Existence and uniqueness of the inverse problem solutions

Considering the importance of these three questions for inverse problem solution, the famous French mathematician lladamard expressed the opinion that a certain mathematical problem was formulated correctly if all three cpicstions loosed above had a positive answer. In other words, the mathematical problem wa.s said to be well-posed, if its solution did exist, was unique and was stable. 

To develop an efficient method of solving geophysical inverse problems, it is important to better understand the properties of these operators and to study the general principles of inverse problem solution. We will discuss this issue in the next chapter. Before moving to the theoretical foundations of inversion, let us take a quick look at the questions of the existence, uniqueness, and stability of the geophysical inverse problems. 

For this formulation, the solution to the inverse problem is unique [7] however, there still exists the problem of continuity of the solution on the data. The linear algebraic counterpart to the elliptic boundary value problem is often useful in discussing this problem of noncontinuity. The numerical solution to all elliptic boundary value problems (such as the Poisson and Laplace problems) can be formulated in terms of a set of linear equations, = b. For the solution of Laplace s equation, the system can be reformulated as ... 

Equation (22) is particularly useful when a concentration gradient in depth exists. In this case, several spectra at different values of 9 are taken and the analysis is called angle resolved X-ray photoelectron spectroscopy. However, for a maximum efficiency, a flat surface (at an atomic level) is needed to avoid shade effects as shown by Fadley in his early works in the 1970s [44]. An additional problem exists the extraction of concentration profiles, cg(x), from Eq. (22) is an inverse problem the intensity as a function of the analysis angle is the Laplace transform of the composition depth profile of the sample [43] and does not have a unique solution. Several algorithms to solve the inversion problem were developed and tested [46]. They are all very unstable and sensitive to small statistical... 

In the case of necrosis, internal surfaces appear. A double layer equal to cr. on each such surface then acts a source in addition to the double layer on the heart surface. A solution to the inverse problem will then yield values of r and fx which are a superposition of their values for the surface cells and an effective value reflecting the internal surfaces. These values on 5 are still unique. If, however, the internal surfaces are unknown (their existance is probably not known) then interpretation of the values of r and fx obtained from the inverse solution becomes more difficult. If an internal necrotic volume is small, it will act approximately as a dipole whose moment is the integral of the double layer over the internal surface. 

Note that the well-posed inverse problem possesses all the properties of the good solution discussed in the previous chapter the solution exists, is unique, and is stable. 

In terms of considerations given in Section 3, ill-posed problems have a non-unique and/or an unstable solution. For a non-unique solution, the matrix K C K on the left side of Eq. (11c) has linearly dependent rows (and columns, since it is symmetrical), i.e. det(K C K)=0 (degenerated matrix) and the inverse operator (K C K)" does not exist. For a quasi-degenerated matrix (det(K C K) 0) the inverse operator (K C K)" exists. However, in... 

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