Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)

 
 

 

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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Given a first-kind integral equation \[ \mathcal {K}u(x) = \int _0^1 {K(x,t)u(t) dt = f(x)} \] 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 $(1/n)\sum {(\mathcal {K}} u({x_i}) - {d_i}{)^2} + \alpha \left \| u \right \|_W^2$. It is shown that in any one of a wide class of norms, which includes ${\left \| \cdot \right \|_W}$, 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 $\alpha$.


References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Mathematics of Computation with MSC: 65R20, 41A25, 45L05

Retrieve articles in all journals with MSC: 65R20, 41A25, 45L05


Additional Information

Article copyright: © Copyright 1988 American Mathematical Society