Inversion problem

The inverse problem would be well defined if we knew the temperature or the harmonic function... 

In integrated photoelasticity it is impossible to achieve a complete reconstruction of stresses in samples by only illuminating a system of parallel planes and using equilibrium equations of the elasticity theory. Theory of the fictitious temperature field allows one to formulate a boundary-value problem which permits to determine all components of the stress tensor field in some cases. If the stress gradient in the axial direction is smooth enough, then perturbation method can be used for the solution of the inverse problem. As an example, distribution of stresses in a bow tie type fiber preforms is shown in Fig. 2 [2]. 

Puro, A. The inverse problem of thermoelasticity of optical tomography. J. Appl. Maths. Mechs. 1993, 57(1) I4I-I45. 

On certain Inverse Problems of Synthesis and Control in Optics. 

Where F and are some functional spaces, and the operator A connects (p and f We have a typical inverse problem [1,2]. There are two widely used models in optics, geometrical optics and Fresnel approximation... 

A V Goncharsky and V.V.Stepanov, Inverse Problems in Synthesis of Optical Elements, Ill-Posed Problems in the Natural Sciences, MIR Publishers, Moscow, (1987), pp.318-340. 

Secondly, the linearized inverse problem is, as well as known, ill-posed because it involves the solution of a Fredholm integral equation of the first kind. The solution must be regularized to yield a stable and physically plausible solution. In this apphcation, the classical smoothness constraint on the solution [8], does not allow to recover the discontinuities of the original object function. In our case, we have considered notches at the smface of the half-space conductive media. So, notche shapes involve abrupt contours. This strong local correlation between pixels in each layer of the half conductive media suggests to represent the contrast function (the object function) by a piecewise continuous function. According to previous works that we have aheady presented [14], we 2584... 

This paper is structured as follows in section 2, we recall the statement of the forward problem. We remind the numerical model which relates the contrast function with the observed data. Then, we compare the measurements performed with the experimental probe with predictive data which come from the model. This comparison is used, firstly, to validate the forward problem. In section 4, the solution of the associated inverse problem is described through a Bayesian approach. We derive, in particular, an appropriate criteria which must be optimized in order to reconstruct simulated flaws. Some results of flaw reconstructions from simulated data are presented. These results confirm the capability of the inversion method. The section 5 ends with giving some tasks we have already thought of. 

JW. Enquire, WE. Deeds, and CV. Dodd. Alternating current distribution between planar conductors. Journal of Applied Physics, 41(10) 3983-3991, September 1970. C. De Mol M. Bertero and E.R. Pike. Linear inverse problems with discrete data. li. stability and regularization. Inverse Problems, 4 pp. 573-594, 1987. 

H.A. Sabbagh and R.G. Lautzenheiser. Inverse problems in electromagnetic nondestructive evaluation. International Journal of Applied Electromagnetics in Materials, 3 253-2614, 1993. 

Progress in mean of modelisation and inverse problem solving [1] let us hope to dispose soon of these tools for flaws 3D imaging in Non Destructive Control with eddy current sensors. This will achieve a real improvement of the actual methods, mainly based upon signature analysis. But the actual eddy current probes used for steam generators tubes inspection in nuclear industry do not produce the adequate measurements and/or are not modelisable. 

More accurately, as the inverse problem process computes a quadratic error with every point of a local area around a flaw, we shall limit the sensor surface so that the quadratic error induced by the integration lets us separate two close flaws and remains negligible in comparison with other noises or errors. An inevitable noise is the electronic noise due to the coil resistance, that we can estimate from geometrical and physical properties of the sensor. Here are the main conclusions ... 

The equation system of eq.(6) can be used to find the input signal (for example a crack) corresponding to a measured output and a known impulse response of a system as well. This way gives a possibility to solve different inverse problems of the non-destructive eddy-current testing. Further developments will be shown the solving of eq.(6) by special numerical operations, like Gauss-Seidel-Method [4]. 

Setting specific conditions for the examined material, discontinuity position and transducer construction, and using relation (6) one can calculate the transducer response to different discontinuities. These data can be used to determine the model matrix if one wishes to determine the discontinuity location by solving the inverse problem [10]. 

The deconvolution is the numerical solution of this convolution integral. The theory of the inverse problem that we exposed in the previous paragraph shows an idealistic character because it doesn t integrate the frequency restrictions introduced by the electro-acoustic set-up and the mechanical system. To attenuate the effect of filtering, we must deconvolve the emitted signal and received signal. 

Lefebvre, J.P., Progress in linear inverse scattering imaging NDE application of Ultrasonic Reflection Tomography, in Inverse Problem in Engineering Mechanies, pp 371-375, (A.A.Balkema/ Rotterdam rookfleld, 1994). 

An undesirable side-effect of an expansion that includes just a quadratic and a cubic term (as is employed in MM2) is that, far from the reference value, the cubic fimction passes through a maximum. This can lead to a catastrophic lengthening of bonds (Figure 4.6). One way to nci iimmodate this problem is to use the cubic contribution only when the structure is ,utficiently close to its equilibrium geometry and is well inside the true potential well. MM3 also includes a quartic term this eliminates the inversion problem and leads to an t". . 11 better description of the Morse curve. 

Two-Dimensional Representation of Chemical Structures. The lUPAC standardization of organic nomenclature allows automatic translation of a chemical s name into its chemical stmcture, or, conversely, the naming of a compound based on its stmcture. The chemical formula for a compound can be translated into its stmcture once a set of semantic rules for representation are estabUshed (26). The semantic rules and their appHcation have been described (27,28). The inverse problem, generating correct names from chemical stmctures, has been addressed (28) and explored for the specific case of naming condensed benzenoid hydrocarbons (29,30). 

The single most severe drawback to reflectivity techniques in general is that the concentration profile in a specimen is not measured directly. Reflectivity is the optical transform of the concentration profile in the specimen. Since the reflectivity measured is an intensity of reflected neutrons, phase information is lost and one encounters the e-old inverse problem. However, the use of reflectivity with other techniques that place constraints on the concentration profiles circumvents this problem. 

It is notoriously diffieult to solve the so-ealled inverse problem and extraet kinetie data using the population balanee. Muralidar and Ramkrishna (1986) deseribe a proeedure to obtain agglomeration frequeneies from measured size distributions without the kinetie proeesses of nueleation, growth and disruption. The authors point out that even if the experimental data are very aeeurate, it is not always possible to estimate the aggregation frequeney satisfaetorily and to distinguish between different meehanisms. 

To extract the agglomeration kernels from PSD data, the inverse problem mentioned above has to be solved. The population balance is therefore solved for different values of the agglomeration kernel, the results are compared with the experimental distributions and the sums of non-linear least squares are calculated. The calculated distribution with the minimum sum of least squares fits the experimental distribution best. 

Muralidar, R. and Ramkrishna, D., 1986. An inverse problem in agglomeration kinetics. Journal of Colloid and Interfacial Science, 112, 348-361. 

Wright, H. and Ramkrishna, D., 1992. Solutions of inverse problems in population balance aggregation kinetics. Computers and Chemical Engineering, 16(2), 1019-1030. 

Only the inversion problem will be described, since the application to finding a single vector x will be obvious. The method is equivalent to the factorization... 

Although long-time Debye relaxation proceeds exponentially, short-time deviations are detectable which represent inertial effects (free rotation between collisions) as well as interparticle interaction during collisions. In Debye s limit the spectra have already collapsed and their Lorentzian centre has a width proportional to the rotational diffusion coefficient. In fact this result is model-independent. Only shape analysis of the far wings can discriminate between different models of molecular reorientation and explain the high-frequency pecularities of IR and FIR spectra (like Poley absorption). In the conclusion of Chapter 2 we attract the readers attention to the solution of the inverse problem which is the extraction of the angular momentum correlation function from optical spectra of liquids. 

Eq. (2.68) may also be used to solve the inverse problem. The recovery of g(a>) from experimentally obtained optical spectra may prompt the origin of the maximum. To find g(oj), it is necessary to determine from the correlation function K( not only Ge but also... 

It has been suggested [115, 116] to solve the inverse problem using the simplified relation which is actually a high-frequency limit of Eq. (2.73). This relation can be found, if we take into account that... 

Despite such a fault, the attempt to utilize the experimental data undertaken in [117] proves that the inverse problem is solvable. It is even simpler to do starting from the differential kinetic equation (2.25) whose integral is [118, 119]... 

Abstract Wavefront sensing for adaptive optics is addressed. The most popular wavefront sensors are described. Restoring the wavefront is an inverse problem, of which the bases are explained. An estimator of the slope of the wavefront is the image centroid. The Cramer-Rao lower bound is evaluated for several probability distribution function... 

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