Moment discretization for ill-posed problems with discrete weakly bounded noise
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Moment discretization for ill-posed problems with discrete weakly bounded noise P. P. B. Eggermont · V. N. LaRiccia · M. Z. Nashed
Received: 22 February 2012 / Accepted: 27 March 2012 / Published online: 9 April 2012 © Springer-Verlag 2012
Abstract We study moment discretization for compact operator equations in Hilbert space with discrete noisy data. Instead of assuming that the error in the data converges strongly to 0, we only assume weak convergence of the noise as introduced by Eggermont et al. (Inverse Probl 25:115018, 2009). A specific instance would be random noise. Under the usual source conditions, we derive optimal convergence rates for Phillips-Tikhonov regularization. The analysis is based on the comparison of the discrete problem with a semi-discrete version of the problem, which is made possible by virtue of a quadrature result in a suitable reproducing kernel Hilbert space. Some numerical results using strong and weak discrepancy principles for the selection of the regularization parameter are presented. Keywords Ill-posed problems · Moment discretization · Discrete weakly bounded noise · Tikhonov regularization · Reproducing kernel Hilbert spaces Mathematics Subject Classification 47A52 · 45Q05 · 45R05 · 46E25 · 60G35
P. P. B. Eggermont (B) · V. N. LaRiccia Food and Resource Economics, University of Delaware, Newark, DE 19717, USA e-mail: [email protected] V. N. LaRiccia e-mail: [email protected] M. Z. Nashed Department of Mathematics, University of Central Florida, Orlando, FL 32143, USA e-mail: [email protected]
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Int J Geomath (2012) 3:155–178
1 Introduction Let X and Y be Hilbert spaces, and let K : X → Y be a bounded linear operator with nonclosed range. Under these circumstances, the operator equation Kx = y
(1.1)
with y ∈ Y the given data, is ill-posed in the sense of Hadamard: The solution may not exist, and if it does, it will not depend continuously on the data. The standard example of an ill-posed problem is given by a Fredholm integral equations of the first kind when K is a compact integral operator. In the model (1.1), one usually assumes that y = K xo + δ
(1.2)
for some unknown xo ∈ X and unknown noise δ which is assumed to be small in some suitable sense. The classical assumption is that η = δ Y is small, def
(1.3)
and one investigates how well one can recover xo when η −→ 0. The hope is that this will allow one to make inferences regarding the small noise case. This approach originated with Tikhonov (1943) and started seriously with Phillips (1962), Tikhonov (1963), and Twomey (1963). See Morozov (1984), Groetsch (1984), Tikhonov et al. (1995) and Engl et al. (1996). In the context of partial differential equations, an early reference on restoring continuity of the inverse operator by means of constraints on the solution is John (1960). One may object to the model (1.3) on several grounds. One objection is that the “data” are usually finite-dimensional and not infinite-dimensional. Although one may argue that the model (1.2)–(1.3) is merely an abstraction, t
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