Practical Approximate Solutions to Linear Operator Equations When the Data are Noisy

Wahba, Grace

doi:10.1137/0714044

Cited by 698 publications

(402 citation statements)

References 19 publications

Supporting

Mentioning

396

Contrasting

Unclassified

Order By: Relevance

“…This is an inherently ill-posed problem and requires some constraints to produce a stable solution. Here we use Tikhonov regularisation [52,53] with the smoothing parameter determined by the Generalized Cross Validation (GCV) method [54,55]. In addition, scanning electron microscopy (SEM) images were acquired to analyse the pore structure.…”

Section: Methodsmentioning

confidence: 99%

Magnetic resonance imaging studies of spontaneous capillary water imbibition in aerated gypsum

Song¹,

Mitchell²,

Jaffel³

et al. 2011

J. Phys. D: Appl. Phys.

View full text Add to dashboard Cite

In this paper we investigate both capillary water imbibition and the sorptivity of aerated gypsum plaster, and how these sorption characteristics are related to the pore structure of the material. These characteristics are examined by monitoring mass change using the conventional gravimetric method and by obtaining water content profiles using non-destructive magnetic resonance imaging (MRI) techniques during capillary imbibition of water. Here, three different gypsum samples are investigated: one non-aerated reference gypsum sample and two aerated gypsum samples produced with different volumetric air fractions. The capillary water absorption into the reference sample follows t 1/2 kinetics (Fickian diffusion), where t is the time of ingress. However, in the aerated gypsum samples there are deviations from t 1/2 kinetics. The MRI results show unambiguously that two wetting fronts advance through the aerated structure; an observation that cannot be made from the gravimetric data alone. The water content profiles of the aerated gypsum samples are therefore analysed by treating them as the sum of two separate absorption processes using sharp front analysis. The capillary water absorption properties of this material are well described as a parallel combination of fast absorption into fine matrix pores and slow absorption into a modified structure of matrix pores inter-connected to air voids introduced into the slurry by aeration.

show abstract

Section: Methodsmentioning

confidence: 99%

Magnetic resonance imaging studies of spontaneous capillary water imbibition in aerated gypsum

Song¹,

Mitchell²,

Jaffel³

et al. 2011

J. Phys. D: Appl. Phys.

View full text Add to dashboard Cite

show abstract

“…(5) becomes which should minimize the inner product of Eq. (8) with itself, IIK·f-SII· (10) Unfortunately, these solutions are numerically unstable, because Eq. (8) is ill posed.…”

Section: Conversion To a Fredholm Integral Equation Of Tile First mentioning

confidence: 99%

On the numerical solution of the Hill–Wheeler equation

Wampler

Wilets

1988

Computers in Physics

View full text Add to dashboard Cite

A particular numerical method is discussed for solving the Hill–Wheeler equation, which is an integral equation that arises in the generator coordinate method analysis of problems in nuclear physics. The method is applicable to scattering problems with finite range potentials and exploits prior knowledge of the asymptotic form of the solution to convert the Hill–Wheeler equation into a Fredholm integral equation of the first kind. The resulting Fredholm equation is ill-posed, which often results in numerically unstable solutions. The method of regularization is used to produce numerically stable, approximate solutions. Optimizing the choice of regularization parameter is discussed and calculations are presented for an example problem.

show abstract

“…Morozov's discrepancy principle [9], generalized crossvalidation [10], the L-curve method [11]) we want to emphasize the Lepskiibalancing principle [12] which has been shown to be adaptive in the sense that it adapts automatically within a scale of Hilbert spaces in order to select the optimal regularization parameter in a minimax sense. We mention in particular [13] and [14].…”

Section: Introductionmentioning

confidence: 99%