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

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper we derive conditions under which the finite-dimensional constrained Tikhonov-regularized solutions of an ill-posed linear operator equation (i.e., is the minimizing element of the functional , in the closed convex set , 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 . We present numerical results that verify the theoretical results.

**[1]**R. W. Cottle & G. B. Dantzig, "Complementary pivot theory of mathematical programming,"*Linear Algebra Appl.*, v. 1, 1968, pp. 103-125. MR**0226929 (37:2515)****[2]**H. W. Engl & A. Neubauer, "An improved version of Marti's method for solving ill-posed linear integral equations,"*Math. Comp.*, v. 45, 1985, pp. 405-416. MR**804932 (86j:65177)****[3]**R. Fletcher,*Practical Methods of Optimization*, Vol. 2, Wiley, New York, 1981. MR**633058 (83i:65055b)****[4]**C. W. Groetsch,*The Theory of Tikhonov Regularization for Fredholm Equations of the First Kind*, Pitman, Boston, 1984. MR**742928 (85k:45020)****[5]**C. W. Groetsch, J. T. King & D. Murio, "Asymptotic analysis of a finite element method for Fredholm equations of the first kind," in*Treatment of Integral Equations by Numerical Methods*(C. T. H. Baker and G. F. Miller, eds.), Academic Press, London, 1982, pp. 1-11. MR**755337 (85k:65107)****[6]**J. T. Marti, "On a regularization method for Fredholm equations of the first kind using Sobolev spaces," in*Treatment of Integral Equations by Numerical Methods*(C. T. H. Baker and G. F. Miller, eds.), Academic Press, London, 1982, pp. 59-66. MR**755342****[7]**V. A. Morozov,*Methods for Solving Incorrectly Posed Problems*, Springer, New York, 1984. MR**766231 (86d:65005)****[8]**U. Mosco, "Convergence of convex sets and of solutions of variational inequalities,"*Adv. in Math.*, v. 3, 1969, pp. 510-585. MR**0298508 (45:7560)****[9]**A. Neubauer, "Tikhonov-regularization of ill-posed linear operator equations on closed convex sets,"*J. Approx. Theory*. (To appear.) MR**947434 (89h:65094)****[10]**A. Neubauer,*Tikhonov-Regularization of Ill-Posed Linear Operator Equations on Closed Convex Sets*, Ph. D. Thesis, University of Linz, 1985, will appear as a book in VWGÖ (= Verlag der Wissenschaftlichen Gesellschaften Österreichs), Vienna. MR**944529 (89h:65093)****[11]**J. T. Oden & E. B. Pires, "Error estimates for the approximation of a class of variational inequalities arising in unilateral problems with friction,"*Numer. Funct. Anal. Optim.*, v. 4, 1981/82, pp. 397-412. MR**673320 (83m:49011)**

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

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

Additional Information

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

Article copyright:
© Copyright 1987
American Mathematical Society