Eds.): Inverse Problems, Image Analysis and Medical Imaging. Contemporary Mathematics, vol. 313, American Mathematical Society, Providence, RI (2002) · Zbl 1003.00013
[188] Nashed, MZ; Votruba, FG; Nashed, MZ (ed.), A unified operator theory of generalized inverses, 1-109, (1976), New York
[189] Nashed, MZ; Wahba, G, Generalized inverses in reproducing kernel spaces: an approach to regularization of linear operator equations, SIAM J. Math. Anal., 5, 974-987, (1974) · Zbl 0287.47009
[190] Nashed, MZ; Wahba, G, Approximate regularized pseudosolution of liner operator equations when the data-vector is not in the range of the operator, Bull. Am. Math. Soc., 80, 1213-1218, (1974) · Zbl 0309.47012
[191] Nashed, MZ; Wahba, G, Convergence rates of approximate least squares solutions of linear integral and operator equations of the first kind, Math. Comput., 28, 69-80, (1974) · Zbl 0273.45012
[192] Natanson, I.P.: Constructive Function Theory. Frederick Ungar Publ. Co., New York (1965) · Zbl 0178.39701
[193] Natterer, F, The finite element method for ill-posed problems, RAIRO Anal. Numer., 11, 271-278, (1977) · Zbl 0369.65012
[194] Natterer, F, Regularisierung schlecht gestellter probleme durch projektionsverfahren, Numer. Math., 28, 329-341, (1977) · Zbl 0364.65042
[195] Natterer, F, Error bounds for Tikhonov regularization in Hilbert scales, Appl. Anal., 18, 29-37, (1984) · Zbl 0504.65031
[196] Neubauer, A, On converse and saturation results for Tikhonov regularization of linear ill-posed problems, SIAM J. Numer. Anal., 34, 517-527, (1997) · Zbl 0878.65038
[197] Novikoff, P, Sur le problème inverse du potentiel, C. R. Acad. Sci. l’URSS, 18, 165-168, (1938) · Zbl 0018.30901
[198] Ortega, JM; Rheinboldt, WC, On discretization and differentiation of operators with applications to newton’s method, SIAM J. Numer. Anal., 3, 143-156, (1966) · Zbl 0143.17001
[199] Parker, RL, The theory of ideal bodies for gravity interpretation, Geophys. J. R. Astr. Soc., 42, 315-334, (1975)
[200] Payne, L.E.: Improperly Posed Problems in Partial Differential Equations. SIAM Publications, Philadelphia (1975) · Zbl 0302.35003
[201] Penrose, R, A generalized inverse for matrices, Proc. Camb. Philos. Soc., 51, 406-413, (1955) · Zbl 0065.24603
[202] Penrose, R, On best approximate solutions of linear matrix equations, Proc. Camb. Philos. Soc., 25, 17-19, (1956) · Zbl 0070.12501
[203] Pereverzev, SV; Schock, E, On the adaptive selection of the parameter in regularization of ill-posed problems, SIAM J. Numer. Anal., 43, 2060-2076, (2005) · Zbl 1103.65058
[204] Perry, WL, On the bojarski-Lewis inverse scattering method, IEEE Trans. Antennas Propag., 6, 826-829, (1974)
[205] Perry, WL, Approximate solution of inverse problems with piecewise continuous solution, Radio Sci., 12, 634-642, (1977)
[206] Petryshyn, WV, On generalilzed inverses and uniform convergence of \((I-β K)^n\) with applications to iterative methods, J. Math. Anal. Appl., 18, 417-439, (1967) · Zbl 0189.47502
[207] Petrov, GI, Appliation of galerkin’s method to a problem of the stability of the flow of a viscous fluid (in Russian), Priklad. Mate. Mekh., 4, 3-12, (1940)
[208] Phillips, BL, A technique for the numerical solution of certain integral equations of the first kind, J. Assoc. Comput. Math., 9, 84-97, (1962) · Zbl 0108.29902
[209] Plackett, RL, An historical note on the method of least squares, Biometrika, 36, 458-460, (1949) · Zbl 0041.46802
[210] Plato, R, Optimal algorithms for linear ill-posed problems yielding regularization methods, Numer. Funct. Anal. Optim., 11, 111-118, (1990) · Zbl 0691.65040
[211] Rado, R, Note on generalized inverses of matrices, Proc. Camb. Philos. Soc., 52, 600-601, (1956) · Zbl 0071.24702
[212] Rao, C.R., Mitra, S.K.: Generalized Inverse of Matrices and Its Applications. Wiley, New York (1971) · Zbl 0236.15004
[213] Rayleigh, L.: The Theory of Sound. Oxdord University Press, Oxdord (1896) · JFM 27.0701.05
[214] Reid, W.T.: Generalized inverses of differential and integral operators. Theory and applications of generalized inverses of matrices (T.L. Boullion and P.L. Odell Eds.). In: Symposium Proceedings, Texas Tech University Mathematics Series, Vol. 4., Lubbock; Texas (1968)
[215] Ribiere, G, Regularisation d’operateurs, Rev. Inf. Rech. Oper., 1, 57-79, (1967) · Zbl 0184.37003
[216] Richter, GR, Numerical solution of integral equations of the first kind with non-smooth kernels, SIAM J. Numer. Anal., 15, 511-522, (1978) · Zbl 0388.65056
[217] Robinson, DW, Gauss and generalized inverses, Hist. Math., 7, 118-125, (1980) · Zbl 0438.15005
[218] Rieder, A.: Keine Probleme mit Inversen Problemen. Vieweg, Braunschweig (2003) · Zbl 1057.65035
[219] Ritz, W, Über lineare funktionalgleichungchungen, Acta Math., 41, 71-98, (1918)
[220] Rudin, L.I.: Functional Analysis. Mc Graw-Hill, New York (1973) · Zbl 0253.46001
[221] Rudin, LI; Osher, S; Fatemi, E, Nonlinear total variation based noise removal algorithms, Phys. D, 60, 259, (1992) · Zbl 0780.49028
[222] Rummel, R; Balmino, G; Johannessen, J; Visser, P; Woodworth, P, Dedicated gravity field missions—principles and aims, J. Geodyn., 33, 3-20, (2002)
[223] Saitoh, S.: Theory of Reproducing Kernels and its Applications. Longman, New York (1988) · Zbl 0652.30003
[224] Scherzer, O. (ed.): Handbook of Mathematical Methods in Imaging. Springer, New York (2015) · Zbl 1322.68001
[225] Scherzer, O., Grasmair, M., Grossauer, H., Haltmeier, M., Lenzen, F.: Variational Methods in Imaging. Applied Mathematical Sciences. Springer, New York (2009) · Zbl 1177.68245
[226] Schuster, T.: The Method of Approximate Inverse: Theory and Applications, Lecture Notes in Mathematics. Springer, Berlin (2007)
[227] Schuster, T., Kaltenbacher, B., Hofmann, B., Kazimierski, K.S.: Regularization Methods in Banach Spaces, Radon Series on Computational and Applied Mathematics, 10. De Gruyter, Berlin (2012) · Zbl 1259.65087
[228] Seidmann, TI, Non-convergence results for the application of least squares estimation onto ill-posed problems, J. Optim. Theory Appl., 30, 535-547, (1980)
[229] Shinozaki, SM; Ranabe, K, Numerical algorithms for the Moore-Penrose inverse of a matrix: direct methods, Ann. Inst. Stat. Math., 24, 193-203, (1972)
[230] Showalter, DW; Ben-Israel, B, Representation and computation of the generalized inverse of a bounded linear operator between two Hilbert spaces, Atti Accad. Naz. Kincei Rend. Cl. Sci, Fis. Mat. Natur. (8), 48, 184-194, (1970) · Zbl 0202.41703
[231] Shure, L; Parker, RL; Backus, GE, Harmonic splines for geomagnetic modelling, Phys. Earth Planet. Inter., 28, 215-229, (1982)
[232] Siegel, CL, Über die analytische theorie der quadratischen formen. III, Ann. Math., 38, 212-291, (1937) · JFM 63.0120.01
[233] Söberg, L.: Station adjustment of derictions using generalized inverses. In: Borre, K., Welsch, W. (Eds.), International Federation of Surveyors—FIG—Proceedings Survey Control Networks Meeting of Study Group 5B, 7th - 9th July, 1982, Aalborg University Centre, Denmark. Schriftenreihe des Wissenschaftlichen Studiengangs Vermessungswesen der Hochschule der Bundeswehr München, Heft 7, pp. 381-399 (1982)
[234] Song, M.: Regularization-Projection Methods and Finite Element Approximations for Ill-Posed Linear Operator Equations. Ph.D. Thesis, University Michigan (1978)
[235] Strand, ON, Theory and methods related to the singular function expansion and landweber’s iteration for integral equations of the first kind, SIAM J. Numer. Anal., 11, 798-825, (1974) · Zbl 0305.65079
[236] Sudakov, V.N., Khalfin, L.A.: A Statistical Approach to the Correctness of the Problems of Mathematical Physics. Dokl Akad Nauk SSSR 157-1058-1060 (1964) · Zbl 0132.33302
[237] Szegö, G.: Orthogonal Polynomials. American Mathematical Society Colloquium Publications, vol. 23. American Mathematical Society, Providence (1959) · JFM 61.0386.03
[238] Tadmor, E; Nezzar, S; Vese, L, A multiscale image representation using hierarchical (BV, \(L^2\)) decompositions, Multiscale Model. Simul., 2, 554-579, (2004) · Zbl 1146.68472
[239] Taylor, A.E., Lay, D.: Functional Analysis. Wiley, New York (1979)
[240] Tikhonov, AN, On the stability of inverse problems, Dokl. Akad. Nauk SSSR, 39, 195-198, (1943)
[241] Tikhonov, AN, On the solution of incorrectly formulated problems and the regularization method, Dokl. Akad. Nauk SSSR, 151, 501-504, (1963) · Zbl 0141.11001
[242] Tikhonov, AN, On methods of solving incorrect problems, Am. Math. Soc. Transl., 2, 222-224, (1968)
[243] Tikhonov, A.N., Arsenin, V.Y.: Solutions of Ill-Posed Problems. Wiley, Washington, DC (1977) · Zbl 0354.65028
[244] Tikhonov, A.N., Goncharsky, A.V., Stepanov, V.V., Yagola, A.G.: Numerical Methods for the Solution of Ill-Posed Problems. Kluwer, Dordrecht (1995) · Zbl 0831.65059
[245] Tikhonov, A.N., Goncharsky, A., Stepanov, V., Yagola, A.G.: Nonlinear Ill-Posed Problems, Vol. 1, 2, Applied Mathematics and Mathematical Computation. Chapman & Hall, London (1998). (Translated from the Russian)
[246] Twomey, S, On the numerical solution of Fredholm integral equations of the first kind by the inversion of the linear system produced by quadrature, J. Assoc. Comput. Mach., 10, 97-101, (1963) · Zbl 0125.36102
[247] Vainberg, M.M.: Variational Method and Method of Monotone Operators. Wiley, New York (1973) · Zbl 0279.47022
[248] Varah, J, On the numerical solution of ill-conditioned linear systems with applications to ill-posed problems, SIAM J. Numer. Anal., 10, 257-267, (1973) · Zbl 0261.65034
[249] Vese, L.A., Le Guyader, C.: Variational Methods in Image Processing. Chapman & Hall/CRC Mathematical and Computational Imaging Sciences. CRC Press, Boca Raton (2016) · Zbl 1332.94003
[250] Vogel, C.R.: Computational Methods for Inverse Problems. SIAM, Philadelphia (2002) · Zbl 1008.65103
[251] Vogel, CR; Oman, ME, Iterative methods for total variation denoising, SIAM J. Sci. Comput., 17, 227-238, (1996) · Zbl 0847.65083
[252] Wahba, G, Convergence rates of certain approximate solutions ot Fredholm integral equations of the first kind, J. Approx. Theory, 7, 167-185, (1973) · Zbl 0252.65100
[253] Weiner, H.W. (ed.): Reproducing Kernel Hilbert Spaces. Applications in Statistical Signal Procssing. Hutchinson Ross, Stroudsburg, PA (1982)
[254] Werner, J.: Numerische Mathematik 1. Vieweg Studium, Braunschweig (1991)
[255] Wolf, H.: Ausgleichungsrechnung. Formeln zur praktischen Anwendung. Dümmler Verlag, Bonn (1975) · Zbl 0346.65007
[256] Xia, XG; Nashed, MZ, The backus-Gilbert method for signals in reproducing Hilbert spaces and wavelet subspaces, Inverse Prob., 10, 785-804, (1994) · Zbl 0805.44006
[257] Xia, XG; Nashed, MZ, A modified minimum norm solution method for band-limited signal extrapolation with inaccurate data, Inverse Prob., 13, 1641-1661, (1997) · Zbl 0888.65142
[258] Yao, K, Applications of reproducing kernel Hilbert spaces-bandlimited signal models, Inf. Control, 11, 429-444, (1967) · Zbl 0207.19004
[259] Yosida, K.: Functional Analysis, 5th edn. Springer, Berlin (1965) · Zbl 0126.11504
[260] Zhou, L; Li, X; Pan, F, Gradient-based iterative identification for Wiener nonlinear systems with non-uniform sampling, Nonlinear Dyn., 76, 627-634, (2014) · Zbl 1319.62196
[261] Zwaan, M, Approximation of the solution to the moment problem in a Hilbert space, Numer. Funct. Anal. Optim., 11, 601-612, (1990) · Zbl 0831.44005
[262] Zwaan, M, MRI reconstruction as a moment problem, Math. Methods Appl. Sci., 15, 661-675, (1992) · Zbl 0782.42026
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