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Assessing regularised solutions

Published online by Cambridge University Press:  17 February 2009

M. A. Lukas
M. A. Lukas
Affiliation:
School of Mathematical and Physical Sciences, Murdoch University, Murdoch, W. A. 6150, Australia.
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Abstract

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Consider the prototype ill-posed problem of a first kind integral equation ℛ with discrete noisy data di, = f(xi) + εi, i = 1, …, n. Let u0 be the true solution and unα a regularised solution with regularisation parameter α. Under certain assumptions, it is known that if α → 0 but not too quickly as n → ∞, then unα converges to u0. We examine the dependence of the optimal sequence of α and resulting optimal convergence rate on the smoothness of f or u0, the kernel K, the order of regularisation m and the error norm used. Some important implications are made, including the fact that m must be sufficiently high relative to the smoothness of u0 in order to ensure optimal convergence. An optimal filtering criterion is used to determine the order where is the maximum smoothness of u0. Two practical methods for estimating the optimal α, the unbiased risk estimate and generalised cross validation, are also discussed.

Type
Research Article
Information
The ANZIAM Journal , Volume 30 , Issue 1 , July 1988 , pp. 24 - 42
Copyright
Copyright © Australian Mathematical Society 1988

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