Inverse problems with

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. 

Note also that expression (10.119) for the log-anomalous apparent resistivity is an approximation. Experiments with synthetic and real data show that the maximum of log anomalous apparent resistivity is of the order of 0.2. For this value, linearization (10.118) holds with an accuracy of 0.01. The estimated noise in the 3-D inverse problem with respect to log-anomalous apparent resistivity is also on the order of 0.01. Thus, we can safely use the notion of log-anomalous apparent resistivity if its maximum value is less than 0.2. 

We formulate the inverse problem with respect to a vector anomalous conductivities of the cells ... 

We have introduced above the formulation of a magnetotelluric inverse problem with respect to the anomalous conductivity vector, denoted by <7. However, during the minimization we may obtain anomalous conductivity values such that the total conductivity becomes negative. Thus, we have to transform the anomalous conductivity into a new space of model parameters with the property that the total conductivity always remains positive. We have already introduced above the conventional way of solving this problem, which uses the logarithm of the total conductivity as a model parameter ... 

The majority of the known methods of solving the direct and inverse problems with moving boundaries in ECM were elaborated within the framework of the so-called model of ideal processes, ignoring the variation of the electrolyte properties in the machining zone owing to heat and gas generation and also the peculiarities of mass transfer in the diffusion boundary layer ([9] and references cited therein, [34-42], etc.). In this case, the distribution of current density over the WP surface is determined solely by the distribution of electric potential over the machining zone. 

However, the inverse problem, with the film on the underside of the solid substrate, now features a competition between the stabilizing (leveling) effect of capillary forces against the destabilizing effect of gravity. This corresponds to a well-known stability problem, called Rayleigh-Taylor instability, applied to the thin film. In this case,... 

A key problem is the evaluation of the relevance of observed seismicity patterns. First, it is important to decide whether an observed pattern has a physical origin or is an artifact, arising for example from inhomogeneous reporting or from man-made seismicity, like quarry blasts or explosions. Second, the non-artifical events have to be analyzed with respect to their underlying mechanisms. This leads to an inverse problem with a nonunique solution, which can be illustrated for the most pronounced observed seismicity pattern, the occurrence of aftershocks. It is empirically known... 

QUADRATIC INTEGRALS IN INVERSE PROBLEMS WITH MULTIPLE SCATTERING... 

Quadratic Integrals in Inverse Problems with Multiple Scattering 127... 

Previously (Klcnin et al., 1977a) the stability-of-solution of the inverse problem with respect to polydispersity was shown. 

FIGURE 6.2.2 Comparison of the aggregation frequency from the inverse problems with the actual (constant) frequency for various values of the regularization parameter when the self-similar distribution is known exactly. Note that the most accurate estimate of the aggregation frequency is obtained with no regularization. (From Wright and Ramkrishna, 1992. Reprinted with permission from Elsevier Science.)... 

Figure 6.2.8 shows the results obtained by Wright and Ramkrishna (1992) for a(0, )/ from the solution of the inverse problem with no added error. It further shows that the best estimate of the aggregation frequency with error-free input data is obtained with no regularization, i.e., = 0. On the... 

Figure 12.16 Concentration dependencies of the diffusion coefficient. Here curve 1 illustrates the initial concentration profile used in calculations curves 2-7 show the profiles calculated by solving the inverse problem with the initial profile distorted with amplitudes of 0, 0.1, 1, 3, 5, and 10%, respectively (a) the concentration profiles calculated for the initial profiles with perturbations <5 = 1 (b), 5 (c), and 10% (d) (curve 1 shows the initial profile, curve 2 shows the distorted profile, and curve 3 shows the profile calculated on the basis of the obtained dependencies b c)).

In the sections on stochastic interpolation the theory of random functions was introduced. To apply the theory to inverse problem solutions it will be assumed that the formalism of Bayesian statistics is a soimd foimdation. There is no space here to discuss this contention, and there is no need, as an extensive literature exists. A recommended overview of the subject is [122]. A practical and complete review of inverse problems, with a distinct Bayesian flavour is [131]. 

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