An Iteration Formula for Fredholm Integral Equations of the First Kind

Landweber, L.

doi:10.2307/2372313

Cited by 968 publications

(502 citation statements)

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“…If one only considers the first order variations of the spectral flux though, one has to approximate the inverse of C. The degree of approximation can be controlled by using an iterative algorithm, such as that of Landweber (Landweber 1951). In that case, one can impose a correlation coefficient between the input and the output parameters to be equal to the desired degree of approximation.…”

Section: The Matisse Methodsmentioning

confidence: 99%

Automatic stellar spectra parameterisation in the IR Ca ii triplet region

Kordopatis

Recio–Blanco

Laverny

et al. 2011

A&A

124

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Context. Galactic archaeology aims to determine the evolution of the Galaxy from the chemical and kinematical properties of its individual stars. This requires the analysis of data from large spectroscopic surveys, with sample sizes in tens of thousands at present, with millions of stars being reached in the near future. Such large samples require automated analysis techniques and classification algorithms to obtain robust estimates of the stellar parameter values. Several on-going and planned spectroscopic surveys have selected their wavelength region to contain the IR Ca ii triplet (∼λλ 8500 Å) and the work presented in this paper focuses on the automatic analysis of such spectra. Aims. We aim to develop and test an automatic method by which one can obtain estimates of values of the stellar atmospheric parameters (effective temperature, surface gravity, overall metallicity) from a stellar spectrum. We also explore the degeneracies in parameter space, estimate the uncertainties in the derived parameter values and investigate the consequences of these limitations for achieving the goals of galactic archaeology. Methods. We investigated two algorithms, both of which compare the observed spectrum to a grid of synthetic spectra, but each uses a different mathematical approach for finding the optimum match and hence the best values of the stellar parameters. Our investigation of these algorithms' robustness can be widely applied because it amplifies the main problems that the other methods can encounter. The first algorithm, MATISSE, derives the values of each stellar parameter through a local fit to the spectrum such that each pixel in wavelength space is treated separately. The sensitivity of the flux at each wavelength to the value of a given stellar parameter is determined from the synthetic spectra. The observed spectrum is then projected using these sensitivity vectors to give an estimated value of the stellar parameters. This value depends on finding the true minimum in the fit and the algorithm must avoid being trapped in false local minima. The second algorithm, DEGAS, uses a pattern-recognition approach and consequently has a more global vision of the parameter space. The best-fit synthetic spectrum is derived through a series of comparisons between the observed and synthetic spectra, summed over wavelength pixels, with additional refinements in the set of synthetic spectra after each stage, i.e. a decision tree. Results. We identified physical degeneracies in different regions of the H − R diagram: hot dwarf and giant stars share the same spectral signatures. Furthermore, it is very difficult to determine an accurate value for the surface gravity of cooler dwarfs. These effects are intensified when the lack of information increases, which happens for low-metallicity stars or spectra with low signal-tonoise ratios (SNRs). Our results demonstrate that the local projection method is preferred for spectra with high SNR, whereas the decision-tree method is preferred for spectra of lower SNR. We therefo...

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Section: The Matisse Methodsmentioning

confidence: 99%

Automatic stellar spectra parameterisation in the IR Ca ii triplet region

Kordopatis

Recio–Blanco

Laverny

et al. 2011

A&A

124

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“…The presence of the projection T is necessary because σ n might not be in S. Using the definition of J we can show that the optimal control algorithm (4.1) is nothing else than the following projected Landweber iteration [9,11,14] given by…”

Section: The Reconstruction Methods 41 Optimization Schemementioning

confidence: 99%

Determining anisotropic conductivity using diffusion tensor imaging data in magneto-acoustic tomography with magnetic induction

et al. 2017

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In this paper we present a mathematical and numerical framework for a procedure of imaging anisotropic electrical conductivity tensor by integrating magneto-acoutic tomography with data acquired from diffusion tensor imaging. Magneto-acoustic Tomography with Magnetic Induction (MAT-MI) is a hybrid, non-invasive medical imaging technique to produce conductivity images with improved spatial resolution and accuracy. Diffusion Tensor Imaging (DTI) is also a noninvasive technique for characterizing the diffusion properties of water molecules in tissues. We propose a model for anisotropic conductivity in which the conductivity is proportional to the diffusion tensor. Under this assumption, we propose an optimal control approach for reconstructing the anisotropic electrical conductivity tensor. We prove convergence and Lipschitz type stability of the algorithm and present numerical examples to illustrate its accuracy and feasibility.

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“…One commonly proposed solution [14,15] is to take the gradient of | | with respect to and iteratively subtract a scalar multiple of it from an initial guess . The result is a form of gradient descent called Landweber iteration:…”

Section: A Landweber Iterationmentioning

confidence: 99%

Optimisation of a tomography sensor for imaging of temperature in a gas turbine engine

Wood

Ozanyan

2013

2013 Ieee Sensors

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Abstract-We present a numerical study of tomographic reconstruction of gas temperature distributions from synthetic data of laser absorption measurements through the imaging plane. A mathematical expression which predicts the reconstruction error as a function of the beam configuration is developed, and a global optimization scheme is employed to minimize this predicted error subject to a constraint on the maximum number of permitted beams. This optimization methodology is applied to an annular measurement region and compared to conventional regular fan-beam arrangements in temperature tomography simulations. Synthetic data generated over the optimized beams led to temperature reconstructions with superior accuracy, provided Tikhonov inversion was used as the method of reconstruction.

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An Iteration Formula for Fredholm Integral Equations of the First Kind

Cited by 968 publications

References 0 publications

Automatic stellar spectra parameterisation in the IR Ca ii triplet region

Automatic stellar spectra parameterisation in the IR Ca ii triplet region

Determining anisotropic conductivity using diffusion tensor imaging data in magneto-acoustic tomography with magnetic induction

Optimisation of a tomography sensor for imaging of temperature in a gas turbine engine

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