The Inversion Problem

The inverse problem would be well defined if we knew the temperature or the harmonic function... 

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]. 

Puro, A. The inverse problem of thermoelasticity of optical tomography. J. Appl. Maths. Mechs. 1993, 57(1) I4I-I45. 

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 ... 

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]. 

The deconvolution is the numerical solution of this convolution integral. The theory of the inverse problem that we exposed in the previous paragraph shows an idealistic character because it doesn t integrate the frequency restrictions introduced by the electro-acoustic set-up and the mechanical system. To attenuate the effect of filtering, we must deconvolve the emitted signal and received signal. 

An undesirable side-effect of an expansion that includes just a quadratic and a cubic term (as is employed in MM2) is that, far from the reference value, the cubic fimction passes through a maximum. This can lead to a catastrophic lengthening of bonds (Figure 4.6). One way to nci iimmodate this problem is to use the cubic contribution only when the structure is ,utficiently close to its equilibrium geometry and is well inside the true potential well. MM3 also includes a quartic term this eliminates the inversion problem and leads to an t". . 11 better description of the Morse curve. 

Two-Dimensional Representation of Chemical Structures. The lUPAC standardization of organic nomenclature allows automatic translation of a chemical s name into its chemical stmcture, or, conversely, the naming of a compound based on its stmcture. The chemical formula for a compound can be translated into its stmcture once a set of semantic rules for representation are estabUshed (26). The semantic rules and their appHcation have been described (27,28). The inverse problem, generating correct names from chemical stmctures, has been addressed (28) and explored for the specific case of naming condensed benzenoid hydrocarbons (29,30). 

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. 

Only the inversion problem will be described, since the application to finding a single vector x will be obvious. The method is equivalent to the factorization... 

Although long-time Debye relaxation proceeds exponentially, short-time deviations are detectable which represent inertial effects (free rotation between collisions) as well as interparticle interaction during collisions. In Debye s limit the spectra have already collapsed and their Lorentzian centre has a width proportional to the rotational diffusion coefficient. In fact this result is model-independent. Only shape analysis of the far wings can discriminate between different models of molecular reorientation and explain the high-frequency pecularities of IR and FIR spectra (like Poley absorption). In the conclusion of Chapter 2 we attract the readers attention to the solution of the inverse problem which is the extraction of the angular momentum correlation function from optical spectra of liquids. 

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... 

It has been suggested [115, 116] to solve the inverse problem using the simplified relation which is actually a high-frequency limit of Eq. (2.73). This relation can be found, if we take into account that... 

Despite such a fault, the attempt to utilize the experimental data undertaken in [117] proves that the inverse problem is solvable. It is even simpler to do starting from the differential kinetic equation (2.25) whose integral is [118, 119]... 

The a priori penalty prior(x) oc — log Pr x allows us to account for additional constraints not carried out by the data alone (i.e. by the likelihood term). For instance, the prior can enforce agreement with some preferred (e.g. smoothness) and/or exact (e.g. non-negativity) properties of the solution. At least, the prior penalty is responsible of regularizing the inverse problem. This implies that the prior must provide information where the data alone fail to do so (in particular in regions where the noise dominates the signal or where data are missing). Not all prior constraints have such properties and the enforced a priori must be chosen with care. Taking into account additional a priori constraints has also some drawbacks it must be realized that the solution will be biased toward the prior. 

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. 

While the ECG is an invaluable tool for the observation of heart rate and rhythm, as well as for the diagnosis of conduction abnormalities, ischaemia, and infarcts, its detailed interpretation is not without pitfalls. One reason for this is that different changes in cardiac cellular behaviour may give rise to very similar effects on the ECG. This makes it difficult to draw conclusions from a patient s ECG to the underlying (sub-)cellular mechanisms. This issue is usually referred to as the inverse problem. ... 

Today s heart models do not yet possess the power to solve the inverse problem. They do, however, aid the understanding and interpretation of the EGG by repeatedly solving forward problems to study the effects of cellular modifications on the calculated ECG. Model reconstruction of a normal ECG is therefore a necessary first step towards developing a better understanding of the information hidden in it. Figure 8.3(a) illustrates this. 

It is also of interest to study the "inverse" problem. If something is known about the symmetry properties of the density or the (first order) density matrix, what can be said about the symmetry properties of the corresponding wave functions In a one electron problem the effective Hamiltonian is constructed either from the density [in density functional theories] or from the full first order density matrix [in Hartree-Fock type theories]. If the density or density matrix is invariant under all the operations of a space CToup, the effective one electron Hamiltonian commutes with all those elements. Consequently the eigenfunctions of the Hamiltonian transform under these operations according to the irreducible representations of the space group. We have a scheme which is selfconsistent with respect to symmetty. 

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... 

Later, we will add one more element, caused by the fact that the field containing information about the parameters of a body (useful signal), is never known exactly. Inasmuch as in the process of gravity interpretation every step is reasonably well defined, we may arrive at the impression that the solution of the inverse problem is... 

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. 

---