The Direct and Inverse Problem

The fundamental problem under consideration is scattering and absorption by single particles in natural environments, however, we are usually confronted with collections of very many particles. Even in the laboratory, where it is possible to do experiments with single particles, it is more usual to make measurements on many particles. A rigorous theoretical treatment of scattering by many particles is indeed formidable (see, e.g., Borghese et al., 1979). But if certain conditions are satisfied, a collection poses no more analytical problems than does a single isolated particle. 

We shall assume, in addition to single scattering, that the particles are many and their separations random, which implies incoherent scattering. That is, there is no systematic relation among the phases of the waves scattered by the individual particles thus, the total irradiance scattered by the collection is just the sum of the irradiances scattered by the individual particles. Even, however, in a collection of randomly separated particles, the scattering is coherent in the forward direction, a subject to which we shall return in Chapter 3. 

There are two general classes of problems in the theory of the interaction of an electromagnetic wave with a small particle. 

The Direct Problem. Given a particle of specified shape, size, and composition, which is illuminated by a beam of specified irradiance, polarization, ami 

The Inverse Problem. By a suitable analysis of the scattered field, describe the particle or particles that are responsible for the scattering. This is the hard problem it consists of describing a dragon from an examination of its tracks (Fig. 1.56). 

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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. 

The transfer processes in the gap within the quasi-steady-state approximation are calculated similarly for both the direct and inverse problems. To simplify the calculation of transfer processes in the gap, the boundary-layer approximation is commonly used. According to this approximation, the current density is calculated separately in the bulk gap and in the near-electrode diffusion layers, and their congruence is provided via the boundary conditions. The transfer processes in the... 

The direct and inverse problems are frequently encountered in practice. A solution of the integral equation allows one to find the relation between the shape of the overall adsorption isotherm and the adsorption energy distribution function. 

The mathematical description of ECM can be used for solving the direct and inverse electrochemical shaping problems [4, 9]. In the first case, there is a need to determine the shape and dimensions of the machined WP surface. In this case, the geometry of the TE working surface is prescribed. In the second case, there is a need to determine the shape and dimensions of the TE working surface, in order to produce the WP surface with a required geometry. 

In the general case, in order to solve the direct and inverse ECM problems, it is necessary to obtain the simultaneous solution of the basic equations describing the WP surface evolution [9] and a set of equations for the electrode reactions kinetics and transfer processes in the gap, which determines the distribution of current density over the WP surface. 

When solving the direct and inverse ECM problems, it is necessary to take into account the peculiarities of the shaping scheme used and the machining conditions. In most cases of ECM, the WP surface is formed under steady-state or nonsteady-state conditions according to one of the following schemes. 

The quasi-steady-state approximation may be used, because the rates of the transfer processes in the I EG (meters per second) are considerably higher than the rate of the variation of the WP surface (millimeters per minute). Within the framework of the quasi-steady-state approximation, it is possible to divide the initial problem into two subproblems (1) Calculation of the transfer processes in the I EG and the determination of the ECM rate field Va. (2) Calculation of the WP surface evolution for the direct problem or correction of the TE surface for the inverse problem. However, even under this simplification, solving the direct and inverse ECM problems, especially for sculptured WPs, involves great difficulties. 

Within the local, one-dimensional approximation, the solution of the direct and inverse ECM problems is considerably simplified. 

Currently there is no commonly accepted classification, which would include and systematize all existing hydrogeochemical models and methods of solving them. If anything, the only commonly accepted position is associated with the distinction between direct and inverse problems. 

The most significant stoichiometric matrix tool, which plays an important role in solving kinetic problems, is its rank. As it is known, a matrix rank defines the number of its linearly independent rows or columns. Using of the notion of a matrix rank allows to reduce the number of differential equations in a reaction mathematical model and, thereby, to make solving the direct and inverse kinetic problems easier. For example, let us consider a reaction scheme ... 

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. 

Inverse problems are very common in experimental and observational sciences. Typically, they are encountered when a large number of parameters (as many as or more than measurements) are to be retrieved from measured data assuming a model of the data - also called the direct model. Such problems are ill-conditioned in the sense that a simple inversion of the direct model applied directly to the data yields a solution which exhibits significant, or even dominant, features which are completely different for a small change of the input data (for instance due to a different realization of the noise). Since the objective constraints set by the data alone are not sufficient to provide a unique and... 

In Chap. E, photoelectron spectroscopic methods, in recent times more and more employed to the study of actinide solids, are reviewed. Results on metals and on oxides, which are representative of two types of bonds, the metallic and ionic, opposite with respect to the problem itineracy vs. localization of 5f states, are discussed. In metals photoemission gives a photographic picture of the Mott transition between Pu and Am. In oxides, the use of photoelectron spectroscopy (direct and inverse photoemission) permits a measurement of the intra-atomic Coulomb interaction energy Uh. 

These formulae can serve as a new effective quasi-anaJ3d ical tool in the solution of both direct and inverse 2-D electromagnetic problems. Numerical tests demonstrate that these approximations produce a very accurate result for 2-D models (Dmitriev et al., 1999). 

Fig. 1. Input and output for solving the direct and the inverse eigenvalue problem...

The presence of two scales enables modeling the mass transfer processes both in the extent of their completion and in time. When the data of mass transfer velocities are absent, used is only the degree of the processes completion. In this case in the USA within the framework of reviewed models are distinguished direct problems, i.e., reaction path models, and inverse problems, i.e., mass balance models. 

The world, not being in equilibrium, presents a complex spectacle of changes varying from the almost instantaneous to the imperceptibly slow. The rates of chemical transformations offer a more intricate problem than equilibria. If the speeds of direct and inverse reactions are known, equilibria can be calculated, but the converse proposition does not hold. Infinitely numerous pairs of values for the rates are consistent with the same equilibrium constant. In fact, alternative routes to the same chemical equilibrium are not only possible in principle but followed in practice, often simultaneously. 

In this section the joint process and control design problem of batch processes is addressed. The problem is formulated within an optimization framework, including the search of the equipment, the motion, and the controller. As stated in the introduction, the emphasis will be placed on the motion and control problem design via the inverse optimality, while the complementary role of the direct optimization framework will be outlined only, in the understanding that the corresponding tools are known and have been employed in batch process studies [3-5, 6]. First, the problem is stated. Then, a passivated dynamical inversion is drawn, and the result is applied to construct the output-feedback controller, and to set the algorithm to design the nominal batch motion. 

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