Other Inverse Problems

There are other examples of inverse problems in the literature that are worthy of mention. Thus, the determination of growth rates of cells from size distributions during balanced growth of a microbial culture is an interesting example which we treat briefly in the next section. 

1 Growth Rates of Cells during Balanced Growth 

FIGURE 6.3.6 Comparison of estimated particle growth rate with actual value used in simulations. Size-dependent part (a) time-dependent part (b). (From Mahoney, 2000.) 

We are concerned with the determination of the functions L(/), h(/), and p l l ), from suitably designed population data to be determined from experiments. We assume that the cells are growing in a batch culture under balanced exponential growth conditions. Under these circumstances we have fid t) = where /(/) is the time-invariant probability dis- 

2) is integrated with respect to cell length, we obtain the result that 

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The other inverse problem generally encountered is the sp>ecification problem, schematized in Figure 3 ... 

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 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. 

Abstract Either because observed images are blurred by the instrument and transfer medium or because the collected data (e.g. in radio astronomy) are not in the form of an image, image reconstmction is a key problem in observational astronomy. Understanding the fundamental problems underlying the deconvolution (noise amplification) and the way to solve for them (regularization) is the prototype to cope with other kind of inverse problems. 

The Wiener filter therefore avoids noise amplification and provides the best solution according to some quality criterion. We will see that these features are common to all other methods which correctly solve the deconvolution inverse problem. The result of applying Wiener inverse-filter to the simulated image is shown in Fig. 3b. 

We have seen how to properly solve for the inverse problem of image de-convolution. But all the problems and solutions discussed in this course are not specific to image restoration and apply for other problems. 

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... 

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. 

We have reviewed here the implementation of the inverse method for going from densities to potentials, based on local-scaling transformations. For completeness, let us mention, however, that several other methods have also been advanced to deal with this inverse problem [101-111]. Consider the decomposition of into orbits Such orbits are characterized by the fact that... 

As already emphasized, the initial physical problem and successful solution is often referred to as the inverse problem in the applied mathematics literature. In the engineering literature, the subsequent repeated application of a developed numerical approach to other examples from the same general class of problems is frequently referred to as system identification. 

The experienced catalytic chemist or chemical reaction engineer will immediately recognize that the study of a new catalytic reaction system using an in situ spectroscopy, has a great deal in common with the concepts of inverse problems and system identification. First, there is a physical system which cannot be physically disassembled, and the researcher seeks to identify a model for the chemistry involved. The inverse in situ spectroscopic problem can be denoted by Eq. (2). Secondly, the physical system evolves in time and spectroscopic measurements as a function of time are a must. There are realistic limitations to the spectroscopic measurements performed. For this reason as well as for various other reasons, the inverse problem is ill-posed (see Section 4.3.6). Third, signal processing will be needed to filter and correct the raw data, and to obtain a model of the system. The ability to have the individual pure component spectra of the species present in... 

The concentration of the radioactive nuclide (reactant, such as Sm) decreases exponentially, which is referred to as radioactive decay. The concentration of the daughter nuclides (products, including Nd and He) grows, which is referred to as radiogenic growth. Note the difference between Equations l-47b and l-47c. In the former equation, the concentration of Nd at time t is expressed as a function of the initial Sm concentration. Hence, from the initial state, one can calculate how the Nd concentration would evolve. In the latter equation, the concentration of Nd at time t is expressed as a function of the Sm concentration also at time t. Let s now define time t as the present time. Then [ Nd] is related to the present amount of Sm, the age (time since Sm and Nd were fractionated), and the initial amount of Nd. Therefore, Equation l-47b represents forward calculation, and Equation l-47c represents an inverse problem to obtain either the age, or the initial concentration, or both. Equation l-47d assumes that there are no other ot-decay nuclides. However, U and Th are usually present in a rock or mineral, and their contribution to " He usually dominates and must be added to Equation l-47d. 

An analogous method for solving the inverse problem allows us to determine the constants in a kinetic equation using other experimental data for example, the results of measurements of the T(t) dependence at different points on an article (or model sample) can be successfully used for this purpose. 

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. 

Generally, software sensors are typical solutions of so-called inverse problems. A so-called forward problem is one in which the parameters and starting conditions of a system, and the kinetic or other equations which govern its behavior, are known. In a complex biological system, in particular, the things which are normally easiest to measure are the variables, not the parameters. In the case of metabolism, the usual parameters of interest are the enzymatic rate and affinity constants, which are difficult to measure accurately in vitro and virtually impossible in vivo [93,118,275,384]. Yet to describe, understand, and simulate the system of interest we need knowledge of the parameters. In other words, one must go backwards from variables such as fluxes and metabolite concentrations, which are relatively easy to measure, to the parameters. Such problems, in which the inputs are the variables and the outputs the parameters, are known as system identification problems or as so-called inverse problems. 

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