Advertisement

Operator-theoretic and regularization approaches to ill-posed problems

  • Willi Freeden
  • M. Zuhair Nashed
Original Paper

Abstract

A general framework of regularization and approximation methods for ill-posed problems is developed. Three levels in the resolution processes are distinguished and emphasized in this expository-research paper: philosophy of resolution, regularization–approximation schema, and regularization algorithms. Dilemmas and methodologies of resolution of ill-posed problems and their numerical implementations are examined with particular reference to the problem of finding numerically minimum weighted-norm least squares solutions of first kind integral equations (and more generally of linear operator equations with non-closed range). An emphasis is placed on the role of constraints, function space methods, the role of generalized inverses, and reproducing kernels in the regularization and stable computational resolution of these problems. The thrust of the contribution is devoted to the interdisciplinary character of operator-theoretic and regularization methods for ill-posed problems, in particular in mathematical geoscience.

Keywords

Ill-psed problems Inverse problems Regularization 

Mathematics Subject Classification

35R30 47A52 65J20 65N21 81A40 

References

  1. Acar, R., Vogel, C.R.: Analysis of bounded variation penalty methods for ill-posed problems. Inverse Prob. 10, 1217–1229 (1994)MathSciNetzbMATHCrossRefGoogle Scholar
  2. Adams, R.A.: Sobolev Spaces. Academic Press, New York (1975)zbMATHGoogle Scholar
  3. Alber, Y.I.: The solution of nonlinear equations with monotone operators in Banach spaces. Sib. Math. J. 16, 1–8 (1975)MathSciNetCrossRefGoogle Scholar
  4. Alber, Y.I.: Iterative regularization in Banach spaces. Soviet Math. (Iz. VUZ) 30, 1–8 (1986)Google Scholar
  5. Alber, Y.I.: The regularization method for variational inequalities with nonsmooth unbounded operators in Banach spaces. Appl. Math. Lett. 6, 63–68 (1993)MathSciNetzbMATHCrossRefGoogle Scholar
  6. Alber, Y.I.: Generalized projection operators in Banach spaces: properties and applications. Funct. Differ. Equ. Proc. Isr. Semin. 1, 1–21 (1994)MathSciNetzbMATHGoogle Scholar
  7. Alber, Y.I.: Metric and generalized projection operators in Banach spaces: properties and applications. In: Kartsatos, A. (ed.) Theory and Applications of Nonlinear Operators of Monotone and Assertive Type, pp. 15–50. Marcel Dekker, New York (1996)Google Scholar
  8. Alber, Y.I., Notik, A.: Perturbed unstable variational inequalities with unbounded operator on approximately given sets. Set Valued Anal. 1(4), 393–402 (1993)MathSciNetzbMATHCrossRefGoogle Scholar
  9. Alber, Y.I., Reich, S.: An iterative method for solving a class of nonlinear operator equations in Banach spaces. Panamerian Math. J. 4, 39–54 (1994)MathSciNetzbMATHGoogle Scholar
  10. Alber, Y.I., Rjazanceva, I.: Variational inequalities with discontinuous monotone mappings. Sov. Math. Dokl. 25, 206–210 (1982)Google Scholar
  11. Albert, A.: Regressions and the Moore–Penrose Pseudoinverse. Academic Press, New York (1972)zbMATHGoogle Scholar
  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)Google Scholar
  13. Anger, G.: A characterization of inverse gravimetric source problem through extremal measures. Rev. Geophys. Space Phys. 19, 299–306 (1981)MathSciNetzbMATHCrossRefGoogle Scholar
  14. Anger, G.: Inverse Problems in Differential Equations. Akademie-Verlag, Berlin (1990)zbMATHGoogle Scholar
  15. Aronszajn, N.: Theory of reproducing kernels. Trans. Am. Math. Soc. 68, 337–404 (1950)MathSciNetzbMATHCrossRefGoogle Scholar
  16. Aubert, G., Kornprobst, P.: Mathematical Problems in Image Processing, Applied Mathematical Sciences, vol. 147, 2nd edn. Springer, New York (2006)zbMATHGoogle Scholar
  17. Baart, M.L.: Methods for Solution of Fredholm Integral Equations of the First Kind. CSIR, Pretoria, Technical Report, vol. 104 (1979)Google Scholar
  18. Baer, R.: Linear Algebra and Projective Geometry. Academic Press, New York (1952)zbMATHGoogle Scholar
  19. Backus, G.E., Gilbert, F.: Numerical applications of a formalism for geophysical inverse problems. Geophys. J.R. Astron. Soc. 13, 247–276 (1967)CrossRefGoogle Scholar
  20. Bakusinskii, A.B.: 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)MathSciNetCrossRefGoogle Scholar
  21. Bakusinskii, A.B.: On the Principle of Iterative Regularization. U.S.S.R. Comput. Math. Meth. Phys. 19, 256–260 (1979)MathSciNetCrossRefGoogle Scholar
  22. Barzaghi, R., Sansò, F.: Remarks on the inverse gravimetric problem. Boll. Geod. Scienze Affini 45, 203–216 (1986)zbMATHGoogle Scholar
  23. Baumeister, J.: Stable Solution of Inverse Problems. Vieweg, Braunschweig (1987)zbMATHCrossRefGoogle Scholar
  24. Ben-Israel, A., Greville, T.N.E.: Generalized Inverses: Theory and Applications. Wiley-Interscience, New York (1974)zbMATHGoogle Scholar
  25. Bertero, M., de Mol, C., Viano, G.A.: Linear inverse problems with discrete data. Inverse Prob. 4: 573–594 (1985/88)Google Scholar
  26. Bertero, M., Brianzi, P., Pike, E.R., Rebolia, L.: Linear regularizing algorithms for positive solutions of linear inverse problems. Proc. R. Soc. Lond. A 415, 257–275 (1988)MathSciNetzbMATHCrossRefGoogle Scholar
  27. Bertero, M., DeMol, C., Viano, G.A.: The stability of inverse problems. In: Baltes, H.P. (ed.) Inverse Scattering Problems in Optics. Springer, Berlin (1979)Google Scholar
  28. Binder, A., Engl, H.W., Groetsch, C.W., Neubauer, A., Scherzer, O.: Weakly closed nonlinear operators and parameter identification in parabolic equations by Tiknonov regularization. Appl. Anal. 55, 215–234 (1994)MathSciNetzbMATHCrossRefGoogle Scholar
  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)MathSciNetzbMATHCrossRefGoogle Scholar
  30. Bjerhammar, A.: Rectangular reciprocal matrices, with special reference to geodetic calculations. Bull. Géod. 25, 188–220 (1951)MathSciNetzbMATHCrossRefGoogle Scholar
  31. Bjerhammar, A.: Theory of Errors and Generalized Matrix Inverses. Elsevier Scientific Publ. Co., Amsterdam (1973)zbMATHGoogle Scholar
  32. Boullion, T.L., Odell, P.L.: Generalized Inverse Matrices. Wiley-Interscience, New York (1971)zbMATHGoogle Scholar
  33. Bruck, R.E.: A strongly convergent iterative solution of \(0\in Ux\) for a maximal montone operator \(U\) in Hilbert space. J. Math. Anal. Appl. 48, 114–126 (1974)MathSciNetzbMATHCrossRefGoogle Scholar
  34. Burger, M., Kaltenbacher, B.: Regularizing Newton–Kaczmarz methods for nonlinear ill-posed problems. SIAM J. Numer. Anal. 44, 1775–1797 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  35. Cavalier, L., Golubev, G.K.: Risk hull method and regularization by projections of illposed inverse problems. Ann. Stat. 34, 1653–1677 (2006)zbMATHCrossRefGoogle Scholar
  36. Cavalier, L., Golubev, G.K., Picard, D., Tsybakov, A.B.: Oracle inequalities for inverse problems. Ann. Stat. 30, 843–874 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  37. Craven, B.D., Nashed, M.Z.: Generalized implicit function theorems when the derivative has no bounded inverse: theory, methods, and applications. Nonlinear Anal. 6, 375–387 (1982)MathSciNetzbMATHCrossRefGoogle Scholar
  38. Davis, P.J.: Interpolation and Approximation. Blaisdell, New York (1963)zbMATHGoogle Scholar
  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)Google Scholar
  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)MathSciNetzbMATHGoogle Scholar
  41. Diaz, J.B., Metcalf, F.T.: 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)zbMATHGoogle Scholar
  42. Dicken, V., Maass, P.: Wavelet–Galerkin methods for ill-posed problems. J. Inverse Ill-posed Probl. 4, 203–222 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
  43. Dimitiev, V.I., Il’inski, A.S., Svenshnikov, A.G.: The developments of mathematical methods for the study of direct and inverse problems in electrodynamics. Russ. Math. Surv. 31, 133–152 (1976)CrossRefGoogle Scholar
  44. Dobson, D.C., Scherzer, O.: Analysis of regularized total variation penalty methods for denoising. Inverse Prob. 12, 601–617 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
  45. Donoho, D.L.: Nonlinear solution of linear inverse problems by Wavelet–Vaguelette decomposition. Appl. Comput. Harm. Anal. 2, 101–126 (1995)MathSciNetzbMATHCrossRefGoogle Scholar
  46. Donoho, D.L., Johnstone, I.M.: Minimax estimation via wavelet shrinkage. Ann. Stat. 26, 879–921 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  47. Dunford, N., Schwarz, J.: Linear Operators, vol. II. Wiley-Interscience, New York (1963)Google Scholar
  48. Duris, C.S.: Optimal quadrature formulas using generalized inverses. Part I. General theory and minimum variance formulas. Math. Comput. 25, 495–504 (1971)zbMATHGoogle Scholar
  49. Eggermont, P.P.B.: Maximum entropy regularization for Fredholm integral equations of the first kind. SIAM J. Math. Anal. 24, 1557–1576 (1993)MathSciNetzbMATHCrossRefGoogle Scholar
  50. Eggermont, P.P.B., LaRicca, V.N.: Maximum Penalized Likelihood Estimation. Volume II. Regression. Springer, New York (2009)Google Scholar
  51. Eggermont, P.P.B., LaRicca, V.N., Nashed, M.Z.: On weakly bounded noise in ill-posed problems. Inverse Prob. 25, 115018–115032 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  52. Eggermont, P.P.B., LaRicca, V.N., Nashed, M.Z.: Moment discretization of ill-posed problems with discrete weakly bounded noise. Int. J. Geomath. 3, 155–178 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  53. Eggermont, P.N., LaRiccia, V., Nashed, M.Z.: Noise models for ill-posed problems. In: Freeden, W., Nashed, M.Z., Sonar, T. (eds.) Handbook of Geomathematics, vol. 2, 2nd edn, pp. 1633–1658. Springer, New York (2015)CrossRefGoogle Scholar
  54. Eicke, B.: Iteration methods for convexly constrained ill-posed problems in hilbert space. Numer. Funct. Anal. Optim 13, 413–429 (1992)MathSciNetzbMATHCrossRefGoogle Scholar
  55. Eisenhart, C.: Carl Friedrich Gauss, vol. VI, pp. 74–81. International Encyclopedia of Social Sciences, New York (1986)Google Scholar
  56. Ekeland, I., Temam, R.: Convex Analysis and Variational Problems. North Holland, Amsterdam (1976)zbMATHGoogle Scholar
  57. Elden, L.: Algorithms for the regularization of ill-conditioned least squares problems. BIT 17, 134–145 (1977)MathSciNetzbMATHCrossRefGoogle Scholar
  58. Elden, L.: An algorithm for the regularization of ill-conditioned banded least squares problems. SIAM J. Sci. Stat. Comput. 5, 237–254 (1984)MathSciNetzbMATHCrossRefGoogle Scholar
  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)MathSciNetzbMATHCrossRefGoogle Scholar
  60. Engl, H.: Integralgleichungen. Springer Lehrbuch Mathematik, Wien (1997)zbMATHCrossRefGoogle Scholar
  61. Engl, H.W., Nashed, M.Z.: Stochastic projectional schemes for random linear operator equations of the first and second kinds. Numer. Funct. Anal. Optim. 1, 451–473 (1979)MathSciNetzbMATHCrossRefGoogle Scholar
  62. Engl, H.W., Nashed, M.Z.: New extremal characterizations of generalized inverses of linear operators. J. Math. Anal. Appl. 82, 566–586 (1981)MathSciNetzbMATHCrossRefGoogle Scholar
  63. Engl, H.W., Hanke, M., Neubauer, A.: Regularization of Inverse Problems. Kluwer, Dordrecht (1996)zbMATHCrossRefGoogle Scholar
  64. Engl, H.W., Kunisch, K., Neubauer, A.: Convergence rates for Tikhonov regularisation of non-linear ill-posed problems. Inverse Prob. 5, 523–540 (1989)zbMATHCrossRefGoogle Scholar
  65. Engl, H., Louis, A.K., Rundell, W. (eds.): Inverse Problems in Geophysical Applications. SIAM, Philadelphia (1997)Google Scholar
  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)MathSciNetzbMATHCrossRefGoogle Scholar
  67. Frankenberger, H., Hanke, M.: Kernel polynomials for the solution of indefinite and ill-posed problems. Numer. Algorithms 25, 197–212 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  68. Franklin, J.N.: Well-posed stochastic extensions to ill-posed linear problems. J. Math. Anal. Appl. 31, 682–716 (1970)MathSciNetzbMATHCrossRefGoogle Scholar
  69. Franklin, J.N.: On Thikhonov’s method for ill-posed problems. Math. Comput. 28, 889–907 (1974)zbMATHGoogle Scholar
  70. Fredholm, I.: Sur une classe d’equations functionelles. Acta Math. 27, 365–390 (1903)MathSciNetzbMATHCrossRefGoogle Scholar
  71. Freeden, W.: On approximation by harmonic splines. Manuscr. Geod. 6, 193–244 (1981)zbMATHGoogle Scholar
  72. Freeden, W.: Multiscale Modelling of Spaceborne Geodata. B.G. Teubner, Stuttgart, Leipzig (1999)zbMATHGoogle Scholar
  73. Freeden, W., Gutting, M.: Special Functions of Mathematical (Geo)Physics. Birkhäuser, Basel (2013)zbMATHCrossRefGoogle Scholar
  74. Freeden, W., Maier, T.: Spectral and multiscale signal-to-noise thresholding of spherical vector fields. Comput. Geosci. 7(3), 215–250 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  75. Freeden, W., Michel, V.: Multiscale Potential Theory (with Applications to Geoscience). Birkhäuser, Boston (2004)zbMATHCrossRefGoogle Scholar
  76. Freeden, W., Nutz, H.: Satellite gravity gradiometry as tensorial inverse problem. Int. J. Geomath. 2, 177–218 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  77. Freeden, W., Schneider, F.: Regularization wavelets and multiresolution. Inverse Prob. 14, 493–515 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  78. Freeden, W., Schreiner, M.: Satellite gravity gradiometry (SGG): from scalar to tensorial solution. In: Freeden, W., Nashed, M.Z., Sonar, T. (eds.) Handbook of Geomathematics, 2nd edn. Springer, New-York (2015)CrossRefGoogle Scholar
  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)MathSciNetCrossRefGoogle Scholar
  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)MathSciNetzbMATHCrossRefGoogle Scholar
  81. Freeden, W., Nashed, Z., Sonar, T. (Eds.): Handbook of Geomathematics, vols. 1,2, and 3, 2nd edn. Springer, New-York (2015)Google Scholar
  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)Google Scholar
  83. Friedrich, K.: Allgemeine für die Rechenpraxis geeignete Lösung für die Aufgaben der kleinsten Absolutsumme und der günstigsten Gewichtsverteilung. Z. Vermess. 337–358 (1937)Google Scholar
  84. Gajewski, H., Gröger, K., Zacharias, K.: Nichtlineare Operatorgleichungen und Operatordifferentialglei-chungen. Akademie-Verlag, Berlin (1974)zbMATHGoogle Scholar
  85. Galerkin, B.G.: Expansions in stability problems for elastic rods and plates (in Russian). Vestn. Inzkenorov 19, 897–908 (1915)Google Scholar
  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)Google Scholar
  87. Gauss, C.F.: Theoria Combinationis Observationum Erroribus Minimis Obnoxiae, vol. 1. Teil, Göttingen (1821)Google Scholar
  88. Gebbauer, B., Scherzer, O.: Impedance-acoustic tomography. SIAM J. Appl. Math. 69, 565–576 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  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)Google Scholar
  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 (1987a)MathSciNetzbMATHCrossRefGoogle Scholar
  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 (1987b)zbMATHCrossRefGoogle Scholar
  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)Google Scholar
  93. Gholami, A., Siahkoohi, H.R.: Regularization of linear and nonlinear geophysical ill-posed problems with joint sparsity constraints. Geophys. J. Int. 180, 871–882 (2010)CrossRefGoogle Scholar
  94. Goldstine, H.H.: A History of Numerical Analysis from the 16th Through the 19th Century. Springer, New York (1977)zbMATHCrossRefGoogle Scholar
  95. Golub, G.H., Van Loan, C.F.: Marix Computations, 3rd edn. The John Hopkins University Press, Baltimore, MD (1996)Google Scholar
  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)Google Scholar
  97. Grafarend, E.W., Awange, J.L.: Applications of Linear and Nonlinear Models. Springer, Berlin (2012)zbMATHCrossRefGoogle Scholar
  98. Graves, J., Prenter, P.M.: On generalized iterative filters for ill-posed problems. Numer. Math. 30, 281–299 (1978)MathSciNetzbMATHCrossRefGoogle Scholar
  99. Grenander, U.: Abstract Inference. Wiley, New York (1981)zbMATHGoogle Scholar
  100. Groetsch, C.W.: Generalized Inverses of Linear Operators. Marcel Dekker. Inc., New York (1977)zbMATHGoogle Scholar
  101. Groetsch, C.W.: The Theory of Tikhonov Regularization for Fredholm Equations of the First Kind. Pitman, London (1984)zbMATHGoogle Scholar
  102. Groetsch, C.W.: Inverse Problems in the Mathematical Science. Vieweg, Braunschweig (1993)zbMATHCrossRefGoogle Scholar
  103. Groetsch, C.W.: Comments on Morozov’s discrepancy principle. In: Hämmerlin, G., Hoffmann, K.H. (eds.) Improperly Posed Problems and Their Numerical Treatment, pp. 97–104. Birkhäuser, Basel (1983)CrossRefGoogle Scholar
  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)Google Scholar
  105. Hadamard, J.: Sur les problémes aux dérivés partielles et leur signification physique. Princeton Univ. Bull. 13, 49–52 (1902)Google Scholar
  106. Hadamard, J.: Lectures on the Cauchy Problem in Linear Partial Differential Equations. Yale University Press, New Haven (1923)zbMATHGoogle Scholar
  107. Haddad, R.A., Parsons, T.W.: Digital Signal Processing: Theory, Applications and Hardware. Computer Science Press, Rockville (1991)Google Scholar
  108. Hanke, M.: Conjugate Gradient Type Methods for Ill-Posed Problems. Pitman Research Notes in Mathematics. Longman House, Harlow (1995)zbMATHGoogle Scholar
  109. Hanke, M., Hansen, P.C.: Regularization methods for large scale problems. Surv. Math. Ind. 3, 253–315 (1993)MathSciNetzbMATHGoogle Scholar
  110. Hanke, M., Scherzer, O.: Inverse probelms light: numerical differentiation. Am. Math. Mon. 108, 512–521 (2001)zbMATHCrossRefGoogle Scholar
  111. Hanke, M., Vogel, C.R.: Two-level preconditioners for regularized inverse problems. Numer. Math. 83, 385–402 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  112. Hansen, P.C.: Analysis of discrete Ill-posed problems by means of the L-curve. SIAM Rev. 34, 561–580 (1992)MathSciNetzbMATHCrossRefGoogle Scholar
  113. Hanson, R.J.: A numerical methods for solving Fredholm inegral equations of the first kind. SIAM J. Numer. Anal. 8, 616–662 (1971)MathSciNetzbMATHCrossRefGoogle Scholar
  114. Hegland, M.: Variable Hilbert scales and their interpolation inequalities with applications to Tikhonov regularization. Appl. Anal. 59, 207–223 (1995)MathSciNetzbMATHCrossRefGoogle Scholar
  115. Helmert, F.: Die Ausgleichungsrechnung nach der Methode der kleinsten Quadrate. Teubner, Berlin (1907)zbMATHGoogle Scholar
  116. Heuser, H.: Funktionalanalysis. 4. Auflage, Teubner (1975)Google Scholar
  117. Hilbert, D.: Grundzüge einer allgemeinen Theorie der linearen Integralgleichungen. Teubner, Leipzig (1912)zbMATHGoogle Scholar
  118. Hille, E.: Introduction to the general theory of reproducing kernels. Rocky Mt. J. Math. 2, 321–368 (1972)MathSciNetzbMATHCrossRefGoogle Scholar
  119. Hofmann, B.: Inverse Probleme. Teubner, Leipzig (1999)zbMATHGoogle Scholar
  120. Hofmann, B., Mathé, P., von Weiszäcker, H.: Regularisation in Hilbert space under unbounded operators and general source conditions. Inverse Prob. 25, 115–130 (2009)CrossRefGoogle Scholar
  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)MathSciNetzbMATHCrossRefGoogle Scholar
  122. Hurwitz, W.A.: On the pseudo-resolvent to the kernel of an integral equation. Trans. Am. Math. Soc. 13, 405–418 (1912)MathSciNetzbMATHGoogle Scholar
  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)zbMATHCrossRefGoogle Scholar
  124. Ivanov, V.K., 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)MathSciNetGoogle Scholar
  125. Jacobsen, M., Hansen, P.C., Saunders, M.A.: Subspace preconditioned LSQR for discrete ill-posed problems. BIT Numer. Math. 43, 975–989 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  126. Jorgensen, P., Tian, F.: Graph Laplacians and discrete reproducing kernel Hilbert spaces from restrictions. Stochastic Analysis and Applications 34, 722–747 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  127. Kaipio, J., Somersalo, E.: Statistical and Computational Inverse Problems. Springer, Berlin (2005)zbMATHGoogle Scholar
  128. Kammerer, W.J., Nashed, M.Z.: The convergence of the conjugate gradient method for singular linear operator equations. SIAM J. Numer. Anal. 9, 165–181 (1972a)MathSciNetzbMATHCrossRefGoogle Scholar
  129. Kammerer, W.J., Nashed, M.Z.: Iterative methods for best approximate solutions of linear integral equation of the first and second kind. J. Math. Anal. Appl. 40, 547–573 (1972b)MathSciNetzbMATHCrossRefGoogle Scholar
  130. Kantorowitsch, L.W., Akilow, G.P.: Funktionalanalysis in Normierten Räumen. Akademie-Verlag, Berlin (1964)zbMATHGoogle Scholar
  131. Kato, T.: Perturbation theory for nullity definciency and other quantities of linear operators. J. Anal. Math. 6, 271–322 (1958)CrossRefGoogle Scholar
  132. Kirsch, A.: An Introduction to the Mathematical Theory of Inverse Problems, 2nd edn. Springer, Heidelberg (1996)zbMATHCrossRefGoogle Scholar
  133. Kowar, R., Scherzer, O.: Convergence analysis of a Landweber–Kaczmarz method for sovling nonlinear ill-posed problems. In: Romanov, S., Kabanikhin, S.I., Anikonov, Y.E., Bukhgeim, A.L. (eds.) Ill-Posed and Inverse Problems. VSP Publishers, Zeist (2002)zbMATHGoogle Scholar
  134. Kress, R.: Linear Integral Equations, 2nd edn. Springer, Berlin (1989)zbMATHCrossRefGoogle Scholar
  135. Kreyszig, E.: Introductory Functional Analysis with Applications. Wiley, New York (1978)zbMATHGoogle Scholar
  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)Google Scholar
  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)Google Scholar
  138. Lieusternik, L.A., Sobolev, V.J.: Elements of Functional Analysis. Ungar, New York (1961)Google Scholar
  139. Lin, Y., Brown, L.D.: Statistical properties of the method of regularization with periodic Gaussian reproducing kernel. Ann. Stat. 32(4), 1723–1743 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  140. Liskovets, O.A.: Regularization of variational inqualities with pseudo-monotone operators on approximately given sets. Differ. Equ. 11, 1970–1977 (1989)Google Scholar
  141. Liu, F., Nashed, M.Z.: Tikhonov regularization of nonlinear ill-posed problems with closed operators in Hilbert scales. J. Inverse Ill-Posed Prob. 5, 363–376 (1997)MathSciNetzbMATHGoogle Scholar
  142. Locker, J., Prenter, P.M.: Regularization with differential operators. J. Math. Anal. Appl. 74, 504–529 (1980)MathSciNetzbMATHCrossRefGoogle Scholar
  143. Louis, A.K.: Inverse und schlecht gestellte Probleme. Teubner, Stuttgart (1989)zbMATHCrossRefGoogle Scholar
  144. Louis, A.K., Maass, P.: A mollifier method for linear equations of the first kind. Inverse Prob. 6, 427–440 (1989)MathSciNetzbMATHCrossRefGoogle Scholar
  145. Louis, A.K., Maass, P., Rieder, A.: Wavelets: Theorie und Anwendungen. B. G. Teubner Studienbücher, Stuttgart (1998)zbMATHCrossRefGoogle Scholar
  146. Mair, B.A., Ruymgaart, F.H.: Statistical inverse estimation in Hilbert scales. SIAM J. Appl. Math. 56, 1424–1444 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
  147. Mathé, P.: The lepskii principle revisited. Inverse Prob. 22, 111–115 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  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, S.V.: The discretized discrepancy principle under general source conditions. J. Complex. 22, 371–381 (2006a)MathSciNetzbMATHCrossRefGoogle Scholar
  150. Mathé, P., Pereverzev, S.V.: Regularization of some linear ill-posed problems with discretized random noisy data. Math. Comput. 75, 1913–1929 (2006b)MathSciNetzbMATHCrossRefGoogle Scholar
  151. Mathé, P., Pereverzev, S.V.: Geometry of linear ill-posed problems in variable Hilbert scales. Inverse Prob. 19, 789–803 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  152. Marti, J.T.: An algorithm for computing minimum norm solutions of fredholm integral equaions of the first kind. SIAM J. Numer. Anal. 15, 1071–1076 (1978)MathSciNetzbMATHCrossRefGoogle Scholar
  153. Marti, J.T.: On the convergence of an algorithm computing minimum-norm solutions of ill-posed problems. Math. Comput. 34, 521–527 (1980)MathSciNetzbMATHCrossRefGoogle Scholar
  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)Google Scholar
  155. Meissl, P.A.: Hilbert spaces and their applications to geodetic least squares problems. Boll. Geod. Sci. Aff. 1, 181–210 (1976)MathSciNetzbMATHGoogle Scholar
  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)Google Scholar
  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 (2002a)MathSciNetzbMATHCrossRefGoogle Scholar
  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)Google Scholar
  159. Miller, K.: Least squares methods for ill-posed problems with a prescribed bounded. SIAM J. Math. Anal. 1, 52–74 (1970)MathSciNetzbMATHCrossRefGoogle Scholar
  160. Moore, E.H.: On the reciprocal of the general algebraic matrix. Bull. Am. Math. Soc. 26, 394–395 (1920)Google Scholar
  161. Moore, E.H.: General Analysis. Mem. Am. Math. Soc. 1, 197–209 (1935)Google Scholar
  162. Moritz, H.: Advanced Physical Geodesy. Herbert Wichmann Verlag, Karlsruhe (1980)Google Scholar
  163. Morozov, V.A.: On the solution of functional equations by the method of regularization. Sov. Math. Doklady 7, 414–41 (1966)MathSciNetzbMATHGoogle Scholar
  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)Google Scholar
  165. Morozov, V.A.: Methods for Solving Incorrectly Posed Problems. Springer, New York (1984)CrossRefGoogle Scholar
  166. Nagel, H.-H.: On the estimation of optical flow: relations between new approaches and some new results. Artif. Intell. 33, 299–324 (1987)CrossRefGoogle Scholar
  167. Nashed, M.Z.: Steepest descent for singular linear opertor equations. SIAM J. Numer. Anal. 7, 358–362 (1970)MathSciNetzbMATHCrossRefGoogle Scholar
  168. Nashed, M.Z.: Generalized inverses, normal solvability and iteration for singular operator equations. In: Rall, L.B. (ed.) Nonlinear Functional Analysis and Applications, pp. 311–359. Academic Press, New York (1971a)CrossRefGoogle Scholar
  169. Nashed, M.Z.: Differentiability and related properties of nonlinear operators: some aspects of the role of differentials in nonlinear functional analyis. In: Rall, L.B. (ed.) Nonlinear Functional analysis and Applications, pp. 103–309. Academic Press, New York (1971b)CrossRefGoogle Scholar
  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)Google Scholar
  171. Nashed, M.Z.: Approximate regularized solutions to improperly posed linear integral and operator equations. In: Colton, D.L., Gilbert, R.P. (eds.) Constructive and Computational Methods for Differential and Integral Equations, vol. 430, pp. 289–322. Springer, New York (1974)CrossRefGoogle Scholar
  172. Nashed, M.Z. (ed.): Generalized Inverses and Applications. Academic Press, New York, San Francisco, London (1976a)Google Scholar
  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)Google Scholar
  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)Google Scholar
  175. Nashed, M.Z.: On moment-discretization and least squares solutions of linear integration equations of the first kind. J. Math. Anal. Appl. 53, 359–366 (1976d)MathSciNetzbMATHCrossRefGoogle Scholar
  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)Google Scholar
  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)Google Scholar
  178. Nashed, M.Z.: Continuous and semicontinuous analogous of iterative method of cimmino and kaczmarz with applications to the inverse radon transform. In: Herman, G.T., Natterer, F. (eds.) Mathematical Aspects of Computerized Tomography, pp. 160–178. Springer, New York (1981a)CrossRefGoogle Scholar
  179. Nashed, M.Z.: Operator-theoretic and computational approaches to ill-posed problems with applications to antenna theory. IEEE Trans. Antennas Propag. 29, 220–231 (1981b)MathSciNetzbMATHCrossRefGoogle Scholar
  180. Nashed, M.Z.: A new approach to classification and regularization of ill-posed operator equations. In: Engl, H., Groetsch, C.W. (eds.) Inverse and Ill-Posed Problems, Band 4, Notes and Reports in Mathematics and Science and Engineering. Academic Press, Boston (1987a)Google Scholar
  181. Nashed, M.Z.: Inner, outer, and generalized inverses in banach and Hilbert spaces. Numer. Funct. Anal. Optim. 9, 261–326 (1987b)MathSciNetzbMATHCrossRefGoogle Scholar
  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)Google Scholar
  183. Nashed, M.Z., Engl, H.W.: Random generalized inverses and approximate solution of random operator equations. In: Bharucha-Reid, A.T. (ed.) Approximate Solution of Random Equations, pp. 149–210. North Holland, New York (1979)zbMATHGoogle Scholar
  184. Nashed, M.Z., Lin, F.: On nonlinear ill-posed problems ii: monotone operator equaions and monotone variational inequalities. In: Kartsatos, A. (ed.) Theory and Applications of Nonlinear Operators of Monotone and Assertive Type, pp. 223–240. Marcel Dekker, New York (1996)Google Scholar
  185. Nashed, M.Z., Scherzer, O.: Stable approximation of nondifferentiable optimization problems with variational inequalities. Contemp. Math. 204, 155–170 (1997a)MathSciNetzbMATHCrossRefGoogle Scholar
  186. Nashed, M.Z., Scherzer, O.: Stable approximation of a minimal surface problem with variational inequalities. Abst. Appl. Anal. 2, 137–161 (1997b)MathSciNetzbMATHCrossRefGoogle Scholar
  187. Nashed, M.Z., Scherzer, O. (Eds.): Inverse Problems, Image Analysis and Medical Imaging. Contemporary Mathematics, vol. 313, American Mathematical Society, Providence, RI (2002)Google Scholar
  188. Nashed, M.Z., Votruba, F.G.: A unified operator theory of generalized inverses. In: Nashed, M.Z. (ed.) Generalized Inverses und Applications, pp. 1–109. Academic Press, New York (1976)zbMATHGoogle Scholar
  189. Nashed, M.Z., Wahba, G.: Generalized inverses in reproducing kernel spaces: an approach to regularization of linear operator equations. SIAM J. Math. Anal. 5, 974–987 (1974a)MathSciNetzbMATHCrossRefGoogle Scholar
  190. Nashed, M.Z., 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 (1974b)zbMATHCrossRefGoogle Scholar
  191. Nashed, M.Z., Wahba, G.: Convergence rates of approximate least squares solutions of linear integral and operator equations of the first kind. Math. Comput. 28, 69–80 (1974c)MathSciNetzbMATHCrossRefGoogle Scholar
  192. Natanson, I.P.: Constructive Function Theory. Frederick Ungar Publ. Co., New York (1965)zbMATHGoogle Scholar
  193. Natterer, F.: The finite element method for ill-posed problems. RAIRO Anal. Numer. 11, 271–278 (1977a)MathSciNetzbMATHCrossRefGoogle Scholar
  194. Natterer, F.: Regularisierung schlecht gestellter Probleme durch Projektionsverfahren. Numer. Math. 28, 329–341 (1977b)MathSciNetzbMATHCrossRefGoogle Scholar
  195. Natterer, F.: Error bounds for Tikhonov regularization in Hilbert scales. Appl. Anal. 18, 29–37 (1984)MathSciNetzbMATHCrossRefGoogle Scholar
  196. Neubauer, A.: On converse and saturation results for Tikhonov regularization of linear ill-posed problems. SIAM J. Numer. Anal. 34, 517–527 (1997)MathSciNetzbMATHCrossRefGoogle Scholar
  197. Novikoff, P.: Sur le problème inverse du potentiel. C. R. Acad. Sci. l’URSS 18, 165–168 (1938)zbMATHGoogle Scholar
  198. Ortega, J.M., Rheinboldt, W.C.: On discretization and differentiation of operators with applications to Newton’s method. SIAM J. Numer. Anal. 3, 143–156 (1966)MathSciNetzbMATHCrossRefGoogle Scholar
  199. Parker, R.L.: The theory of ideal bodies for gravity interpretation. Geophys. J. R. Astr. Soc. 42, 315–334 (1975)CrossRefGoogle Scholar
  200. Payne, L.E.: Improperly Posed Problems in Partial Differential Equations. SIAM Publications, Philadelphia (1975)zbMATHCrossRefGoogle Scholar
  201. Penrose, R.: A generalized inverse for matrices. Proc. Camb. Philos. Soc. 51, 406–413 (1955)zbMATHCrossRefGoogle Scholar
  202. Penrose, R.: On best approximate solutions of linear matrix equations. Proc. Camb. Philos. Soc. 25, 17–19 (1956)zbMATHCrossRefGoogle Scholar
  203. Pereverzev, S.V., Schock, E.: On the adaptive selection of the parameter in regularization of ill-posed problems. SIAM J. Numer. Anal. 43, 2060–2076 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  204. Perry, W.L.: On the Bojarski–Lewis inverse scattering method. IEEE Trans. Antennas Propag. 6, 826–829 (1974)MathSciNetCrossRefGoogle Scholar
  205. Perry, W.L.: Approximate solution of inverse problems with piecewise continuous solution. Radio Sci. 12, 634–642 (1977)MathSciNetCrossRefGoogle Scholar
  206. Petryshyn, W.V.: On generalilzed inverses and uniform convergence of \((I-\beta K)^n\) with applications to iterative methods. J. Math. Anal. Appl. 18, 417–439 (1967). MR 34, 8191MathSciNetzbMATHCrossRefGoogle Scholar
  207. Petrov, G.I.: 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)Google Scholar
  208. Phillips, B.L.: A technique for the numerical solution of certain integral equations of the first kind. J. Assoc. Comput. Math. 9, 84–97 (1962)MathSciNetzbMATHCrossRefGoogle Scholar
  209. Plackett, R.L.: An historical note on the method of least squares. Biometrika 36, 458–460 (1949)MathSciNetzbMATHCrossRefGoogle Scholar
  210. Plato, R.: Optimal algorithms for linear ill-posed problems yielding regularization methods. Numer. Funct. Anal. Optim. 11, 111–118 (1990)MathSciNetzbMATHCrossRefGoogle Scholar
  211. Rado, R.: Note on generalized inverses of matrices. Proc. Camb. Philos. Soc. 52, 600–601 (1956)MathSciNetzbMATHCrossRefGoogle Scholar
  212. Rao, C.R., Mitra, S.K.: Generalized Inverse of Matrices and Its Applications. Wiley, New York (1971)zbMATHGoogle Scholar
  213. Rayleigh, L.: The Theory of Sound. Oxdord University Press, Oxdord (1896)zbMATHGoogle Scholar
  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)Google Scholar
  215. Ribiere, G.: Regularisation d’operateurs. Rev. Inf. Rech. Oper. 1, 57–79 (1967)MathSciNetzbMATHGoogle Scholar
  216. Richter, G.R.: Numerical solution of integral equations of the first kind with non-smooth kernels. SIAM J. Numer. Anal. 15, 511–522 (1978)MathSciNetzbMATHCrossRefGoogle Scholar
  217. Robinson, D.W.: Gauss and generalized inverses. Hist. Math. 7, 118–125 (1980)MathSciNetzbMATHCrossRefGoogle Scholar
  218. Rieder, A.: Keine Probleme mit Inversen Problemen. Vieweg, Braunschweig (2003)zbMATHCrossRefGoogle Scholar
  219. Ritz, W.: Über lineare Funktionalgleichungchungen. Acta Math. 41, 71–98 (1918)Google Scholar
  220. Rudin, L.I.: Functional Analysis. Mc Graw-Hill, New York (1973)zbMATHGoogle Scholar
  221. Rudin, L.I., Osher, S., Fatemi, E.: Nonlinear total variation based noise removal algorithms. Phys. D 60, 259 (1992)MathSciNetzbMATHCrossRefGoogle Scholar
  222. Rummel, R., Balmino, G., Johannessen, J., Visser, P., Woodworth, P.: Dedicated gravity field missions—principles and aims. J. Geodyn. 33, 3–20 (2002)CrossRefGoogle Scholar
  223. Saitoh, S.: Theory of Reproducing Kernels and its Applications. Longman, New York (1988)zbMATHGoogle Scholar
  224. Scherzer, O. (ed.): Handbook of Mathematical Methods in Imaging. Springer, New York (2015)Google Scholar
  225. Scherzer, O., Grasmair, M., Grossauer, H., Haltmeier, M., Lenzen, F.: Variational Methods in Imaging. Applied Mathematical Sciences. Springer, New York (2009)zbMATHGoogle Scholar
  226. Schuster, T.: The Method of Approximate Inverse: Theory and Applications, Lecture Notes in Mathematics. Springer, Berlin (2007)CrossRefGoogle Scholar
  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)zbMATHCrossRefGoogle Scholar
  228. Seidmann, T.I.: Non-convergence results for the application of least squares estimation onto ill-posed problems. J. Optim. Theory Appl. 30, 535–547 (1980)MathSciNetCrossRefGoogle Scholar
  229. Shinozaki, S.M., Ranabe, K.: Numerical algorithms for the Moore–Penrose inverse of a matrix: direct methods. Ann. Inst. Stat. Math. 24, 193–203 (1972)MathSciNetCrossRefGoogle Scholar
  230. Showalter, D.W., 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)zbMATHGoogle Scholar
  231. Shure, L., Parker, R.L., Backus, G.E.: Harmonic splines for geomagnetic modelling. Phys. Earth Planet. Inter. 28, 215–229 (1982)CrossRefGoogle Scholar
  232. Siegel, C.L.: Über die analytische Theorie der quadratischen Formen. III. Ann. Math. 38, 212–291 (1937)MathSciNetzbMATHCrossRefGoogle Scholar
  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)Google Scholar
  234. Song, M.: Regularization-Projection Methods and Finite Element Approximations for Ill-Posed Linear Operator Equations. Ph.D. Thesis, University Michigan (1978)Google Scholar
  235. Strand, O.N.: 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)MathSciNetzbMATHCrossRefGoogle Scholar
  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)Google Scholar
  237. Szegö, G.: Orthogonal Polynomials. American Mathematical Society Colloquium Publications, vol. 23. American Mathematical Society, Providence (1959)Google Scholar
  238. Tadmor, E., Nezzar, S., Vese, L.: A multiscale image representation using hierarchical (BV, \(L^2\)) decompositions. Multiscale Model. Simul. 2, 554–579 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  239. Taylor, A.E., Lay, D.: Functional Analysis. Wiley, New York (1979)Google Scholar
  240. Tikhonov, A.N.: On the stability of inverse problems. Dokl. Akad. Nauk SSSR 39, 195–198 (1943)Google Scholar
  241. Tikhonov, A.N.: On the solution of incorrectly formulated problems and the regularization method. Dokl. Akad. Nauk SSSR 151, 501–504 (1963)MathSciNetzbMATHGoogle Scholar
  242. Tikhonov, A.N.: On methods of solving incorrect problems. Am. Math. Soc. Transl. 2, 222–224 (1968)Google Scholar
  243. Tikhonov, A.N., Arsenin, V.Y.: Solutions of Ill-Posed Problems. Wiley, Washington, DC (1977)zbMATHGoogle Scholar
  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)zbMATHCrossRefGoogle Scholar
  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)Google Scholar
  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)MathSciNetzbMATHCrossRefGoogle Scholar
  247. Vainberg, M.M.: Variational Method and Method of Monotone Operators. Wiley, New York (1973)zbMATHGoogle Scholar
  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)MathSciNetzbMATHCrossRefGoogle Scholar
  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)Google Scholar
  250. Vogel, C.R.: Computational Methods for Inverse Problems. SIAM, Philadelphia (2002)zbMATHCrossRefGoogle Scholar
  251. Vogel, C.R., Oman, M.E.: Iterative methods for total variation denoising. SIAM J. Sci. Comput. 17, 227–238 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
  252. Wahba, G.: Convergence rates of certain approximate solutions ot fredholm integral equations of the first kind. J. Approx. Theory 7, 167–185 (1973)zbMATHCrossRefGoogle Scholar
  253. Weiner, H.W. (ed.): Reproducing Kernel Hilbert Spaces. Applications in Statistical Signal Procssing. Hutchinson Ross, Stroudsburg, PA (1982)Google Scholar
  254. Werner, J.: Numerische Mathematik 1. Vieweg Studium, Braunschweig (1991)Google Scholar
  255. Wolf, H.: Ausgleichungsrechnung. Formeln zur praktischen Anwendung. Dümmler Verlag, Bonn (1975)zbMATHGoogle Scholar
  256. Xia, X.G., Nashed, M.Z.: The Backus–Gilbert method for signals in reproducing Hilbert spaces and wavelet subspaces. Inverse Prob. 10, 785–804 (1994)MathSciNetzbMATHCrossRefGoogle Scholar
  257. Xia, X.G., Nashed, M.Z.: A modified minimum norm solution method for band-limited signal extrapolation with inaccurate data. Inverse Prob. 13, 1641–1661 (1997)zbMATHCrossRefGoogle Scholar
  258. Yao, K.: Applications of reproducing kernel Hilbert spaces–bandlimited signal models. Inf. Control 11, 429–444 (1967)zbMATHCrossRefGoogle Scholar
  259. Yosida, K.: Functional Analysis, 5th edn. Springer, Berlin (1965)zbMATHCrossRefGoogle Scholar
  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)MathSciNetzbMATHCrossRefGoogle Scholar
  261. Zwaan, M.: Approximation of the solution to the moment problem in a Hilbert space. Numer. Funct. Anal. Optim. 11, 601–612 (1990)MathSciNetzbMATHCrossRefGoogle Scholar
  262. Zwaan, M.: MRI reconstruction as a moment problem. Math. Methods Appl. Sci. 15, 661–675 (1992)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2017

Authors and Affiliations

  1. 1.Geomathematics GroupUniversity of KaiserslauternKaiserslauternGermany
  2. 2.Mathematics DepartmentUniversity of Central FloridaOrlandoUSA

Personalised recommendations