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Convergence rates for regularized solutions


Author: Mark A. Lukas
Journal: Math. Comp. 51 (1988), 107-131
MSC: Primary 65R20; Secondary 41A25, 45L05
DOI: https://doi.org/10.1090/S0025-5718-1988-0942146-8
MathSciNet review: 942146
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Abstract: Given a first-kind integral equation

$\displaystyle \mathcal{K}u(x) = \int_0^1 {K(x,t)u(t)\,dt = f(x)} $

with discrete noisy data $ {d_i} = f({x_i}) + {\varepsilon _i}$, $ i = 1,2, \ldots ,n$, let $ {u_{n\alpha }}$ be the minimizer in a Hilbert space W of the regularization functional $ (1/n)\sum {(\mathcal{K}} u({x_i}) - {d_i}{)^2} + \alpha \left\Vert u \right\Vert _W^2$. It is shown that in any one of a wide class of norms, which includes $ {\left\Vert \cdot \right\Vert _W}$, if $ \alpha \to 0$ in a certain way as $ n \to \infty $, then $ {u_{n\alpha }}$ converges to the true solution $ {u_0}$. Convergence rates are obtained and are used to estimate the optimal regularization parameter $ \alpha $.

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DOI: https://doi.org/10.1090/S0025-5718-1988-0942146-8
Article copyright: © Copyright 1988 American Mathematical Society

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