Are linear algorithms always good for linear problems?
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We exhibit linear problems for which every linear algorithm has infinite error, and show a (mildly) nonlinear algorithm with finite error. The error of this nonlinear algorithm can be arbitrarily small if appropriate information is used. We illustrate these examples by the inversion of a finite Laplace transform, a problem arising in remote sensing.
- Bakhvalov, N. S.,On the optimality of linear methods for operator approximation in convex classes of functions. U.S.S.R. Computational Math. and Math. Phys.11 (1971), 244–249.
- Dunn, H. S.,A generalization of the Laplace transform. Proc. Cambridge Philos. Soc.63 (1967), 155–160.
- Osipenko, K. Yu,Best approximation of analytic functions from information about their values at a finite number of points. Math. Notes19 (1976), 17–23.
- Packel, E. W.,Linear problems (with extended range) have linear optimal algorithms. Computer Science Department Research Report CUCS-106-84, Columbia University, New York, 1984. Aequationes Math.30 (1986), 18–25.
- Traub, J. F. andWoźniakowski, H.,A General theory of optimal algorithms. Academic Press, New York, 1980.
- Twomey, S.,Introduction to the mathematics of inversion in remote sensing and indirect measurement. Developments in Geomathematics3, Elsevier Scientific Publ., Amsterdam, 1977.
- Werschulz, A. G.,What is the complexity of ill-posed problems?. In preparation.
- Werschulz, A. G.,Optimal residual algorithms. In preparation.
- Are linear algorithms always good for linear problems?
Volume 31, Issue 1 , pp 202-212
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- Author Affiliations
- 1. Department of Computer Science, Columbia University, 10027, New York, NY, USA
- 2. Department of Computer Science, Columbia University, 10027, New York, NY, USA
- 3. Division of Science and Mathematics, Fordham University, College at Lincoln Center, 113 West 60th Street, 10023, New York, NY, USA
- 4. Institute of Informatics, University of Warsaw, PKiN, Warsaw, Poland