×

A generalization of the maximal entropy method for solving ill-posed problems. (English. Russian original) Zbl 0970.47051

Sib. Math. J. 41, No. 4, 716-724 (2000); translation from Sib. Mat. Zh. 41, No. 4, 863-872 (2000).
Let \(Z(T)\) be a space of functions \(z(x)\), \(x\in\mathbb R^N\), defined on a closed bounded domain. The space \(Z(T)\) is furnished with some sequential convergence topology \(\tau\). Let \(A\) be an operator from \(Z(T)\) into a normed space \(U\). A nonempty set \(D\subset Z(T)\) is fixed and the following operator equation \[ Az = u,\quad z = z(x),\;u\in U, \] is considered on the set \(D\). It is assumed that the set \(Z^* = \arg\inf\{ \|Az - u\|_U: z\in D\}\) of quasisolutions to this equation is nonempty. For finding special quasisolutions to the equation, some functional \(\Omega(z)\) on \(D\) is used. This functional is selected by the so-called \(\Omega\)-optimal quasisolution to the equation, i.e., functions \(\overline z(x)\in Z^*\) for which \[ \Omega(\overline z) = \inf\{\Omega(z): z\in Z^*\}. \] Denote the set of such functions by \(\overline{Z}\). The above-mentioned equation represents an ill-posed problem in which some approximations \(\{A_h, u_{\delta}\}\) with accuracy \(\eta = \{h, \delta\}\) are known instead of the exact data \(\{A, u\}\). In this case, given \(\{A_h, u_{\delta}, h, \delta\}\), it is required to construct a stable approximate solution to the problem, i.e., a function \(z_{\eta}\in D\) such that \(z_{\eta}(x)\overset\tau\rightarrow\overline{Z}\) as \(\eta\to 0\). These are the specific properties of the regularizing functional \(\Omega(z)\) on \(D\) that are well known to guarantee solvability of the problem and possibility of finding stable approximate solutions \(z_{\eta}(x)\) to the inverse problem. In this case the problem under consideration with \(\Omega(z) = \Omega_E(z)\), \(\Omega_E(z) = \int_Tz(x)\ln z(x) dx\) is equivalent to maximizing the entropy functional \(E(z) = -\Omega_E(z)\) on the set of all nonnegative quasisolutions to the equation. The properties of the functional \(\Omega_E\) were studied by U. Amato and W. Hughes in [Inver. Prob. 7, No. 6, 793-808 (1991; Zbl 0752.65091)] and by P. P. B. Eggermont [SIAM J. Math. Anal. 24, No. 6, 1557-1576 (1993; Zbl 0791.65099)] in connection with solving ill-posed linear problems by the residual method and the Tikhonov regularization method. The main result of these articles is the theorem on strong convergence of the approximate solutions \(z_{\eta}(x)\) obtained by these methods to a unique exact solution to the equation with maximal entropy in the space \(L_1(T)\) as \(\eta\to 0\).
It turns out that such a result can be obtained for functionals different from \(\Omega_E\) in the problem. Convergence of approximate solutions in \(L_1(T)\) to \(\Omega\)-optimal quasisolutions in this space can be also obtained for a rather wide class of integral functionals \(\Omega(z)\) and for the general case of nonlinear operator equations which are studied by the author. In the article under review, the author introduces and studies such functionals \(\Omega(z)\). The corresponding problem (see the above-mentioned problem) represents a generalization of the maximal entropy method.

MSC:

47J06 Nonlinear ill-posed problems
47J35 Nonlinear evolution equations
47B06 Riesz operators; eigenvalue distributions; approximation numbers, \(s\)-numbers, Kolmogorov numbers, entropy numbers, etc. of operators
PDFBibTeX XMLCite
Full Text: DOI EuDML

References:

[1] Tikhonov A. N. andArsenin V. Ya., Methods for Solving Ill-Posed Problems [in Russian], Nauka, Moscow (1979).
[2] Ivanov V. K., Vasin V. V., andTanana V. P., Theory of Linear Ill-Posed Problems and Its Applications [in Russian], Nauka, Moscow (1978). · Zbl 0489.65035
[3] Morozov V. A., Regular Methods for Solving Ill-Posed Probelms [in Russian], Nauka, Moscow (1987).
[4] Tikhonov A. N., Leonov A. S., andYagola A. G., Nonlinear Ill-Posed Problems [in Russian], Nauka, Moscow (1995). · Zbl 0843.65041
[5] Wernecke S. J. andD’Addario L. R., ”Maximum entropy image reconstruction,” IEEE Trans. Comput., C-26, 351–64 (1977). · Zbl 0359.62078 · doi:10.1109/TC.1977.1674845
[6] Amato U. andHughes W., ”Maximum entropy regularization of Fredholm integral equations of the first kind,” Inverse Problems,7, 793–808 (1991). · Zbl 0752.65091 · doi:10.1088/0266-5611/7/6/004
[7] Eggermont P. P. B., ”Maximum entropy regularization for Fredholm integral equations of the first kind,” SIAM J. Math. Anal.,24, No. 6, 1557–1576 (1993). · Zbl 0791.65099 · doi:10.1137/0524088
[8] Kolmogorov A. N. andFomin S. V., Elements of the Theory of Functions and Functional Analysis [in Russian], Nauka, Moscow (1968).
[9] Vasil’ev F. P., Methods for Solving Extremal Problems [in Russian], Nauka, Moscow (1981).
[10] Krasnosel’skiî M. A., Zabreîko P. P., Pustyl’nik E. I., andSobolevskiî P. E., Integral Operators in Spaces of Summable Functions [in Russian], Nauka, Moscow (1966).
[11] Edwards R. D., Functional Analysis. Theory and Applications [Russian translation], Mir, Moscow (1969). · Zbl 0189.12103
[12] Leonov A. S., ”Some algorithms for solving ill-posed extremal problems,” Mat. Sb.,129, No. 2, 218–231 (1986). · Zbl 0604.65042
[13] Leonov A. S., ”Functions of several variables with bounded variation in ill-posed problems,” Zh. Vychisl. Mat. i Mat. Fiziki,36, No. 9, 35–49 (1996). · Zbl 0915.65057
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.