Ill-Posed Problems Research Group

Publications

I. General theory of ill-posed problems

G. Vainikko (1991). Regularisierung Nichtkorrekter Aufgaben. Volume 200 of Fachbereich Mathematik: Preprint. 80 pp. Universität Kaiserslautern. Pdf
G. M. Vainikko, A. Yu. Veretennikov (1986). Iteration Procedures in Ill-Posed Problems. 183 pp. Nauka, Moscow. In Russian. Pdf
G. Vainikko (1984). About a class of regularization methods when a priori information about the solution is given. Acta et Commentationes Universitatis Tartuensis, 672, 3–9. In Russian. Pdf of issue
G. Vainikko (1982). Methods for Solution of Ill-Posed Problems in Hilbert Spaces. 110 pp. University of Tartu. In Russian. Pdf

II. Self-regularization of ill-posed problems by projection methods

U. Hämarik, U. Kangro (2018). On self-regularization of ill-posed problems in Banach spaces by projection methods. In: Hofmann, Bernd; Leitao, Antonio; Zubelli, Jorge P. (Ed.). New Trends in Parameter Identification for Mathematical Models, Birkhäuser - De Gruyter, 89−105, 978-3-319-70824-9_5. Pdf
U. Hämarik, B. Kaltenbacher, U. Kangro, E. Resmerita (2016). Regularization by discretization in Banach spaces (vol 32, 035004, 2016). Inverse Problems, 32 (3), 035004, doi:10.1088/0266-5611/32/3/035004.
A. Ganina, U. Hämarik, U. Kangro (2014). On the self-regularization of ill-posed problems by the least error projection method. Mathematical Modelling and Analysis, 19(3),299-308.
U. Hämarik, E. Avi, A. Ganina (2002). On the solution of ill-posed problems by projection methods with a posteriori choice of the discretization level. Mathematical Modelling and Analysis, 7 (2), 241–252. Article on journal's webpage
J. Saranen, G. Vainikko (2002). Periodic Integral and Pseudodifferential Equations with Numerical Approximation. Springer Monographs in Mathematics. xi+452 pp. Springer. doi:10.1007/978-3-662-04796-5
R. Plato, G. Vainikko (2001). On the fast and fully discretized solution of integral and pseudo-differential equations on smooth curves. Calcolo, 38 (1), 25–48. doi:10.1007/PL00011226
R. Plato, G. Vainikko (2001). The fast solution of periodic integral and pseudodifferential equations by GMRES. Computational Methods in Applied Mathematics, 1 (4), 383–397. Article on journal's webpage
G. Vainikko (1997). Trigonometric Galerkin fast solvers for periodic integral equation of the first kind. Functional Differential Equations, 4 (3–4), 419–441.
G. Bruckner, S. Prössdorf, G. Vainikko (1996). Error bounds of discretization methods for boundary integral equations with noisy data. Applicable Analysis 63 (1–2), 25–37. doi:10.1080/00036819608840494
U. Hämarik (1990). On the self-regularization by solving ill-posed problems by projection methods. Acta et Commentationes Universitatis Tartuensis, 913, 65–72. Pdf of issue
U. Hämarik (1988). About self-regularization solving ill-posed problems by projection methods. Acta et Commentationes Universitatis Tartuensis, 833, 91–96. In Russian. Pdf of issue
G. Vainikko, U. Hämarik (1988). Self-regularization solving ill-posed problems by projection methods. In B. A. Beltyukov, V. P. Bulatov (Eds.), Models and Methods in Operational Research, 157–164. Nauka, Novosibirsk. In Russian. Pdf
G. M. Vainikko, U. A. Khyamarik (1985). Projection methods and self-regularization in incorrect problems. Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 10, 3–17. In Russian. Article on mathnet.ru Translation: Soviet Mathematics, 29 (10), 1–20. Pdf
U. Hämarik (1984). Regularized projection methods for ill-posed problems. Proceedings of the Estonian Academy of Sciences. Physics. Mathematics, 33 (3), 266–276. In Russian.
U. Hämarik (1984). The discrepancy principle for choice of dimension solving ill-posed problems by projection methods. Acta et Commentationes Universitatis Tartuensis, 672, 27–34. In Russian. Pdf of issue
U. Hämarik (1983). Projection methods for the regularization of linear ill-posed problems. Proceedings of the Computing Centre of Tartu State University, 50, 69–90. In Russian.
U. Hämarik (1983). Projection methods for the regularization of linear ill-posed problems. In M. M. Lavrentiev. (Ed.), Theory and Methods for Solving Ill-Posed Problems and Their Applications, 79–81. Nauka, Novosibirsk. In Russian.

III. Extrapolation of Tikhonov and Lavrentiev methods

U. Hämarik, R. Palm, T. Raus (2010). Extrapolation of Tikhonov regularization method. Mathematical Modelling and Analysis, 15 (1), 55–68. doi:10.3846/1392-6292.2010.15.55-68
R. Palm (2010). Numerical comparison of regularization algorithms for solving ill-posed problems. PhD thesis. 105 pp. University of Tartu. Pdf
U. Hämarik, R. Palm, T. Raus (2008). Extrapolation of Tikhonov and Lavrentiev regularization methods. Journal of Physics: Conference Series, 135, 012048, 8 pp. doi:10.1088/1742-6596/135/1/012048
U. Hämarik, R. Palm, T. Raus (2007). Use of extrapolation in regularization methods. Journal of Inverse and Ill-Posed Problems, 15 (3), 277–294. doi:10.1515/jiip.2007.015
U. Hämarik (1993). On the parameter choice in the regularized Ritz-Galerkin method. Proceedings of the Estonian Academy of Sciences. Physics. Mathematics, 42 (2), 133–143. Pdf

IV. Optimality and quasioptimality of the choice of the regularization parameter

T. Raus, U. Hämarik (2012). On the quasi-optimal rules for the choice of the regularization parameter in case of a noisy operator. Advances in Computational Mathematics, 36 (2), 221–233. doi:10.1007/s10444-011-9203-6
T. Raus, U. Hämarik (2009). On numerical realization of quasioptimal parameter choices in (iterated) Tikhonov and Lavrentiev regularization. Mathematical Modelling and Analysis, 14 (1), 99–108. Article on journal's webpage
T. Raus, U. Hämarik (2007). On the quasioptimal regularization parameter choices for solving ill-posed problms. Journal of Inverse and Ill-Posed Problems, 15 (4), 419–439. doi:10.1515/jiip.2007.023
U. Hämarik, T. Raus (2002). On the choice of the stopping index in iteration methods for solving problems with noisy data. In E. A. Lipitakis (Ed.), HERCMA 2001: Proceedings of the 5th Hellenic-European Conference on Computer Mathematics and Its Applications. September 20–22, 2001, Athens University of Economics and Business, Athens, Hellas, 524–529. LEA Publishers, Athens. Pdf
R. Plato, U. Hämarik (1996). On pseudo-optimal parameter choices and stopping rules for regularization methods in Banach spaces. Numerical Functional Analysis and Optimization, 17 (1–2), 181–195. doi:10.1080/01630569608816690
T. Raus (1992). About regularization parameter choice in case of approximately given error bounds of data. Acta et Commentationes Universitatis Tartuensis, 937, 77–89. Pdf of issue
T. Raus (1990). An a-posteriori choice of the regularization parameter in case of approximately given error bound of data. Acta et Commentationes Universitatis Tartuensis, 913, 73–87. Pdf of issue
G. Vainikko (1987). The optimal choice of regularization parameter in the Tikhonov method. Acta et Commentationes Universitatis Tartuensis, 762, 3–8. In Russian. Pdf of issue
G. Vainikko (1987). On the optimality of regularization methods. In H. W. Engl, C. W. Groetsch (Eds.), Inverse and Ill-Posed Problems, 77–95. Academic Press, New York. Pdf
G. Vainikko (1987). On the optimality of methods for ill-posed problems. Zeitschrift für Analysis und ihre Anwendungen 6 (4), 351–362.
T. Raus (1985). Discrepancy principle for ill-posed problems with non-selfadjoint operators. Acta et Commentationes Universitatis Tartuensis, 715, 12–20. In Russian. Pdf of issue
G. Vainikko (1985). On the concept of the optimality of approximate methods for ill-posed problems. Acta et Commentationes Universitatis Tartuensis, 715, 3–11. In Russian. Pdf of issue
T. Raus (1984). Discrepancy principle for ill-posed problems. Acta et Commentationes Universitatis Tartuensis, 672, 16–26. In Russian. Pdf of issue

V. Heuristic rules for choice of the regularization parameter

T. Raus, U. Hämarik (2020). Q-curve and area rules for choosing heuristic parameter in Tikhonov regularization. Mathematics, 8 (7, 1166).
U. Hämarik, U. Kangro, S. Kindermann, K. Raik (2019). Semi-heuristic parameter choice rules for Tikhonov regularisation with operator perturbations. J. Inverse Ill-Posed Probl. 27 (1), 117–131.
T. Raus, U. Hämarik (2018). Heuristic parameter choice in Tikhonov method from minimizers of the quasi-optimality function. In: Hofmann, Bernd, Leitao, Antonio, Zubelli, Jorge P. (Ed.). New Trends in Parameter Identification for Mathematical Models, Birkhäuser - De Gruyter, 227−244, 978-3-319-70824-9_12. Pdf
U. Hämarik, R. Palm, T. Raus (2011). Comparison of parameter choices in regularization algorithms in case of different information about noise level. Calcolo, 48 (1), 47–59. doi:10.1007/s10092-010-0027-4
U. Hämarik, R. Palm, T. Raus (2009). On minimization strategies for choice of the regularization parameter in ill-posed problems. Numerical Functional Analysis and Optimization, 30 (9&10), 924–950. doi:10.1080/01630560903392941
U. Hämarik, R. Palm (2007). On rules for stopping the conjugate gradient type methods in ill-posed problems. Mathematical Modelling and Analysis, 12 (1), 61–70. doi:10.3846/1392-6292.2007.12.61-70
U. Hämarik, R. Palm (2005). Comparison of stopping rules in conjugate gradient type methods for solving ill-posed problems. In MMA2005 Proceedings: 10th International Conference "Mathematical Modelling and Analysis" & 2nd International Conference "Computational Methods in Applied Mathematics", Trakai, Lithuania, June 1–5, 2005, 285–291. Technika, Lithuania. Pdf
M. Hanke, T. Raus (1996). A general heuristic for choosing the regularization parameter in ill-posed problems. SIAM Journal on Scientific Computing, 17 (4), 956–972. doi:10.1137/0917062

See also III Palm (2010).

VI. Choice of the regularization parameter in the case of inexact noise level of the data

U. Hämarik, R. Palm, T. Raus (2012). A family of rules for parameter choice in Tikhonov regularization of ill-posed problems with inexact noise level. Journal of Computational and Applied Mathematics, 236 (8), 2146–2157. doi:10.1016/j.cam.2011.09.037
U. Hämarik, R. Palm, T. Raus (2012). A family of rules for the choice of the regularization parameter in the Lavrentiev method in the case of rough estimate of the noise level of the data. Journal of Inverse and Ill-Posed Problems, 20 (5–6), 831–854. doi:10.1515/jip-2012-0061
U. Hämarik, U. Kangro, R. Palm, T. Raus (2012). On parameter choice in the regularization of ill-posed problems with rough estimate of the noise level of the data. In Th. E. Simos, G. Psihoyios, Ch. Tsitouras, Z. Anastassi (Eds.), ICNAAM 2012: International Conference on Numerical Analysis and Applied Mathematics 2012, Kos (Greece), 19–25 September 2012, vol 1479 of AIP Conference Proceedings, 2332–2335. American Institute of Physics, New York. doi:10.1063/1.4756661
T. Raus, U. Hämarik (2009). New rule for choice of the regularization parameter in (iterated) Tikhonov method. Mathematical Modelling and Analysis, 14 (2), 187–198. doi:10.3846/1392-6292.2009.14.187-198
U. Hämarik, T. Raus (2006). On the choice of the regularization parameter in ill-posed problems with approximately given noise level of data. Journal of Inverse and Ill-Posed Problems, 14 (3), 251–266. doi:10.1515/156939406777340928
U. Hämarik, T. Raus (2005). Choice of the regularization parameter in ill-posed problems with rough estimate of the noise level of data. WSEAS Transactions on Mathematics, 4 (2), 76–81. Pdf
U. Hämarik, T. Raus (2005). On the choice of the regularization parameter in the case of the approximately given noise level of data. In D. Lesnic (Ed.), Proceedings of the 5th International Conference on Inverse Problems in Engineering: Theory and Practice, Cambridge, UK, 11–15th July, 2005, 2 (H01), 9 pp. Leeds University Press. Pdf
U. Hämarik, T. Raus (2004). On the choice of the regularization parameter for solving self-adjoint ill-posed problems with the approximately given noise level of data. Proceedings of the Estonian Academy of Sciences. Physics. Mathematics, 53 (2), 77–83. Issue on Google Books
U. Hämarik, T. Raus (2004). On the regularization parameter choice in case of approximately given error bounds of data. In M. Feistauer, V. Dolejši, P. Knobloch; K. Najzar (Eds.), Numerical Mathematics and Advanced Applications. Proceedings of ENUMATH 2003, the 5th European Conference on Numerical Mathematics and Advanced Applications, Prague, Aug 18–22, 2003, 400–409. Springer. doi:10.1007/978-3-642-18775-9_37
T. Raus (1987). About the discrepancy principle when the level of the error of the data is given approximateley. Acta et Commentationes Universitatis Tartuensis, 762, 47–58. In Russian. Pdf of issue

See also III Palm (2010), IV Raus (1992), IV Raus (1990), V Hämarik, Palm, Raus (2011), V Hämarik, Palm, Raus (2009).

VII. Choice of the regularization parameter by the monotone error rule

U. Hämarik, U. Kangro, R. Palm, T. Raus, U. Tautenhahn (2014). Monotonicity of error of regularized solution and its use for parameter choice. Inverse Problems in Science and Engineering, 22 (1), 10–30. doi:10.1080/17415977.2013.827185
U. Hämarik, T. Raus, R. Palm (2008). On the analog of the monotone error rule for parameter choice in the (iterated) Lavrentiev regularization. Computational Methods in Applied Mathematics, 8 (3), 237–252. doi:10.2478/cmam-2008-0017
U. Hämarik, U. Tautenhahn (2003). On the monotone error rule for choosing the regularization parameter in ill-posed problems. In M. M. Lavrent'ev, S. I. Kabanikhin (Eds.), Ill-Posed and Non-Classical Problems of Mathematical Physics and Analysis. Proceedings of the International Conference, Samarkand, Uzbekistan, September 2000, vol 41 of Inverse and Ill-Posed Problems Series, 27–55. VSP, Utrecht-Boston. Pdf
U. Hämarik, U. Tautenhahn (2001). On the monotone error rule for parameter choice in iterative and continuous regularization methods. BIT Numerical Mathematics, 41 (5), 1029–1038. doi:10.1023/A:1021945429767
U. Hämarik, U. Tautenhahn (2000). The monotone error rule for parameter choice in regularization methods. In P. Neittaanmäki, T. Tiihonen, P. Tarvainen (Eds.), Numerical Mathematics and Advanced Applications. Proceedings of the 3rd European Conference ENUMATH 99, Jyväskylä, Finland, 26–30 July 1999, 518–525. World Scientific Publishing, Singapore. Pdf
U. Tautenhahn, U. Hämarik (1999). The use of monotonicity for choosing the regularization parameter in ill-posed problems. Inverse Problems, 15 (6), 1487–1505. doi:10.1088/0266-5611/15/6/307
U. Hämarik, T. Raus (1999). On the a posteriori parameter choice in regularization metods. Proceedings of the Estonian Academy of Sciences. Physics. Mathematics, 48 (2), 133–145. Issue on Google Books
U. Hämarik (1999). Monotonicity of error and choice of the stopping index in iterative regularization methods. In A. Pedas (Ed.), Differential and Integral Equations. Theory and Numerical Analysis, 15–30. Estonian Mathematical Society, Tartu. Pdf

See also III Palm (2010), IV Hämarik, Raus (2002), V Hämarik, Palm, Raus (2011), V Hämarik, Palm, Raus (2009), V Hämarik, Palm (2007).

VIII. Choice of the regularization parameter by (modified) discrepancy principle and by balancing principle

U. Hämarik, T. Raus (2009). About the balancing principle for choice of the regularization parameter. Numerical Functional Analysis and Optimization, 30 (9&10), 951–970. doi:10.1080/01630560903393139
T. Raus, U. Hämarik (2008). About the balancing principle for choice of the regularization parameter. Journal of Physics: Conference Series, 135, 012087, 8 pp. doi:10.1088/1742-6596/135/1/012087
T. Raus (1999). On the posteriori parameter choice for the determination of a quasisolution. In A. Pedas (Ed.), Differential and Integral Equations. Theory and Numerical Analysis, 81–96, Tartu.
T. Raus (1992). On the regularization parameter choice for ill-posed problems with a quasisolution. In A. N. Tikhonov (Ed.), Ill-Posed Problems in Natural Sciences. Proceedings of the International Conference Held in Moscow, August 19–25, 1991, 144–152. VSP, Utrecht-Moscow. Preview on Google Books
G. Vainikko (1991). Parameter choice in Tikhonov regularization of ill-posed extremal problems. In Estimation and Control of Distributed Parameter Systems. Proceedings of an International Conference on Control and Estimation of Distributed Parameter Systems, Vorau, July 8–14, 1990, vol 100 of International Series of Numerical Mathematics, 355–365. Birkhäuser, Basel. doi:10.1007/978-3-0348-6418-3_25
T. Raus (1989). About the choice of parameter for the class of regularization methods in case of random errors. Acta et Commentationes Universitatis Tartuensis, 863, 86–94. In Russian. Pdf of issue
T. Raus (1988). About the choice of the regularization parameter in case of random errors of initial data. Acta et Commentationes Universitatis Tartuensis, 833, 107–122. In Russian. Pdf of issue
G. Vainikko (1983). The critical level of discrepancy in regularization methods. Zhurnal Vychislitel'noi Matematiki i Matematicheskoi Fiziki 23 (6), 1283–1297. In Russian. Article on mathnet.ru Translation: USSR Computational Mathematics and Mathematical Physics, 23 (6), 1–9.
G. Vainikko (1982). The discrepancy principle for a class of regularization methods. Zhurnal Vychislitel'noi Matematiki i Matematicheskoi Fiziki 22 (3), 499–515. In Russian. Article on mathnet.ru Translation: USSR Computational Mathematics and Mathematical Physics, 22 (3), 1–19.
G. Vainikko (1980). Error estimates of the successive approximation method for ill-posed problems. Avtomatika i telemekhanika, 3 (1), 84–92. In Russian. Article on mathnet.ru Translation: Automation and Remote Control, 41 (3), 356–363.

See also I Vainikko (1991), I Vainikko, Veretennikov (1986), I Vainikko (1984), I Vainikko (1982), II Bruckner, Prössdorf, Vainikko (1996), II Hämarik (1990), II Hämarik (1988), II Vainikko, Hämarik (1988), III Hämarik, Palm, Raus (2010), III Palm (2010), III Hämarik, Palm, Raus (2008), III Hämarik, Palm, Raus (2007), IV Raus, Hämarik (2012), IV Raus, Hämarik (2007), IV Hämarik, Raus (2002), IV Raus (1992), IV Raus (1990), IV Raus (1985), IV Raus (1984), VII Hämarik, Raus (1999).

IX. Regularized projection methods

U. Hämarik (1992). Quasioptimal error estimate for the regularized Ritz-Galerkin method with the a-posteriori choice of the parameter. Acta et Commentationes Universitatis Tartuensis, 937, 63–76. Pdf of issue
U. Hämarik (1992). On the discretization error in regularized projection methods with parameter choice by discrepancy principle. In A. N. Tikhonov (Ed.), Ill-Posed Problems in Natural Sciences. Proceedings of the International Conference Held in Moscow, August 19–25, 1991, 24–28. VSP, Utrecht-Moscow. Preview on Google Books Pdf
U. Hämarik (1992). On the discretization error in the regularized Ritz-Galerkin method for solving ill-posed problems. In M. Lomp et al (Eds.), Numerical Methods and Optimization, vol 3, 12–21. Estonian Academy of Sciences, Institute of Cybernetics, Tallinn.
G. Vainikko (1992). On the discretization and regularization of ill-posed problems with noncompact operators. Numerical Functional Analysis and Optimization 13 (3–4), 381–396. doi:10.1080/01630569208816485
R. Plato, G. Vainikko (1990). On the regularization of projection methods for solving ill-posed problems. Numerische Mathematik 57 (1), 63–79. doi:10.1007/BF01386397
R. Plato, G. Vainikko (1989). On the regularization of the Ritz-Galerkin method for solving ill-posed problems. Acta et Commentationes Universitatis Tartuensis, 863, 3–18. Pdf of issue

See also II Hämarik (1984, Regularized...), III Hämarik (1993).

X. Analysis of specific ill-posed problems

U. Tautenhahn, U. Hämarik, B. Hofmann, Y. Shao (2013). Conditional stability estimates for ill-posed PDE problems by using interpolation. Numerical Functional Analysis and Optimization, 34 (12), 1370–1417. doi:10.1080/01630563.2013.819515
U. Kangro (2012). Solution of three-dimensional electromagnetic scattering problems by interior source methods. In Th. E. Simos, G. Psihoyios, Ch. Tsitouras, Z. Anastassi (Eds.), ICNAAM 2012: International Conference on Numerical Analysis and Applied Mathematics 2012, Kos (Greece), 19–25 September 2012, vol 1479 of AIP Conference Proceedings, 2328–2331. American Institute of Physics. doi:10.1063/1.4756660
G. Vainikko (2012). First kind cordial Volterra integral equations 1. Numerical Functional Analysis and Optimization 33 (6), 680–704. doi:10.1080/01630563.2012.665260
U. Kangro (2010). Convergence of collocation method with delta functions for integral equations of first kind. Integral Equations and Operator Theory, 66 (2), 265–282. doi:10.1007/s00020-010-1748-0
G. Vainikko (2000). Fast solvers of the Lippmann-Schwinger equation. Direct and Inverse Problems of Mathematical Physics, vol 5 of International Society for Analysis, Applications and Computation, 423–440. Springer. doi:10.1007/978-1-4757-3214-6_25
G. Vainikko (1995). Inverse problem of ground water filtration: identifiability, discretization and regularization. In H. W. Engl, W. Rundell (Eds.), Inverse Problems in Diffusion Processes: Proceedings of the GAMM-SIAM Symposium held on Lake St. Wolfgang in the Austrian Alps, June 27–July 1, 1994, 90–107. SIAM, Philadelphia. Collection on Google Books
G. Vainikko, K. Kunisch (1993). Identifiability of the transmissivity coefficient in an elliptic boundary value problem. Zeitschrift für Analysis und ihre Anwendungen, 12 (2), 327–341.
G. Vainikko (1993). Identification of filtration coefficient. In Inverse Problems in Mathematical Physics. Proceedings of The Lapland Conference on Inverse Problems Held at Saariselkä, Finland, 14–20 June 1992, vol 422 of Lecture Notes in Physics, 251–256. Springer. doi:10.1007/3-540-57195-7_29
G. Vainikko (1992). Identification of filtration coefficient. In A. N. Tikhonov (Ed.), Ill-Posed Problems in Natural Sciences. Proceedings of the International Conference Held in Moscow, August 19–25, 1991, 202–213. VSP, Utrecht-Moscow. Preview on Google Books
G. M. Vainikko (1985). Error bounds in regularization methods for normally solvable problems. Zhurnal Vychislitel'noi Matematiki i Matematicheskoi Fiziki 25 (10), 1443–1456. In Russian. Article on mathnet.ru Translation: USSR Computational Mathematics and Mathematical Physics, 25 (5), 107–117.
G. Vainikko (1983). Optimal regularization of normally solvable problems. In Theory and methods for solving ill-posed problems and their applications. All-Union school-seminar abstracts, Samarkand, 1983, 23–29. Novosibirsk State University. In Russian.