Inverse problems determination

The Inverse Problem Determination of the Pore Volume Distribution The analysis presented so far can be used to solve the inverse problem, that is if we know the amount adsorbed versus pressure, the equation (3.9-27) can be used to determine the constants for the pore volume distribution provided that we know the shape of the distribution a-priori. We shall handle this inverse problem by assuming that a mesopore volume distribution can be described by the double Gamma distribution as given in eq.(3.9-22). With this form of distribution, the amount adsorbed can be calculated from eq.(3.9-27), and the result is ... 

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]. 

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]. 

If an inverse is desired, it can be calculated by solving for the LU decomposition and then solving n problems with right-hand sides consisting of all zeroes except one entiy. Thus 4n /2> — n/2> multiplications are required for the inverse. The determinant is given by... 

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... 

Once wells have been drilled into the formation, the local properties of the reservoir rocks and fluids can be determined. To constmct a realistic model of the reservoir, its properties over its total extent— not just at the well sites—must be known. One way of estimating these properties is to match production histories at the wells with those predicted by the reservoir model. This is a classic ill-posed inverse problem that is very difficult to solve. 

Repeating these calculations with different pairs of gx(x) we may increase the accuracy of the evaluation of h. Next, making use of the value of this component at any point, the mass m is evaluated. In the case when only the vertical component is known, the determination of the position of mass and its value is similar. Here it is appropriate to notice the following. Inasmuch as an arbitrary body, located at a large distance from an observation point p, creates a field, known always with some error, often it cannot be practically distinguished from that of an elementary particle, and for this reason we are able to determine only the product of volume and density, mass, but each of them remains unknown. It is the first illustration of the fact that the solution of the inverse problem in gravity, as well as in other geophysical methods, is an ill-posed one, because some parameters of a body... 

The major prerequisite of above assumption was the following the detailed data on the change of electrophysical characteristics of a semiconductor adsorbent caused by adsorption of a certain gas over substantially wide pressure range makes it possible to solve the inverse problem concerning determination of the concentration of this gas in ambient atmosphere, due to detected change in electrophysical characteristics of the adsorbent. 

Our approach to determine the properties of heterogeneous media utilizes mathematical models of the measurement process and, as appropriate, the flow process itself. To determine the desired properties, we solve an associated system and parameter identification problem (also termed an inverse problem) to estimate the properties from the measured data. 

The inverse problem to simulation from a reaction mechanism is the determination of the reaction mechanism from observed kinetics. The process of building a mechanism is an interactive one, with successive changes followed by experimental testing of the model predictions. The purpose is to be able to explain why a reacting system behaves the way it does in order to control it better or to improve it (e g., in reactor performance). 

The inverse problem is to determine the flow rate for a given pressure drop. For turbulent flow, this is not so straightforward because the value of / is unknown until the flow rate, and hence Re, are known. The traditional solution to this problem is to use the plot of /Re2 against Re or ifRe2 against Re shown in Figure 2.2. 

Consistent with the notion of approximating polynomials as in the preceding paragraph, finite-element approaches attempt to simplify the solution process by carefully choosing polynomials so as to minimize the number of coefficients required to determine and simplify the matrix inversion problem. One choice is the Bezier polynomials, defined by... 

Such statistical methods have been used for the inverse problem in the polymer literature, i.e., the formation of a macromolecular network by polymerization (40-44). The model was used to determine what effect the removal of crosslinks in the network have on thermal decomposition of the network. 

Solution of Eq. (2) is only part of the overall problem in mechanistic studies. Indeed, Eq. (2) represents only the first of many sequential inverse problems vthich we commonly lump together and which represent the overall inverse problem in chemical kinetics [110-112]. The next problem is the determination of the moles of all species, the elemental stoichiometries of species, the number of observable reactions, and the balanced stoichiometries for all observable reactions. 

Nnmerical procednres can also be used for determining adsorption isotherms from overloaded profiles. The so-called inverse problem of chromatography consists of calculating the adsorption isotherm from the profiles of overloaded bands [35-37],... 

A specific example of applications in the second category is the dating of rocks. Age determination is an inverse problem of radioactive decay, which is a first-order reaction (described later). Because radioactive decay follows a specific law relating concentration and time, and the decay rate is independent of temperature and pressure, the extent of decay is a measure of time passed since the radioactive element is entrapped in a crystal, hence its age. In addition to the age, the initial conditions (such as initial isotopic ratios) may also be inferred, which is another example of inverse problems. 

This section describes the experimental methods and focuses on the estimation of diffusivity after the experiment. The analytical methods are not described here. Estimation of diffusivity from homogeneous reaction kinetics (e.g., Ganguly and Tazzoli, 1994) is discussed in Chapter 2 and will not be covered here. Determination of diffusion coefficients is one kind of inverse problems in diffusion. This kind of inverse problem is relatively straightforward on the basis of solutions to forward diffusion problems. The second kind of inverse problem, inferring thermal history in thermochronology and geospeedometry, is discussed in Chapter 5. 

Thus far we have examined the determination of a field that will control the quantum many-body dynamics of a system when all that is specified is the initial and final states of the system and the constraints imposed by the equations of motion and physical limitations on the field. When posed in this fashion, the calculation of the control field is an inverse problem that has similarities to the determination of the interaction potential from scattering data. Despite the similarities, the mathematical methods used are very different. Because only the end points of the initial-to-final state transforma-... 

Virtually all inverse problems are ill-posed, in that the finite available data will not permit a full determination of the potential. There-... 

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