Isolation of instability in the Fredholm integral equation of the first kind: Application to the deconvolution of noisy spectra

Price, Glenn L.

doi:10.1063/1.331307

Cited by 2 publications

(2 citation statements)

References 5 publications

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“…The problem of errors in deconvolution methods has been considerred by many authors (e.g. Jones and Misell 1970;Price 1982) and various ways for dealing with solution instability have been proposed.…”

Section: Convolution and Deconvolutionmentioning

confidence: 99%

Powder diffraction

Langford¹,

Louër²

1996

Rep. Prog. Phys.

364

249

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The powder diffraction method, by using conventional X-ray sources, was devised independently in 1916 by Debye and Scherrer in Germany and in 1917 by Hull in the United States. The technique developed steadily and, half a century later, the 'traditional' applications, such as phase identification, the determination of accurate unit-cell dimensions and the analysis of structural imperfections, were well established. There was then a dramatic increase of interest in powder methods during the 1970s, following the introduction by Rietveld in 1967 of his powerful method for refining crystal structures from powder data. This has since been used extensively, initially by using neutron data and later with X-rays, and it was an important step towards extracting 3-dimensional structural information from 1-dimensional powder diffraction patterns, in order to study the structure of crystalline materials. Similarly, techniques which do not involve structural data have been introduced for modelling powder diffraction patterns, to extract various parameters (position, breadth, shape, etc.) which define the individual reflections. These are used in most applications of powder diffraction and are the basis of new procedures for characterizing the microstructural properties of materials. Many subsequent advances have been based on this concept and powder diffraction is now one of the most widely used techniques available to materials scientists for studying the structure and microstructure of crystalline solids. It is thus timely to review progress during the past twenty years or so.Powder data have been used for the identification of unknown materials or mixtures of phases since the late 1930s. This is achieved by comparison of experimental data with standard data in crystallographic databases. The technique has benefited substantially from the revolution in the development of storage media during the last decade and from the introduction of fast search/match algorithms. Phase identification sometimes precedes a quantitative analysis of compounds present in a sample and powder diffraction is frequently the only approach available to the analyst for this purpose. A new development in quantitative analysis is the use of the Rietveld method with multi-phase refinement.A major advance in recent years has occurred in the determination of crystal structures ab initio from powder diffraction data, in cases where suitable single crystals are not available. This is a consequence of progress made in the successive stages involved in structure solution, e.g. the development of computer-based methods for determining the crystal system, cell dimensions and symmetry (indexing) and for extracting the intensities of Bragg reflections, the introduction of high resolution instruments and the treatment of

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Section: Convolution and Deconvolutionmentioning

confidence: 99%

Powder diffraction

Langford¹,

Louër²

1996

Rep. Prog. Phys.

364

249

View full text Add to dashboard Cite

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“…In the presence of noise on the right hand side, the direct inversion of this integral type leads to unbounded errors in the solution. This instability, demonstrated for our problem by the perturbation described above, is well known and the associated difficulties associated with spectral estimation are well described [24,39]. Experimental error is always present when gathering data, and therefore to achieve a stable estimate of the viscoelastic spectrum, we must account for this behavior in our solution method.…”

Section: Stable Fitting Of Stress Responsementioning

confidence: 87%

Stable fitting of noisy stress relaxation data

et al. 2019

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The mechanical behavior of a viscoelastic material can often be described by a spectrum of Maxwell-Wiechert elements. The inverse problem associated with estimating the parameters of this spectrum from the results of viscoelastic relaxation (ramp-and-hold) test is in general ill-posed and unstable, with estimates highly sensitive to initial conditions and noise. Here, we demonstrate stable estimation of a continuous viscoelastic spectrum from stress relaxation experiments using Tikhonov regularization. We assess the effects of noise and sampling frequency on these estimates, and describe regularization parameter selection. We demonstate the algorithm by estimating the viscoelastic relaxation spectra of soft vinyl samples.Keywords Viscoelastic material characterization · Viscoelastic spectrum · Noise · Ill-posed inverse problems · Tikhonov regularization IntroductionThe mechanical responses of nearly all structures and materials, especially soft materials, depend upon both the rate at which they are deformed, and their history of prior loading [19]. Engineers classify such materials as viscoelastic. Engineering models of viscoelastic behavior describe, on a phenomological level, the storage, transmission, and dissipation of energy within the material. Although typical engineering models of viscoelasticity do not attempt to directly describe the mechanisms governing these processes, they have been successfully used to study causal links between these small-scale processes and the overall mechanical response of the material in a variety of fields, ranging from biology to solid state physics [29].In the body, cells and tissues play both structural and mechanical roles. Studies have revealed mechanical responses consistent with the engineering understanding of viscoelasticity in collagen-rich extracellular matrices (ECMs) [41], single cells [32,53], and subcellular structures such as the nucleus [21]. With the individual building blocks displaying viscoelastic

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Isolation of instability in the Fredholm integral equation of the first kind: Application to the deconvolution of noisy spectra

Cited by 2 publications

References 5 publications

Powder diffraction

Powder diffraction

Stable fitting of noisy stress relaxation data

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