Abstract
The gravitational lensing phenomenon can provide us with the information about luminous and dark matter in our Universe. But robust and effective tools are needed to extract that valuable information from observations. In this chapter, two inverse problems arising in gravitational lensing research are considered. Both problems are ill-posed, so regularization is needed. The first problem is the one of image reconstruction. Based on the Tikhonov regularization approach, several modifications of the regularizing algorithm, taking account of specific properties of gravitational lens systems, are proposed. The numerical results show that incorporation of all available a priori information allows reconstructing images of gravitational lens systems quite well. Furthermore, the algorithm decomposes the images into two parts — point sources and smooth background, that is needed for further investigations. The second problem concerns reconstruction of a distant quasar light distribution based on observations of microlensing events. In this case, a gravitational lens system as a gravitational telescope and the Tikhonov regularization method as a tool allow getting valuable information about the distant quasar unresolved by an instrument.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
E. Agol and J. Krolik, Imaging a quasar accretion disk with microlensing, The Astrophysical Journal, 524(1), 49–64.
K. Chang and S. Refsdal, Star disturbances in gravitational lens galaxies, Astronomy and Astrophysics, 132(1), 168–178, 1984.
F. W. Dyson, A. S. Eddington and C. R. Davidson, A determination of the deflection of light by the sun’s gravitational field from observations made at the total eclipse of May 29, 1919. Mem. R. Astron. Soc., 62, 291, 1920.
A. Einstein, Uber den Einflub der Schwerkraft auf die Ausbreitung des Lichtes (On the Influence of Gravity on the Propagation of Light), Annalen der Physik, 35, 898, 1911.
A. V. Goncharskiy, A. S. Leonov and A. G. Yagola, On the solution of two-dimesional first kind Fredholm integral equations with kernel depending on the arguments difference, Zhurnal vychislitelnoy matematiki i matematicheskoy fiziki, 11(5), 1971 (in Russian).
B. Grieger, R. Kayser and T. Schramm, The deconvolution of the quasar structure from microlensing light curves, Astronomy and Astrophysics, 252(2), 508–512, 1991.
J. Hadamard, Lectures on Cauchy’s Problem in Linear Partial Differential Equations, Yale Univ. Press, New Haven, 1923.
C. S. Kochanek, E. E. Falco, C. Impey, J. Lehar, B. McLeod and H.-W. Rix, CfAArizona Space Telescope LEns Survey, http://www.cfa.harvard.edu/glensdata/.
E. Koptelova, E. Shimanovskaya, B. Artamonov, M. Sazhin, A. Yagola, V. Bruevich, O. Burkhonov, Image reconstruction technique and optical monitoring of the QSO2237+0305 from Maidanak Observatory in 2002–2003, Monthly Notices of the Royal Astronomical Society, 356(1), 323–330, 2005.
E. Koptelova, E. Shimanovskaya, B. Artamonov and A. Yagola, Analysis of the Q2237+0305 light-curve variability with regularization technique, Monthly Notices of the Royal Astronomical Society, 381(4), 1655–1662, 2007.
A. S. Leonov, Numerical piecewise-uniform regularization for two-dimensional ill-posed problems, Inverse Problems, 15, 1165–1176, 1999.
A. S. Leonov, A. N. Tikhonov and A. G. Yagola, Nonlinear Ill-Posed Problems, Chapman & Hall, London, 1998.
S. Mineshige and A. Yonehara, Gravitational microlens mapping of a quasar accretion disk, Publ. of the Astronomical Society of Japan, 51, 497–504, 1999.
V. N. Shalyapin, L. J. Goicoechea, D. Alcalde, E. Mediavilla, J. A. Munoz and R. GilMerino, The nature and size of the optical continuum source in QSO 2237+0305, The Astrophysical Journal, 579(1), 127–135, 2002.
A. N. Tikhonov, et al., Numerical Methods for the Solution of Ill-Posed-Problems, Kluwer Academic Press, Dordrecht, 1995.
A. N. Tikhonov, On the solution of ill-posed problems and on the regularization method, Doklady akademii nauk SSSR, 151(3), 1963 (in Russian).
V. G. Vakulik, R. E. Schild, V. N Dudinov, A. A. Minakov, S. N. Nuritdinov, V. S. Tsvetkova, A. P. Zheleznyak, V. V. Konichek, I. Ye. Sinelnikov, O. A. Burkhonov, B. P. Artamonov and V. V. Bruevich, Color effects associated with the 1999 microlensing brightness peaks in gravitationally lensed quasar Q2237+0305, Astronomy and Astrophysics, 420, 447–457, 2004.
J. Wambsganss, H. J. Witt and P. Schneider, Gravitational microlensing — Powerful combination of ray-shooting and parametric representation of caustics, Astronomy and Astrophysics, 258(2), 591–599, 1992.
J. Wambsganss, Gravitational lensing in astronomy, Living Rev. Relativity, 1, http://www.livingreviews.org/lrr-1998-12, 1998.
H. J. Witt, R. Kayser and S. Refsdal, Microlensing predictions for the Einstein cross 2237+0305, Astronomy and Astrophysics, 268(2), 501–510, 1993.
J. S. B. Wyithe, R. L. Webster, E. L. Turner, MNRAS, Interpretation of the OGLE Q2237+0305 microlensing light curve (1997–1999), Monthly Notices of the Royal Astronomical Society, 318(4), 1120–1130, 2000.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2010 Higher Education Press, Beijing and Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Artamonov, B., Koptelova, E., Shimanovskaya, E., Yagola, A.G. (2010). Tikhonov Regularization for Gravitational Lensing Research. In: Wang, Y., Yang, C., Yagola, A.G. (eds) Optimization and Regularization for Computational Inverse Problems and Applications. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-13742-6_14
Download citation
DOI: https://doi.org/10.1007/978-3-642-13742-6_14
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-13741-9
Online ISBN: 978-3-642-13742-6
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)