Termination conditions for approximating linear problems with noisy information

Authors:
B. Z. Kacewicz and L. Plaskota

Journal:
Math. Comp. **59** (1992), 503-513

MSC:
Primary 65J10; Secondary 41A65

DOI:
https://doi.org/10.1090/S0025-5718-1992-1142284-4

MathSciNet review:
1142284

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Abstract | References | Similar Articles | Additional Information

Abstract: We study the diameter termination criterion for approximating linear continuous problems. It is assumed that only nonexact information about the problem is available. We evaluate the quality of the diameter termination criterion by comparing it with the theoretically best stopping condition. The comparison is made with respect to the cost of computing an -approximation. Although the diameter termination criterion is independent of a particular problem, it turns out to be essentially equivalent to the theoretical condition. Optimal information and the best way of constructing an -approximation are exhibited.

**[1]**K. I. Babenko,*Theoretical background and constructing computational algorithms for mathematical-physical problems*, Nauka, Moscow, 1979. (Russian)**[2]**B. Z. Kacewicz and L. Plaskota,*On the minimal cost of approximating linear problems based on information with deterministic noise*, Numer. Funct. Anal. Optim., Nos. 5 and 6 (1990), 511-529. MR**1079289 (92e:65033)****[3]**-,*Noisy information for linear problems in the asymptotic setting*, J. Complexity**7**(1991), 35-57. MR**1096171 (92a:65370)****[4]**A. A. Marchuk and K. Y. Osipenko,*Best approximation of functions specified with an error at a finite number of points*, Math. Notes**17**(1975), 207-212.**[5]**C. A. Micchelli and T. J. Rivlin,*A survey of optimal recovery*, Estimation in Approximation Theory, Plenum Press, New York, 1977, pp. 1-54. MR**0617931 (58:29718)****[6]**J. F. Traub, G. W. Wasilkowski, and H. Woźniakowski,*Information-based complexity*, Academic Press, New York, 1988. MR**958691 (90f:68085)****[7]**G. M. Trojan,*Asymptotic setting for linear problems*, unpublished, see Traub et al. [6, pp. 383-395].

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DOI:
https://doi.org/10.1090/S0025-5718-1992-1142284-4

Article copyright:
© Copyright 1992
American Mathematical Society