Forward problems

To find the potential field in tissue caused by a known signal current source is forward problem. The tissue may be inhomogeneous, anisotropic, and of finite dimensions. 

Examples are given in Section 6.2. Usually there is one unique solution to a posed problem. 

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No a priori information about the unknown profile is used in this algorithm, and the initial profile to start the iterative process is chosen as (z) = 1. Moreover, the solution of the forward problem at each iteration can be obtained with the use of the scattering matrices concept [8] instead of a numerical solution of the Riccati equation (4). This allows to perform reconstruction in a few seconds of a microcomputer time. The whole algorithm can be summarized as follows ... 

Eddy-current non-destructive evaluation is widely used in the aerospace and nuclear power industries for the detection and characterisation of defects in metal components. The ability to predict the probe response to various types of defect is highly valuable since it enables the influence of particular parameters to be studied without recourse to costly and time consuming experiments. The solution of forward problems is also essential in the process of inverting experimental data. 

This paper compares experimental data for aluminium and steel specimens with two methods of solving the forward problem in the thin-skin regime. The first approach is a 3D Finite Element / Boundary Integral Element method (TRIFOU) developed by EDF/RD Division (France). The second approach is specialised for the treatment of surface cracks in the thin-skin regime developed by the University of Surrey (England). In the thin-skin regime, the electromagnetic skin-depth is small compared with the depth of the crack. Such conditions are common in tests on steels and sometimes on aluminium. 

This paper is structured as follows in section 2, we recall the statement of the forward problem. We remind the numerical model which relates the contrast function with the observed data. Then, we compare the measurements performed with the experimental probe with predictive data which come from the model. This comparison is used, firstly, to validate the forward problem. In section 4, the solution of the associated inverse problem is described through a Bayesian approach. We derive, in particular, an appropriate criteria which must be optimized in order to reconstruct simulated flaws. Some results of flaw reconstructions from simulated data are presented. These results confirm the capability of the inversion method. The section 5 ends with giving some tasks we have already thought of. 

These equations are the coupled system of discrete equations that define the rigorous forward problem. Note that we can take advantage of the convolution form for indices (i — I) and (j — J). Then, by exciting the conductive media with a number N/ oi frequencies, one can obtain the multifrequency model. The kernels of the integral equations are described in [13] and [3j. 

In this case, we can conclude that the small sensor is lightly tilted with an angle of 0,25 degrees. We have concluded, during experimentations, that the measurement of the magnetic field is very sensitive to the angle of inclinaison of the sensor. In this way, we validate the computation of the incident field E (r). We can also expect some difficulties for the validation of the forward problem by experimental data. 

Solution Example 4.5 was a reverse problem, where measured reactor performance was used to determine constants in the rate equation. We now treat the forward problem, where the kinetics are known and the reactor performance is desired. Obviously, the results of Run 1 should be closely duplicated. The solution uses the method of false transients for a variable-density system. The ideal gas law is used as the equation of state. The ODEs are... 

Use these new values for k and kc to solve the forward problem in Example 4.6. 

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. 

Here i, j, k are unit vectors along the coordinate axes. Thus, the solution of the forward problem is trivial. In particular, along the line where y = 0 and z — h we have ... 

Inasmuch as Equation (1.6) allows us to solve the forward problem for any distribution of masses, we may say in this sense that the theory of the gravitational method is completely developed. However, in order to understand better the behavior of the field of the earth and sometimes to improve the quality of the... 

Besides, the potential and its first derivatives are continuous at boundaries where a volume density is discontinuous function. It is obvious that in this case the solution of the forward problem is unique. Now consider a completely different situation, when a density 5 q) is given only inside some volume V surrounded by a surface S, Fig. 1.8a. Inasmuch as the distribution of masses outside V is unknown, it is natural to expect that Poisson s equation does not uniquely define the potential U, and in order to illustrate this fact let us represent its solution as a sum ... 

This equality is a special form of Gauss s theorem, and it will allow us to find several boundary conditions, which provide uniqueness of a solution of the forward problem. First, we make three comments ... 

If there is a difference between the measured and calculated fields, all parameters of the first approximation or some of them are changed in such a way that a better fit to these fields is achieved. Thus, we obtain a second approximation of mass distribution. Of course, in those cases when even the new set of parameters does not provide a satisfactory match of these fields, this process of calculation has to be continued. As we see from this process, every step of the interpretation requires application of Newton s law. Let us recall that this procedure, based on the use of Newton s law, is often called the solution of the forward problem of the field of attraction, and by definition we have... 

Substituting values of these parameters into Equation (4.2) we solve the forward problem and compare the measured and calculated fields. 

Performing a solution of the forward problem with the new parameters we again compare fields, and this process can continue until the accuracy of a determination of the parameters satisfies our requirements. 

The second step is a solution of the forward problem, applying Newton s law of attraction... 

As soon as the values of a calculated field are located inside the observation interval further improvement of matching between the measured and calculated fields does not have any meaning, because we do not know where inside the interval the useful signal is located. Therefore, we stop the process of fitting of fields and start a new procedure which also requires a solution of the forward problem. In the last stage of matching, we obtained the set of parameters... 

SOLUTION OF THE FORWARD PROBLEM (A CAECUEATION OF THE FIELD OF ATTRACTION)... 

Similar expressions can be written for horizontal components of the field of attraction. Thus, instead of the volume integral, we have derived an expression for the field that requires integration only over the surface. The formulas described in this section allow us, in many cases, to simplify the solution of the forward problem. 

The forward problem is the calculation of displacement fields from input viscoelasticity parameters. The latter describe correctly the investigated object if the calculated and measured displacement images converge. 

Fig. 8. Reconstruction of Young s modulus map in a simulated object. A 3D breast phantom was first designed in silico from MR anatomical images. Then a given 3D Young s modulus distribution was supposed with a 1 cm diameter stiff inclusion of 200 kPa (A). The forward problem was the computing of the 3D-displacement field using the partial differential equation [Eq. (5)]. The efficiency of the 3D reconstruction (inverse problem) of the mechanical properties from the 3D strain data corrupted with 15% added noise can be assessed in (B). The stiff inclusion is detected by the reconstruction algorithm, but its calculated Young s modulus is about 130 kPa instead of 200 kPa. From Ref. 44, reprinted by permission of Wiley-Liss, Inc., a subsidiary of John Wiley Sons, Inc.

Literature abounds with a rich terminology concerning the possible relationships between observations provided by experiment or analysis and parameters which are the physical quantities needed for a mathematical formulation of a process (the model) to be uniquely determined. A forward problem relates observations to parameters by a relationship such as... 

Solving the forward problem of the isotopic and chemical evolution of n reservoir exchanging a radioactive and its daughter isotope requires the solution of 3n— 1 differential equations (the minus one stems from the closure condition). The parameters are n (n — 1) independent flux factors k for the stable isotope N and n (n — 1) independent M/N fractionation factors D. In addition, the n values of R y the n values of Rh and the n—1 allotments x of the stable isotope among the reservoirs must be assumed at some time, preferably at the beginning of the evolution (e.g., 4.5 Ga ago), or in the modern times, in which case integration is carried out backwards in time. 

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