Inverse problems distributions

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]. 

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. 

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. 

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... 

Unlike prisms, in this class of bodies uniqueness requires knowledge of the density. This theorem was proved by P. Novikov. The simplest example of starshaped bodies is a spherical mass. Of course, prisms are also star-shaped bodies but due to their special form, that causes field singularities at corners, the inverse problem is unique even without knowledge of the density. It is obvious that these two classes of bodies include a wide range of density distributions besides it is very possible that there are other classes of bodies for which the solution of the inverse problem is unique. It seems that this information is already sufficient to think that non-uniqueness is not obvious but rather a paradox. 

Fig. 8. Reconstruction of Young s modulus map in a simulated object. A 3D breast phantom was first designed in silico from MR anatomical images. Then a given 3D Young s modulus distribution was supposed with a 1 cm diameter stiff inclusion of 200 kPa (A). The forward problem was the computing of the 3D-displacement field using the partial differential equation [Eq. (5)]. The efficiency of the 3D reconstruction (inverse problem) of the mechanical properties from the 3D strain data corrupted with 15% added noise can be assessed in (B). The stiff inclusion is detected by the reconstruction algorithm, but its calculated Young s modulus is about 130 kPa instead of 200 kPa. From Ref. 44, reprinted by permission of Wiley-Liss, Inc., a subsidiary of John Wiley Sons, Inc.

From the above equation, the measured capacitance between the source and the detector electrodes of the electrode pair i is determined from the given dielectric constant (permittivity) distribution of the medium under investigation. The processes of finding the capacitance for a given permittivity distribution is referred to as the forward problem. On the other hand, the process of finding the permittivity distribution from a set of capacitance measurements is referred to as the inverse problem. 

Transistors based on a-Si H do not follow the Borkan-Weimer equation (Borkan and Weimer, 1963) since the characteristics of an a-Si H transistor depend on the density of states in the gap. For an exponential density of tail state distribution, appropriate theories were formulated as early as 1975 (Neudeck and Malhotra, 1975, 1976) that have subsequently been developed further (Kishida et al., 1983). If the density of states varies in a nonexponential manner, numerical methods must be used to derive the transistor characteristics. The inverse problem, a derivation of the density of states from field-effect measurements, is discussed in Chapter 2 by Cohen of Volume 21C. 

Actually, the inverse problem should be solved, i.e., given the data n(t) containing errors, obtain a plausible candidate / (h) associated with a known function p(t,h). This function, termed kernel, is assumed to be a retentiontime distribution other than an exponential one otherwise, the problem has a tractable solution by means of the moment generating functions as presented earlier. This part aims to supply some indications on how to select the density of h. For a given probability density function f (h), one has to mix the kernel with / (h) ... 

Since this monograph is devoted only to the conception of mathematical models, the inverse problem of estimation is not fully detailed. Nevertheless, estimating parameters of the models is crucial for verification and applications. Any parameter in a deterministic model can be sensibly estimated from time-series data only by embedding the model in a statistical framework. It is usually performed by assuming that instead of exact measurements on concentration, we have these values blurred by observation errors that are independent and normally distributed. The parameters in the deterministic formulation are estimated by nonlinear least-squares or maximum likelihood methods. 

Let us finally briefly consider the Inverse problem how to find /(AadgU) if Vlp.T] is known from experiment, and the local isotherm is known (or assumed). Mathematically this procedure is equivalent to solving so-called Fredholm integrals of the first kind the local isotherm functions serve as the kernel. Computationally speaking, this is an ill-posed problem because minor variations in the data may cause substantial variations in /(Aads )- n other words, the Inevitable experimental noise thwarts one obtaining a physically significant distribution. In practice the elaboration may follow one of three paths ... 

Crawley et. al. [57] applied the above equations to determine particle size distributions from turbidity measurements. The problems arise in finding a particle size distribution from the measured extinction coefficient due to the ill-defined inversion problem. Scholtz et.al. [58] focused on the problem of analyzing spectra of colloidal solutions, for which the size distribution was known from other methods like electron microscopy and light scattering they termed this transmission spectroscopy. ... 

The inverse problem consists in reconstructing the velocity distribution from the observed pressure field ... 

Numerical solution of the Helmholtz equation for a given velocity distribution describes the forward ])roblem, while the inverse problem is actually aimed at determining the coefficients (velocity c(r)) for the given pres.surc field p r,uj). Both of these problems arc nonlinear. Note that often inverse seismic problems are formulated not for the velocity but for the slowness, which is the inverse velocity ... 

The question of the existence of the inverse problem solution has two aspects. One is the physical existence of some geophysical parameter distribution that generates observed data, and another one is the existence of the mathematical solution of the operator equation (1.34). There is no doubt about the physical existence of the inverse problem solution however, the mathematical existence can be questionable. To understand this phenomenon better, note that measured geophysical data always contain certain errors (5d ... 

Therefore, the body will consist of two spherical layers the inner layer (ball) with a negative density, and the outer layer with a positive density. Thus we come to an idea of the existence of a density distribution that generates a zero external gravity field. This idea is the centerpiece of the non-uniqueness principle for gravity field inverse problems. One can add this kind of density distribution to any given density model and obtain another model generating the same gravity field. 

Novikov (1938) proved the uniqueness theorem for a star-type body with the given homogeneous density distribution p(r) = pg. The theorem states that if it is known that the gravity field is generated by a star-type body with the a given constant density, the gravity inverse problem has a unique solution. In the case of spherical balls, this result is obvious according to formula (1.39), if pj = P2, then R — H-2 ... 

Figure 1-12 Demonstration of instability in inverse problem solution. Smooth solid line shows the true distribution of the gravity potential at a distance 20 m above the material ball. Oscillating line presents a plot of gravity potential analytically continued downward at a distance 20 m above the material ball. One can see that a small, practically invisible noise in the original data results in dramatic oscillations for downward analytically continued data.

Figure 7-3 presents a map of the migration density distribution given by formula (7.49). One can see that this distribution has a local maximum at the position of the point source xq, zq). Thus, migration density does not solve directly the inverse problem but provides an image of the source associated with the maximum of the function (x, z). 

Difficulties arise even in forward modeling because of the huge size of the numerical problem to be solved for adequate representation of the complex 3-D distribution of EM parameters in the media. As a re.sult, computer simulation time and memory requirements could be excessive even for practically realistic models. Additional difficulties are related to EM imaging which is based on EM inverse problems. These problems are nonlinear and ill-posed, because, in general cases, the solutions can be unstable and/or nonunique. In order to overcome these difficulties one should... 

In a solution of the inverse problem we assume that the anomalous field, E , and the background conductivity are given. The goal is to find the anomalous conductivity distribution. Ad. In this case, formula (10.16) has to be treated as a linear equation with respect to Ad. 

After determining m and A it is possible to evaluate the anomalous conductivity distribution Aa from equation (10.63). This inversion scheme reduces the original nonlinear inverse problem to three linear inverse problems the first one (the quasi-Born inversion) for the tensor m, the second one for the tensor A, and the third one (correction of the result of the quasi-Born inversion) for the conductivity Act. This method is called a quasi-linear (QL) inversion. ... 

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