Finite-dimensional approximation of constrained Tikhonov-regularized solutions of ill-posed linear operator equations

Author:
A. Neubauer

Journal:
Math. Comp. **48** (1987), 565-583

MSC:
Primary 65J10; Secondary 65R20

DOI:
https://doi.org/10.1090/S0025-5718-1987-0878691-2

MathSciNet review:
878691

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper we derive conditions under which the finite-dimensional constrained Tikhonov-regularized solutions ${x_{\alpha ,{C_n}}}$ of an ill-posed linear operator equation $Tx = y$ (i.e., ${x_{\alpha ,{C_n}}}$ is the minimizing element of the functional ${\left \| {Tx - y} \right \|^2} + \alpha {\left \| x \right \|^2}$, $\alpha > 0$ in the closed convex set ${C_n}$, which is a finite-dimensional approximation of a closed convex set *C*) converge to the best approximate solution of the equation in *C*. Moreover, we develop an estimate for the approximation error, which is optimal for certain sets *C* and ${C_n}$. We present numerical results that verify the theoretical results.

- Richard W. Cottle and George B. Dantzig,
*Complementary pivot theory of mathematical programming*, Linear Algebra Appl.**1**(1968), no. 1, 103–125. MR**226929**, DOI https://doi.org/10.1016/0024-3795%2868%2990052-9 - Heinz W. Engl and Andreas Neubauer,
*An improved version of Marti’s method for solving ill-posed linear integral equations*, Math. Comp.**45**(1985), no. 172, 405–416. MR**804932**, DOI https://doi.org/10.1090/S0025-5718-1985-0804932-1 - R. Fletcher,
*Practical methods of optimization. Vol. 1*, John Wiley & Sons, Ltd., Chichester, 1980. Unconstrained optimization; A Wiley-Interscience Publication. MR**585160** - C. W. Groetsch,
*The theory of Tikhonov regularization for Fredholm equations of the first kind*, Research Notes in Mathematics, vol. 105, Pitman (Advanced Publishing Program), Boston, MA, 1984. MR**742928** - C. W. Groetsch, J. T. King, and D. Murio,
*Asymptotic analysis of a finite element method for Fredholm equations of the first kind*, Treatment of integral equations by numerical methods (Durham, 1982) Academic Press, London, 1982, pp. 1–11. MR**755337** - J. T. Marti,
*On a regularization method for Fredholm equations of the first kind using Sobolev spaces*, Treatment of integral equations by numerical methods (Durham, 1982) Academic Press, London, 1982, pp. 59–66. MR**755342** - V. A. Morozov,
*Methods for solving incorrectly posed problems*, Springer-Verlag, New York, 1984. Translated from the Russian by A. B. Aries; Translation edited by Z. Nashed. MR**766231** - Umberto Mosco,
*Convergence of convex sets and of solutions of variational inequalities*, Advances in Math.**3**(1969), 510–585. MR**298508**, DOI https://doi.org/10.1016/0001-8708%2869%2990009-7 - A. Neubauer,
*Tikhonov-regularization of ill-posed linear operator equations on closed convex sets*, J. Approx. Theory**53**(1988), no. 3, 304–320. MR**947434**, DOI https://doi.org/10.1016/0021-9045%2888%2990025-1 - Andreas Neubauer,
*Tikhonov-regularization of ill-posed linear operator equations on closed convex sets*, Dissertationen der Johannes-Kepler-Universität Linz [Dissertations of the Johannes Kepler University of Linz], vol. 58, Verband der Wissenschaftlichen Gesellschaften Österreichs (VWGÖ), Vienna, 1986. With a German introduction and summary. MR**944529** - E. B. Pires and J. T. Oden,
*Error estimates for the approximation of a class of variational inequalities arising in unilateral problems with friction*, Numer. Funct. Anal. Optim.**4**(1981/82), no. 4, 397–412. MR**673320**, DOI https://doi.org/10.1080/01630568208816125

Retrieve articles in *Mathematics of Computation*
with MSC:
65J10,
65R20

Retrieve articles in all journals with MSC: 65J10, 65R20

Additional Information

Article copyright:
© Copyright 1987
American Mathematical Society