Literature cited
L. Aleksandrov, “Regularized Newton-Kantorovich computational processes,” Zh. Vychisl. Matem. I Matem. Fiz.,11, No. 1, 36–43 (1971).
B. Aliev, “Regularizing algorithms for finding a stable normal solution of an equation of the second kind on a spectrum,” Zh. Vychisl. Matem. i Matem. Fiz.,10, No. 3, 569–576 (1970).
B. Aliev, “Evaluation of the regularization method for integral equations of the second kind on a spectrum,” Zh. Vychisl. Matem. i Matem. Fiz.,11, No. 2, 505–510 (1971).
B. Aliev, “Two difference approaches to the solution of the Neumann problem in a rectangular domain,” Zh. Vychisl. Matem. i Matem. Fiz.,12, No. 1, 230–236 (1972).
B. A. Andreev, “Calculations of a spatial distribution,” Izv. Akad. Nauk SSSR, Ser. Geogr. i Geofiz., No. 1 (1947).
Yu. E. Anikonov, “Operator equations of the first kind,” Dokl. Akad. Nauk SSSR,207, No. 2, 257–258 (1972).
Yu. T. Antokhin, “Some ill-posed problems in potential theory,” Differents. Uravnen.,2, No. 4, 525–532 (1966).
Yu. T. Antokhin, “Some ill-posed problems in the theory of partial differential equations,” Differents. Uravnen.,2, No. 2, 241–250 (1966).
Yu. T. Antokhin, “Some problems in the analytic theory of equations of the first kind,” Differents. Uravnen.,2, No. 2, 226–240 (1966).
Yu. T. Antokhin, “Ill-posed problems in a Hilbert space and stable methods for their solution,” Differents. Uravnen.,3, No. 7, 1135–1156 (1967).
Yu. T. Antokhin, “Ill-posed problems for equations of the convolution type,” Differents. Uravnen.,4, No. 9, 1691–1704 (1968).
V. Ya. Arsenin, “Discontinuous solutions of equations of the first kind,” Zh. Vychisl. Matem. i Matem. Fiz.,5, No. 5, 922–926 (1965).
V. Ya. Arsenin, “Optimal summation of Fourier series with approximate coefficients,” Dokl. Akad. Nauk SSSR,182, No. 2, 257–260 (1968).
V. Ya. Arsenin, “Algorithms for the solution of ill-posed problems,” in: Digital Computer Software [in Russian], Kiev (1972), pp. 79–106.
V. Ya. Arsenin and V. V. Ivanov, “On optimal regularization,” Dokl. Akad. Nauk SSSR,182, No. 1, 9–12 (1968).
V. Ya. Arsenin and V. V. Ivanov, “Solution of certain convolution-type integral equations of the first kind by the regularization method,” Zh. Vychisl. Matem. i Matem. Fiz.,8, No. 2, 310–321 (1968).
V. Ya. Arsenin and O. A. Liskovets, “Ill-posed problems and their solutions,” in: History of Soviet Mathematics [in Russian], Vol. 4, Kiev (1970), Chap. 2, pp. 134–145.
V. Ya. Arsenin and T. I. Savelova, “Application of the regularization method to convolution-type integral equations of the first kind,” Zh. Vychisl. Matem. i Matem. Fiz.,9, No. 6, 1392–1396 (1969).
A. B. Bakushinskii, “A numerical method for the solution of Fredholm integral equations of the first kind,” Zh. Vychisl. Matem. i Matem. Fiz.,5, No. 4, 744–749 (1965).
A. B. Bakushinskii, “A general technique for the synthesis of regularizing algorithms for a linear ill-posed equation in Hilbert space,” Zh. Vychisl. Matem. i Matem. Fiz.,7, No. 3, 672–676 (1967).
A. B. Bakushinskii, Selected Problems in the Approximate Solution of Ill-Posed Problems [in Russian] (Mimeographed), Vychisl. Tsentr, Moskov. Univ., Moscow (1968).
A. B. Bakushinskii, “Certain properties of regularizing algorithms,” Zh. Vychisl. Matem. i Matem. Fiz.,8, No. 2, 426–428 (1968).
A. B. Bakushinskii, “Synthesis of regularizing algorithms in random interference,” Dokl. Akad. Nauk SSSR,189, No. 2, 231–233 (1969).
A. B. Bakushinskii, “Aspects of the theory of regularizing algorithms,” in: Computational Methods and Programs [in Russian], No. 12, Moskov. Univ., Moscow (1969), pp. 56–79.
A. B. Bakushinskii, “Extension of the residual principle,” Zh. Vychisl. Matem. i Matem. Fiz.,10, No. 1, 210–213 (1970).
A. B. Bakushinskii and V. N. Strakhov, “Solution of certain integral equations of the first kind by the method of successive approximations,” Zh. Vychisl. Matem. i Matem. Fiz.,8, No. 1, 181–185 (1968).
R. V. Borisova, P. N. Zaikin, and V. A. Maslennikov, “Numerical solution of linear automatic control problems,” in: Computational Methods and Programming [in Russian], No. 12, Moskov. Univ., Moscow (1969), pp. 80–86.
Yu. E. Boyarintsev and V. G. Vasil'ev, “Stability of the quasi-reversibility method in the solution of ill-posed evolutionary equations,” Zh. Vychisl. Matem. i Matem. Fiz.,9, No. 4, 951–952 (1969).
Yu. E. Boyarintsev and V. G. Vasil'ev, “Theory of the convergence of approximate solutions of operator equations,” Dokl. Akad. Nauk SSSR,180, No. 5, 1027–1028 (1968).
B. M. Budak and E. M. Berkovich, “Approximation of extremal problems (I, II),” Zh. Vychisl. Matem. i Matem. Fiz.,11, No. 3, 580–596; No. 4, 870–884 (1971).
B. M. Budak, E. M. Berkovich, and Yu. L. Gaponenko, “Construction of a strongly convergent minimizing sequence for a continuous convex functional,” Zh. Vychisl. Matem. i Matem. Fiz.,9, No. 2, 286–299 (1969).
B. M. Budak, A. Vin'oli, and Yu. L. Gaponenko, “A regularization technique for a continuous convex functional,” Zh. Vychisl. Matem. i Mat em. Fiz.,9, No. 5, 1046–1056 (1969).
L. A. Vainshtein, “Numerical solution of integral equations of the first kind with the use of a priori information on the sought-after function,” Dokl. Akad. Nauk SSSR,20, No. 6, 1331–1334 (1972).
V. V. Vasin, “Interrelationship of certain variational methods for the approximate solution of ill-posed problems,” Matem. Zametki,7, No. 3, 265–272 (1970).
V. V. Vasin, “β-Convergence of the projection method for nonlinear operator equations,” Zh. Vychisl. Matem. i Matem. Fiz.,12, No. 2, 492–496 (1972).
V. V. Vasin, “Computation of the values of an unbounded operator in B spaces,” Izv. Vyssh. Uchebn. Zaved., Matem., No. 5, 22–28 (1972).
V. V. Vasin and V. P. Tanana, “Approximate solution of operator equations of the first kind,” Matem. Zap. Ural'sk. Univ.,6, No. 4, 27–37 (1968).
V. A. Vinokurov, “A necessary condition for Tikhonov regularizability,” Dokl. Akad. Nauk SSSR,195, No. 3, 530–531 (1970).
V. A. Vinokurov, “General properties of the error of approximate solutions of linear functional equations,” Zh. Vychisl. Matem. i Matem. Fiz.,11, No. 1, 22–28 (1971).
V. A. Vinokurov, “The residual approximation principle in nonreflexive spaces,” Zh. Vychisl. Matem. i Matem. Fiz.,12, No. 1, 207–212 (1972).
V. V. Voevodin, “A regularization method,” Zh. Vychisl. Matem. i Matem. Fiz.,9, No. 3, 671–673 (1969).
M. K. Gavurin, “A. N. Tikhonov's method for the solution of ill-posed problems,” in: Computational Methods [in Russian], No. 4, Leningrad Univ., Leningrad (1967), pp. 21–25.
V. B. Glasko, “Uniqueness of solution of the problem of reproducing the structure of the earth's crust from the Bayleigh-wave dispersion spectrum,” Dokl. Akad. Nauk SSSR,20, No. 6, 1345–1348 (1972).
V. B. Glasko, N. I. Kulik, and A. N. Tikhonov, “Determination of the geoelectric cross section on the basis of the regularization method,” Zh. Vychisl. Matem. i Matem. Fiz.,12, No. 1, 139–149 (1972).
G. M. Goluzin and V. I. Krylov, “Generalization of the Carleman formula and its application to the analytic continuation of functions,” Matem. Sborn.,40, 144–149 (1933).
A. V. Goncharskii and A. G. Yagola, “Uniform approximation of monotone solutions of ill-posed problems,” Dokl. Akad. Nauk SSSR,184, No. 4, 771–773 (1969).
A. V. Goncharskii, A. G. Yagola, and A. G. Leonov, “Estimates of the rate of convergence of regularized approximations for convolution-type equations,” Zh. Vychisl. Matem. i Matem. Fiz.,12, No. 3, 762–770 (1972).
A. V. Goncharskii, A. G. Yagola, and A. G. Leonov, “A regularizing algorithm for ill-posed problems with an approximately specified operator,” Zh. Vychisl. Matem. i Matem. Fiz.,12, No. 6, 1592–1594 (1972).
A. D. Gorbunov, “Solution of nonlinear boundary-value problems for a system of ordinary differential equations,” in: Computational Methods and Programming [in Russian], Moskov. Univ., Moscow (1967), pp. 186–199.
A. M. Denisov, “Approximation of quasi-solutions of a special type of Fredholm integral equation of the first kind,” Zh. Vychisl. Matem. i Matem. Fiz.,12, No. 6, 1565–1568 (1972).
R. Denchev, “Stability of equations on a compactum,” Zh. Vychisl. Matem. i Matem. Fiz.,7, No. 6, 1367–1369 (1967).
R. Denchev, “A. N. Tikhonov's regularization method for weak solutions of boundary-value problems,” Zh. Vychisl. Matem. i Matem. Fiz.,9, No. 3, 707–711 (1969).
V. I. Dmitriev and E. V. Zakharov, “Numerical solution of certain Fredholm integral equations of the first kind,” in: Computational Methods and Programming [in Russian], No. 10, Moskov. Univ., Moscow (1968), pp. 49–54.
V. I. Dmitriev and A. N. Tikhonov, “Methods for the solution of an inverse problem in antenna theory,” in: Computational Methods and Programming [in Russian], No. 10, Moskov. Univ., Moscow (1968), pp. 209–214.
M. G. Dmitriev and V. S. Poleshchuk, “Regularization of a class of unstable extremal problems,” Zh. Vychisl. Matem. i Matem. Fiz.,12, No. 5, 1316–1318 (1972).
T. F. Dolgopolova and V. K. Ivanov, “On numerical differentiation,” Zh. Vychisl. Matem. i Matem. Fiz.,6, No. 3, 570–575 (1966).
I. N. Dombrovskaya, “Equations of the first kind with a closed operator,” Izv. Vyssh. Uchebn. Zaved., Matem., No. 6, 39–72 (1967).
I. N. Dombrovskaya, “Solution of ill-posed linear equations in Hilbert spaces,” Matem. Zap. Ural'sk. Univ.,4, No. 4, 36–40 (1964).
I. N. Dombrovskaya, “Approximate solution of Fredholm integral equations of the first kind,” Matem. Zap. Ural'sk. Univ.,4, No. 4, 30–35 (1964).
I. N. Dombrovskaya, “Linear operator equations of the first kind,” Izv. Vyssh. Uchebn. Zaved., Matem., No. 2, 74–78 (1964).
I. N. Dombrovskaya and V. K. Ivanov, “Ill-posed linear equations and exceptional cases of convolution-type equations,” Izv. Vyssh. Uchebn. Zaved., Matem., No. 4, 69–74 (1964).
I. N. Dombrovskaya and V. K. Ivanov, “Theory of certain linear equations in abstract spaces,” Sibirsk. Matem. Zh.,6, No. 3, 499–508 (1965).
N. P. Zhidkov, B. M. Shchedrin, N. G. Rambidi, and N. M. Egorova, “Application of the regularization method for the solution of problems in the electron-diffraction analysis of gases,” in: Computational Methods and Programming [in Russian], No. 10, Moskov. Univ., Moscow (1968), pp. 215–222.
E. L. Zhukovskii, “Statistical regularization of algebraic systems of equations,” Zh. Vychisl. Matem. i Matem. Fiz.,12, No. 1, 185–191 (1972).
E. L. Zhukovskii and V. A. Morozov, “Sequential Bayes regularization of algebraic systems of equations,” Zh. Vychisl. Matem. i Matem. Fiz.,12, No. 2, 464–465 (1972).
P. N. Zaikin, “Numerical solution of an inverse problem of operational calculus in a real domain,” Zh. Vychisl. Matem. i Matem. Fiz.,8, No. 2, 411–415 (1968).
V. V. Ivanov, Theory of Approximation Methods and Its Application to the Numerical Solution of Singular Integral Equations [in Russian], Naukova Dumka, Kiev (1968), 287 pages.
V. V. Ivanov and V. Yu. Kudrinskii, “Approximate solution of linear operator equations in a Hilbert space by the least-squares method,” Zh. Vychisl. Matem. i Matem. Fiz.,6, No. 5, 831–841 (1966),
V. K. Ivanov, “Uniqueness theorem for the inverse problem of the logarithmic potential for star sets,” Izv. Vyssh. Uchebn. Zaved., Matem., No. 3, 99–106 (1958).
V. K. Ivanov, “Stability of the inverse problem of the logarithmic potential,” Izv. Vyssh. Zaved., Matem., No. 4, 96–99 (1958).
V. K. Ivanov, “Integral equations of the first kind and an approximate solution of the inverse potential problem,” Dokl. Akad. Nauk SSSR,142, No. 5, 997–1000 (1962).
V. K. Ivanov, “On linear ill-posed problems,” Dokl. Akad. Nauk SSSR,145, No. 2, 270–272 (1962).
V. K. Ivanov, “On ill-posed problems,” Matem. Sborn.,61, No. 2, 211–223 (1963).
V. K. Ivanov, “A type of ill-posed linear equations in vector topological spaces,” Sibirsk. Matem. Zh.,6, No. 4, 832–839 (1965).
V. K. Ivanov, “Approximate solution of operator equations of the first kind,” Zh. Vychisl. Matem. i Matem. Fiz.,6, No. 6, 1089–1093 (1966).
V. I. Ivanov, “Uniform regularization of unstable problems,” Sibirsk. Matem. Zh.,7, No. 3, 546–558 (1966).
V. K. Ivanov, “On Fredholm integral equations of the first kind,” Differents. Uravnen.,3, No. 3, 410–421 (1967).
V. I. Ivanov, “Ill-posed problems in topological spaces,” Sibirsk. Matem. Zh.,10, No. 5, 1065–1074 (1969).
V. K. Ivanov, “Error estimation for the solution of operator equations of the first kind,” in: Proc. Sympos. Aspects of the Computational Error and Efficiency of Algorithms [in Russian], Vol. 2, Kiev (1969).
V. K. Ivanov, “The quasi-reversibility problem for the heat-conduction equation in uniform metric,” Differents. Uravnen.,8, No. 4, 652–658 (1972).
V. K. Ivanov, “Solution of operator equations that do not satisfy well-posedness conditions,” Trudy Matem. Inst. Akad. Nauk SSSR,112, 232–240 (1971).
V. K. Ivanov, “Linear unstable problems with multiple-valued operators,” Sibirsk. Matem. Zh.,11, No. 5, 1010–1016 (1970).
V. K. Ivanov and T. I. Korolyuk, “Error estimation for the solution of linear ill-posed problems,” Zh. Vychisl. Matem. i Matem. Fiz.,9, No. 1, 30–41 (1969).
L. A. Kalyakin, “Approximate solution of ill-posed problems in normed spaces,” Zh. Vychisl. Matem. i Matem. Fiz.,12, No. 5, 1168–1181 (1972).
L. V. Kantorovich, “New approaches to computational methods and the processing of observations,” Sibirsk. Matem. Zh.,3, No. 5, 701–709 (1962).
L. F. Korkina, “Solution of operator equations of the first kind in Hilbert spaces,” Izv. Vyssh. Uchebn. Zaved., Matem., No. 7, 65–69 (1967).
L. F. Korkina, “Regularization of integral equations of the first kind with a Green kernel,” Izv. Vyssh. Uchebn. Zaved., Matem., No. 5, 44–49 (1968).
L. F. Korkina, “Estimate of the modulus of continuity of an inverse operator,” Matem. Zap. Ural'sk. Univ.,7, No. 2, 76–87 (1969).
L. F. Korkina, “Regularization of operator equations of the first kind,” Izv. Vyssh. Uchebn., Zaved., Matem., No. 8, 26–29 (1969).
T. I. Korolyuk, “On the analytic continuation of a harmonic function,” Matem. Zap. Ural'sk. Univ.,6, No. 4, 81–86 (1968).
S. G. Krein, “Classes of solutions for which certain boundary-value problems are well posed,” Dokl. Akad. Nauk SSSR,114, No. 6, 1162–1165 (1957).
S. G. Krein and O. I. Prozorovskaya, “Analytic subgroups and ill-posed problems for evolutionary equations,” Dokl. Akad. Nauk SSSR,133, No. 2, 277–280 (1960).
S. G. Krein and O. I. Prozorovskaya, “Approximate methods for the solution of ill-posed problems,” Zh. Vychisl. Matem. i Matem. Fiz.,3, No. 1, 120–130 (1963).
M. M. Lavrent'ev, “On the Cauchy problem for the Laplace equation,” Dokl. Akad. Nauk SSSR,102, No. 2, 205–206 (1955).
M. M. Lavrent'ev, “On the Cauchy problem for the Laplace equation,” in: Proc. Third All-Union Mathematics Congr. [in Russian], No. 2, Izd. AN SSSR, Moscow (1956), 118 pages.
M. M. Lavrent'ev, “Quantitative refinements of inner uniqueness theorems,” Dokl. Akad. Nauk SSSR,110, No. 5, 731–734 (1956).
M. M. Lavrent'ev, “On the inverse problem of potential theory,” Dokl. Akad. Nauk SSSR,106, No. 3, 389–390 (1956).
M. M. Lavrent'ev, “On the Cauchy problem for second-order linear elliptic equations,” Dokl. Akad. Nauk SSSR,112, No. 2, 195–197 (1957).
M. M. Lavrent'ev, “Mathematical problems in the interpretation of geophysical observations,” in: Methods and Algorithms for the Interpretation of Geophysical Data [in Russian], Nauka, Moscow (1967), pp. 3–8.
M. M. Lavrent'ev, “On integral equations of the first kind,” Dokl. Akad. Nauk SSSR,127, No. 1, 31–33 (1969).
M. M. Lavrent'ev, Ill-Posed Problems of Mathematical Physics [in Russian], Sibirsk. Otd. Akad. Nauk SSSR, Novosibirsk (1962), 92 pages.
M. M. Lavrent'ev and V. G. Vasil'ev, “On the formulation of certain ill-posed problems of mathematical physics,” Sibirsk. Matem. Zh.,7, No. 3, 559–576 (1966).
M. M. Lavrent'ev and V. G. Romanov, Multidimensional Inverse Problems for Differential Equations [in Russian], Nauka, Novosibirsk (1969), 67 pages.
M. M. Lavrent'ev, Yu. E. Anikonov, and F. N. Fazylov, “Approximate solution of certain nonlinear operator equations,” Dokl. Akad. Nauk SSSR,200, No. 4, 770–772 (1971).
E. M. Landis, “Certain properties of the solutions of elliptic equations,” Usp. Matem. Nauk,11, No. 2, 235–237 (1956).
R. Lattès and J. L. Lions, The Method of Quasi Reversibility: Applications to Partial Differential Equations, Amer. Elsevier, New York (1969).
V. I. Lebedev, “Solutions of certain reconstruction problems on compact sets,” Zh. Vychisl. Matem. i Matem. Fiz.,6, No. 6, 1002–1018 (1966).
J. L. Lions, Optimal Control of Systems Governed by Partial Differential Equations, Springer, Berlin-New York (1971).
O. A. Liskovets, “Ill-posed problems with closed noninvertible operators,” Differents. Uravnen.,3, No. 3, 636–646 (1967).
O. A. Liskovets, “Numerical solution of certain ill-posed problems by the method of quasi solutions,” Differents. Uravnen.,4, No. 4, 735–742 (1968).
O. A. Liskovets, “Regularization of equations with a closed linear operator,” Differents. Uravnen.,1, No. 7, 1273–1278 (1970).
O. A. Liskovets, “Stable computation of the values of a closed operator,” Dokl. Akad. Nauk BSSR,15, No. 6, 481–483 (1971).
O. A. Liskovets, “Stability of quasi solutions for equations with a closed operator,” Differents. Uravnen.,7, No. 9, 1707–1709 (1971).
O. A. Liskovets, “A regularization method for nonlinear problems with a closed operator,” Sibirsk. Matem. Zh.,12, No. 6, 1311–1317 (1971).
M. K. Likht, “Computation of functionals on solutions of linear equations of the first kind,” Zh. Vychisl. Matem. i Matem. Fiz.,2, No. 3, 667–672 (1967).
L. V. Maiorov, “On the well-posedness of an inverse problem,” Zh. Vychisl. Matem. i Matem. Fiz.,5, No. 2, 363–365 (1965).
I. G. Malkin, “Determination of the thickness of a homogeneous attractive layer,” in: Transactions of the Steklov Institute [in Russian], Vol. 2, No. 4, Izd. AN SSSR, Moscow (1932).
A. K. Malovichko, Methods for the Analytic Continuation of Anomalies of the Force of Gravity and Applications to Gravitational Prospecting Problems [in Russian], Gostoptekhizdat (1956).
G. I. Marchuk, “On the formulation of certain inverse problems,” Dokl. Akad. Nauk SSSR,156, No. 3, 503–506 (1964).
G. I. Marchuk and S. A. Atanbaev, “Aspects of ‘global’ regularization,” Dokl. Akad. Nauk SSSR,190, No. 3, 527–530 (1970).
G. I. Marchuk and V. G. Vasil'ev “Approximate solution of operator equations,” Dokl. Akad. Nauk SSSR,195, No. 4, 773–775 (1970).
V. P. Maslov, “Regularization of ill-posed problems for singular integral equations,” Dokl. Akad. Nauk SSSR,176, No. 5, 1012–1014 (1967).
V. P. Maslov, “Existence of a solution of an ill-posed problem for the equiconvergence of a regularization process,” Usp. Matem. Nauk,23, No. 3, 183–184 (1968).
I. V. Mel'nikova, “Solution of integral equations of the first kind in M space,” Matem. Zap. Ural'sk. Univ.,6, No. 4, 95–102 (1968).
S. N. Mergelyan, “Harmonic approximation and approximation solution of the Cauchy problem for the Laplace equation,” Usp. Matem. Nauk,11, No. 5, 3–26 (1956).
V. A. Morozov, “Regularization of ill-posed problems and selection of the regularization parameter,” Zh. Vychisl. Matem. i Matem. Fiz.,6, No. 1, 170–175 (1966).
V. A. Morozov, “Solution of functional equations by the regularization method,” Dokl. Akad. Nauk SSSR,167, No. 3, 510–512 (1966).
V. A. Morozov, “Parameter selection for the solution of functional equations by the regularization method,” Dokl. Akad. Nauk SSSR,175, No. 6, 1225–1228 (1967).
V. A. Morozov, Methods of Solving Unstable Problems [in Russian] (mimeographed), Vychisl. Tsentr Moskov. Univ., Moscow (1967).
V. A. Morozov, “Reconstruction of functions by the regularization method,” Zh. Vychisl. Matem. i Matem. Fiz.,7, No. 4, 874–881 (1967).
V. A. Morozov. “The principle of the residual for the solution of operator equations by the regularization method,” Zh. Vychisl. Matem. i Matem. Fiz.,8, No. 2, 295–309 (1968).
V. A. Morozov, “On pseudosolutions,” Zh. Vychisl. Matem. i Matem. Fiz.,9, No. 6, 1381–1392 (1969).
V. A. Morozov, “Regularization of certain classes of extremal problems,” in: Computational Methods and Programming [in Russian], No. 12, Moskov. Univ., Moscow (1969), pp. 24–37.
V. A. Morozov, “Solution of ill-posed problems with unbounded nonlinear operators, by the regularization method,” Differents. Uravnen.,6, No. 8, 1453–1458 (1970).
V. A. Morozov, “A stable method of computing the values of unbounded operators,” Dokl. Acad. Nauk SSSR,185, No. 2, 267–270 (1969).
V. A. Morozov, “Optimal regularization of operator equations,” Zh. Vychisl. Matem. i Matem. Fiz.,10, No. 4, 818–829 (1970).
V. A. Morozov, “Estimates of the error of solution of ill-posed problems with unbounded linear operators,” Zh. Vychisl. Matem. i Matem. Fiz.,10, No. 5, 1081–1091 (1970).
V. A. Morozov, “Theory of splines and the problem of stable computation of the values of an unbounded operator,” Zh. Vychisl. Matem. i Matem. Fiz.,11, No. 3, 545–558 (1971).
V. A. Morozov, “Optimality of the residual criterion in the problem of computing the values of unbounded operators,” Zh. Vychisl. Matem. i Matem. Fiz.,11, No. 4, 1019–1924 (1971).
V. A. Morozov and N. N. Kirsanova, “A generalization of the regularization method,” in: Collected Papers of the Computation Center of Moscow State University [in Russian], Vol. 14 (1970), pp. 40–45.
M. V. Murav'eva, “Finding a pulse transition function by A. N. Tikhonov's regularization method,” Zh. Vychisl. Matem. i Matem. Fiz.,13, No. 1, 194–199 (1973).
P. S. Novikov, “Uniqueness of the inverse problem of potential theory,” Dokl. Akad. Nauk SSSR,19 (1938).
P. S. Novikov, “Inverse problems for differential equations,” in: Proc. All-Union Symposium, Feb. 1–6, 1971 [in Russian], Novosibirsk (1972).
A. Kh. Ostromogil'skii, “Uniqueness of solution of the inverse problem of potential theory,” Zh. Vychisl. Matem. i Matem. Fiz.,10, No. 2, 352–365 (1970).
A. Kh. Ostromogil'skii, “Uniqueness of solution of certain inverse problems,” Zh. Vychisl. Matem. i Matem. Fiz.,11, No. 1, 176–181 (1971).
A. P. Petrov, “Estimates of linear functions for the solution of certain inverse problems,” Zh. Vychisl. Matem. i Matem. Fiz.,7, No. 3, 648–654 (1967).
A. P. Petrov and A. V. Khovanskii, “Lower bound for the error arising in the solution of operator equations of the first kind on compacta,” Zh. Vychisl. Matem. i Matem. Fiz.,9, No. 1, 194–201 (1969).
V. V. Petrov and A. S. Uskov, “Information-theoretic aspects of the regularization problem,” Dokl. Akad. Nauk SSSR,195, No. 4, 780–782 (1970).
B. T. Polyak and E. S. Levitin, “Convergence of minimizing sequences in problems on a conditional extremum,” Dokl. Akad. Nauk SSSR,168, No. 5, 997–1000 (1966).
A. I. Prilepko, “Inverse contact problems for generalized magnetic potentials,” Dokl. Akad. Nauk SSSR,181, No. 5, 1065–1068 (1968).
A. I. Prilepko, “The interior inverse problem of potential theory,” Dokl. Akad. Nauk SSSR,182, No. 3, 503–505 (1968).
A. I. Prilepko, “The exterior inverse three-dimensional potential problem for a body close to a given body,” Dokl. Akad. Nauk SSSR,185, No. 1, 40–42 (1969).
A. I. Prilepko, “Stability and uniqueness of solution of inverse problems for the generalized potentials of a simple layer,” Sibirsk. Matem. Zh.,12, No. 4, 828–836 (1971).
A. G. Ramm, “On equations of the first kind,” Differents. Uravnen.,4, No. 11, 2056–2060 (1968).
A. G. Ramm, “Optimal solution of the synthesis problem for a linear antenna,” Izv. Vyssh. Uchebn. Zaved., Radiofiz.,12, No. 12, 1842–1848 (1969).
I. M. Rapoport, “On the planar inverse problem of potential theory,” Dokl. Akad. Nauk SSSR,28, 305–307 (1940).
I. M. Rapoport, “Stability in the inverse problem of potential theory,” Dokl. Akad. Nauk SSSR,31, 303–305 (1941).
V. G. Romanov, Certain Inverse Problems for an Equation of Hyperbolic Type [in Russian], Nauka, Novosibirsk (1969), 196 pages.
S. B. Stechkin, “Best approximation of linear operators,” Matem. Zametki,1, No. 2, 137–148 (1967).
V. N. Strakhov, “A numerical method for the solution of linear convolution-type integral equations,” Dokl. Akad. Nauk SSSR,153, No. 3, 533–536 (1963).
V. N. Strakhov, “Numerical solution of ill-posed problems represented by convolution-type integral equations,” Dokl. Akad. Nauk SSSR,178, No. 2, 299–302 (1968).
V. N. Strakhov, “Theory of approximate solution of linear ill-posed problems in a Hilbert space and its application in geophysical prospecting (I),” Izv. Akad. Nauk SSSR, Fiz. Zemli., No. 8, 30–53 (1969).
V. N. Strakhov, “Solution of linear ill-posed problems in a Hilbert space,” Differents. Uravnen.,6, No. 8, 1490–1495 (1970).
V. N. Strakhov, “Algorithms for the approximate solution of conditionally well-posed linear problems,” Dokl. Akad. Nauk SSSR,207, No. 5, 1057–1059 (1972).
V. N. Sudakov and L. A. Khalfin, “Statistical approach to ill-posed problems of mathematical physics,” Dokl. Akad. Nauk SSSR,157, No. 5, 1058–1060 (1964).
É. É. Tamme, “Stability of difference schemes in the solution of ill-posed problems by the quasi-reversibility method,” Zh. Vychisl. Matem. i Matem. Fiz.,12, No. 5, 1319–1325 (1972).
V. P. Tanana, “Ill-posed problems and the geometry of Banach spaces,” Dokl. Akad. Nauk SSSR,193, No. 1, 43–45 (1970).
V. P. Tanana, “Approximate solution of operator equations of the first kind and geometric properties of Banach spaces,” Izv. Vyssh. Uchebn. Zaved., Matem., No. 7, 81–93 (1971).
A. N. Tikhonov, “Stability of inverse problems,” Dokl. Akad. Nauk SSSR,39, No. 5, 195–198 (1943).
A. N. Tikhonov, “Solution of ill-posed problems and the regularization method,” Dokl. Akad. Nauk SSSR,151, No. 3, 501–504 (1963).
A. N. Tikhonov, “Regularization of ill-posed problems,” Dokl. Akad. Nauk SSSR,153, No. 1, 49–52 (1963).
A. N. Tikhonov, “Stable methods for the summation of Fourier series,” Dokl. Akad. Nauk SSSR,156, No. 2, 268–271 (1964).
A. N. Tikhonov, “Solution of nonlinear integral equations of the first kind,” Dokl. Akad. Nauk SSSR,156, No. 6, 1296–1299 (1964).
A. N. Tikhonov, “On nonlinear equations of the first kind,” Dokl. Akad. Nauk SSSR,161, No. 5, 1023–1026 (1965).
A. N. Tikhonov, “Methods for the regularization of optimal control problems,” Dokl. Akad. Nauk SSSR,162, No. 4, 763–765 (1965).
A. N. Tikhonov, “Ill-posed problems of linear algebra and a stable method for their solution,” Dokl. Akad. Nauk SSSR,163, No. 3, 591–594 (1965).
A. N. Tikhonov, “Ill-posed optimal planning problems and stable methods for their solution,” Dokl. Akad. Nauk SSSR,164, No. 3, 507–510 (1965).
A. N. Tikhonov, “Stability of algorithms for the solution of singular systems of linear algebraic equations,” Zh. Vychisl. Matem. i Matem. Fiz.,5, No. 4, 718–722 (1965).
A. N. Tikhonov, “Stability of the functional optimization problem,” Zh. Vychisl. Matem. i Matem. Fiz.,6, No. 4, 631–634 (1966).
A. N. Tikhonov, “Ill-posed optimal planning problems,” Zh. Vychisl. Matem. i Matem. Fiz.,6, No. 1, 81–89 (1966).
A. N. Tikhonov, “On ill-posed problems,” in: Computational Methods and Programming [in Russian], No. 8, Moskov. Univ., Moscow (1967), pp. 3–33.
A. N. Tikhonov and F. P. Vasil'ev, “Methods for the solution of ill-posed optimal control problems,” in: Proc. All-Union Conf. Mathematical Optimal Control Theory [in Russian], Baku (1972).
A. N. Tikhonov and V. B. Glasko, “Application of the regularization method to nonlinear problems,” Zh. Vychisl. Matem. i Matem. Fiz.,5, No. 3, 463–473 (1965).
A. N. Tikhonov and V. B. Glasko, “Approximate solution of Fredholm integral equations of the first kind,” Zh. Vychisl. Matem. i Matem. Fiz.,4, No. 3, 564–571 (1964).
A. N. Tikhonov and V. B. Glasko, “On methods of determining the surface temperature of a body,” Zh. Vychisl. Matem. i Matem. Fiz.,7, No. 4, 910–914 (1967).
A. N. Tikhonov, V. Ya. Galkin, and P. N. Zaikin, “Direct methods for the solution of optimal control problems,” Zh. Vychisl. Matem. i Matem. Fiz.,7, No. 2, 416–423 (1967).
A. N. Tikhonov, V. G. Karmanov, and T. L. Rudneva, “Stability of linear programming problems,” in: Computational Methods and Programming [in Russian], No. 12, Moskov. Univ., Moscow (1969), pp. 3–9.
A. N. Tikhonov, V. G. Shevchenko, et al., “System for the continuous automatic processing of experimental results on the cross sections of photonuclear reactions,” in: Collected Papers of the Computation Center of Moscow State University [in Russian], Vol. 14, (1970), pp. 3–26.
V. F. Turchin, “Solution of a Fredholm equation of the first kind in a statistical ensemble of smooth functions,” Zh. Vychisl. Matem. i Matem. Fiz.,7, No. 6, 1270–1284 (1967).
V. F. Turchin, “Selection of an ensemble of smooth functions for the solution of an inverse problem.” Zh. Vychisl. Matem. i Matem. Fiz.,8, No. 1, 230–238 (1968).
V. N. Faddeeva, “A translation for systems with poorly conditioned matrices,” Zh. Vychisl. Matem. i Matem. Fiz.,5, No. 4, 907–911 (1965).
G. I. Furletov, “Stable methods for the summation of Fourier series,” Differents. Uravnen.,7, No. 3, 510–518 (1971).
G. V. Khromova, “Regularization of integral equations of the first kind with a Green kernel,” in: Differential Equations and Computational Mathematics [in Russian], No. 1, Saratov Univ., Saratov (1972), pp. 34–53.
Yu. I. Khudak, “Regularization of solutions of integral equations of the first kind,” Zh. Vychisl. Matem. i Matem. Fiz.,6, No. 4, 766–769 (1966).
Yu. I. Khudak, “Convergence of regularizing algorithms,” Zh. Vychisl. Matem. i Matem. Fiz.,11, No. 1, 29–35 (1971).
Yu. I. Khudak, “Convergence of a family of regularizing algorithms,” Zh. Vychisl. Matem. i Matem. Fiz.,12, No. 2, 497–502 (1972).
A. V. Chechkin, “A special Tikhonov regularizer for integral equations of the first kind,” Zh. Vychisl. Matem. i Matem. Fiz.,10, No. 2, 453–461 (1970).
L. A. Chudov, “Finite-difference schemes and ill-posed problems for partial differential equations,” in: Computational Methods and Programming [in Russian], No. 8, Moskov. Univ., Moscow (1967), pp. 34–62.
V. F. Sholokhovich, “Unstable extremal problems,” Matem. Zap. Ural'sk. Univ.,7, No. 2, 103–111 (1969).
V. F. Sholokhovich. “Unstable extremal problems and the geometric properties of Banach spaces,” Dokl. Akad. Nauk SSSR,195, No. 2, 289–291 (1970).
V. F. Sholokhovich, “Approximate solution of unstable extremal problems,” Izv. Vyssh. Uchebn. Zaved., Matem., No. 5, 101–108 (1971).
R. Arcangeli, “Pseudosolution de l'équation Ax=y,” Compt. Rend.,263, No. 8, A282-A285 (1966).
M. Atteia, “Fonctions ‘splines’ difinies sur un ensemble convexe,” Numer. Math.,12, No. 3, 192–210 (1968).
R. Bellman, R. Kalaba, and J. Lochett, “Dynamic programming and ill-conditioned linear systems,” J. Math. Anal. Appl.,10, 206–215 (1965).
A. Bensonssan and P. Kenneth, “Sur l'analogue entre les méthodes de régularisation et de pénalisation,” Rev. Franc. Inform. et Rech. Opér.,2, No. 13, 13–25 (1968).
H. S. Cabayan and G. G. Belford, “On computing a stable least-squares solution to the inverse problem for a planar Newtonian potential,” SIAM J. Appl. Math.,20, No. 1, 51–61 (1971).
T. Carleman, Les Fonctions Quasi Analytiques, Paris (1926).
S. Cruceanu, “Sur la minimisation des fonctionelles convexes,” Compt. Rend.,273, No. 17, A763-A765 (1971).
S. Cruceanu, “Régularisation pour les problèmes à operateurs monotones et la methode de Galerkine,” Comment. Math. Univ. Carol.,12, No. 1, 1–13 (1971).
J. Douglas, “A numerical method for analytic continuation,” in: Boundary Problems for Differential Equations, Univ. Wisconsin Press, Madison (1960), pp. 179–189.
J. Douglas, “Mathematical programming and integral equations,” in: Sympos. Numerical Treatment of Ordinary Differential Equations,” Integral and Integrodifferential Equations, Birkhäuser (1960), pp. 269–274.
J. Douglas, “Approximate continuation of harmonic and parabolic functions,” in: Numerical Solution of Partial Differential Equations, Academic Press, New York (1966), pp. 353–364.
J. Douglas and T. Gallie, “An approximate solution of an improper boundary-value problem,” Duke Math. J.,26, No. 3, 339–347 (1959).
Ku Fan and I. Glicksberg, “Some geometric properties of the spheres in a normed linear space,” Duke Math. J.,25, No. 4, 553–568 (1958).
J. N. Franklin, “Well-posed stochastic extensions of ill-posed linear problems,” J. Math. Anal. Appl.,31, No. 3, 682–716 (1970).
M. Furi and A. Vignoli, “On the regularization of nonlinear ill-posed problems in Banach spaces,” J. Optimization Theory Appl.,4, No. 3, 206–209 (1969).
J. Hadamard, “Sur les problèmes aux dérivées partielles et leur signification physique,” Bull. Univ. Princeton,13, 49–52 (1902).
J. Hadamard, Le Problème de Cauchy et les Équations aux Dérivées Partielles Linéaires Hyperboliques, Hermann, Paris (1932).
F. John, “Numerical solution of the equation of heat conduction for preceding times,” Ann. Mat. Pura et Appl.,40, 129–142 (1955).
F. John, “A note on ‘improper’ problems in partial differential equations,” Commun. Pure Appl. Math.,8, 591–594 (1955).
F. John, “Continuous dependence on data for solutions with a prescribed bound,” Commun. Pure Appl. Math.,13, No. 4, 551–585 (1960).
F. John, “Numerical solution of problems which are not well posed in the sense of Hadamard,” in: Sympos. Numerical Treatment of Partial Differential Equations with Real Characteristics, Rome (1959), pp. 103–116.
K. Marton and L. Varga, “Regularization of certain operator equations by filters,” Stud. Sci. Math. Hung.,6, Nos. 3–4, 457–465 (1971).
P. Medgyessy, “Inkirrekt matematikai problémák vizsgálatának jelen állásárol uülönös tekintettel. I. Fajú operátoregyenletek megoldására,” Magy. Túd. Akad. Mat. és Fiz. Tud. Oszt. Közl.,20, Nos. 1–2, 97–131 (1971).
K. Miller, “Three circle theorems in partial differential equations and applications to improperly posed problems,” Arch. Rational Mech. Anal.,16, No. 2, 126–154 (1964).
I. P. Nedyalkov, “An approach in the theory of incorrect problems,” Dokl. Bolg. Akad. Nauk,23, 5–8 (1970).
D. L. Phillips, “A technique for the numerical solution of certain integral equations of the first kind,” J. Assoc. Computing Mach.,9, No. 1, 84–97 (1962).
C. Pucci, “Sui problemi Cauchy non ‘ben posti,’,” Atti Acad. Naz. Lincei. Rend. Cl. Sci. Fis. Mat. e Natur.,18, No. 5, 473–477 (1955).
C. Pucci, “Discussione del problema di Cauchy pur le equazioni di tipo elliptico,” Ann. Mat. Pura ed Appl.,46, 131–154 (1958).
J. Replogle, B. D. Ho1comb, and W. R. Burrus, “The use of mathematical programming for solving singular and poorly conditioned systems of equations,” J. Math. Anal.,20, No. 2, 310–324 (1967).
G. Ribiére, “Regularisation des operateurs,” Rev. Franc. Inform. et Rech. Opér.,1, No. 11, 57–79 (1967).
C. B. Shaw, “Improvement of the resolution of an instrument by numerical solution of an integral equation,” J. Math. Anal. Appl.37, No. 1, 83–112 (1972).
J. Singer, “Some remarks on approximative compactness,” Rev. Roumaine Math. Pures et Appl.,9, No. 2, 167–177 (1964).
O. N. Strand and E. R. Westwater, “Statistical estimation of the numerical solution of Fredholm integral equations of the first kind,” J. Assoc. Computing Mach.,15, No. 1, 100–114 (1968).
S. Twomey, “On the numerical solution of Fredholm integral equations of the first kind by the inversion of the linear system produced by quadrature,” J. Assoc. Computing Mach.,10, No. 1, 97–101 (1963).
S. Twomey, “The application of numerical filtering to the solution of integral equations encountered in indirect sensing measurements,” J. Franklin Inst.,279, 95–109 (1965).
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Translated from Itogi Nauki i Tekhniki (Matematicheskii Analiz), Vol. 11, pp. 129–178, 1973.
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Morozov, V.A. Linear and nonlinear ill-posed problems. J Math Sci 4, 706–736 (1975). https://doi.org/10.1007/BF01083885
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DOI: https://doi.org/10.1007/BF01083885