A regularized solution with weighted Bregman distances for the inverse problem of photoacoustic spectroscopy

Neto, Antônio José da Silva; Cella, N.

doi:10.1590/s0101-82052006000200003

Cited by 6 publications

(2 citation statements)

References 42 publications

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“…In this paper we focus on investigating the latter two aspects. Though the structural characterization on which local mechanical properties depend also plays a vital role in inverse problems arising from nanotechnology (see, e.g., Mlinar et al [14], Cidade et al [7,18] and Li et al [11]), it is beyond the scope of this paper. Before proceeding to propose the new model, we review some widely used models for quantitative study of mechanical properties of one dimension uniform nanomaterials.…”

Section: Introductionmentioning

confidence: 99%

Identification of the material properties in nonuniform nanostructures

Bao

2015

Inverse Problems

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This paper is concerned with addressing two significant challenges arising from quantifying mechanical properties of nanomaterials, namely nonuniformity of the nanomaterial and the high noise level of measurements. For nonuniformity, an explicit solution is derived for the general Euler-Bernoulli equation in terms of the Green function for the Poisson equation. Then, by examining a stochastic source, the systematic error may be removed from measurements, which leads to more accurate estimation of mechanical properties. Based on Itô integral properties, three deterministic Fredholm integral equations can be deduced to extract the stiffness and the structure of the random source from measured data. To overcome ill-posedness and high nonlinearity in solving the Fredholm equations, a Tikhonov regularization method is developed with an a priori strategy of choosing the regularization parameter. Moreover, under a regularity assumption for the stiffness coefficient and structures of the random source, the convergence rate can be obtained in the sense of probability. Numerical examples are presented to illustrate the validity and effectiveness of the novel model and regularization method.

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Section: Introductionmentioning

confidence: 99%

Identification of the material properties in nonuniform nanostructures

Bao

2015

Inverse Problems

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“…The atomic force microscope (AFM) and the scanning tunneling microscope (STM) are two versions of scanning probes utilized in nanotechnology, which have made it possible to see structures and deformation at the nanoscale in real space. Though structural characterization on which local mechanical properties depend plays a vital role in inverse problems arising from nanotechnology, Mlinar et al [25], Cidade and Silva et al [9,32] and Li et al [20], we do not intend to discuss it in this paper. Instead, we focus on quantifying the elastic modulus from AFM measurements, which is considered as one of many steps in nanotechnology.…”

Section: Introductionmentioning

confidence: 99%

An inverse random source problem in quantifying the elastic modulus of nanomaterials

Bao

2012

Inverse Problems

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Nanotechnology deals with structures sized between 1 and 100 nanometer and involves manipulating and controlling materials or devices within that scale. Due to its small size, it is difficult to quantify the mechanical properties of nanomaterials which significantly rely on measurement techniques, conditions and environment. In this paper, to describe the elastic deformation of nanobelts obtained from an atomic force microscope (AFM) under contact mode, we propose a novel model based on a Euler-Bernoulli equation with a stochastic source term accounting for effects of initial bending, surface roughness and white noise during measurements. With the random source, the forward problem is demonstrated to have a unique and explicit path-wise solution. Furthermore, the inverse problem consists of identifying the elastic modulus of nanobelts and reconstructing the random source structure, i.e. the mean and the variance. Based upon explicit formulae of direct problems, the corresponding inverse problem can be reduced into a first kind of Fredholm-type integral equation where two kinds of regularization techniques are applied to obtain stable solutions, i.e. Tikhonov regularization (TR) for smooth solutions and a statistical regularization method from Bayes' formula and maximum likelihood estimation for discontinuous solutions, respectively. Numerical examples are presented to illustrate the validity and effectiveness of the proposed methods.

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Interval inverse analysis of hyperbolic heat conduction problem

Tang

Xue

2014

International Communications in Heat and Mass Transfer

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A regularized solution with weighted Bregman distances for the inverse problem of photoacoustic spectroscopy

Cited by 6 publications

References 42 publications

Identification of the material properties in nonuniform nanostructures

Identification of the material properties in nonuniform nanostructures

An inverse random source problem in quantifying the elastic modulus of nanomaterials

Interval inverse analysis of hyperbolic heat conduction problem

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