Ill-posed problems

The classical computer tomography (CT), including the medical one, has already been demonstrated its efficiency in many practical applications. At the same time, the request of the all-round survey of the object, which is usually unattainable, makes it important to find alternative approaches with less rigid restrictions to the number of projections and accessible views for observation. In the last time, it was understood that one effective way to withstand the extreme lack of data is to introduce a priori knowledge based upon classical inverse theory (including Maximum Entropy Method (MEM)) of the solution of ill-posed problems [1-6]. As shown in [6] for objects with binary structure, the necessary number of projections to get the quality of image restoration compared to that of CT using multistep reconstruction (MSR) method did not exceed seven and eould be reduced even further. 

A. N. Tichonov and V. Arsenin Solution of ill posed problems. New York, Wiley, 1977. 

AN.Tikhonov, A.V.Goncharsky, V.V.Stepanov and A.G.Yagola, Numerical Methods for the Solution of Ill-Posed Problems, Kluwer Academic Publishers, Dordrecht / Boston / London, 1995. p251. 

A V Goncharsky and V.V.Stepanov, Inverse Problems in Synthesis of Optical Elements, Ill-Posed Problems in the Natural Sciences, MIR Publishers, Moscow, (1987), pp.318-340. 

If two or more of the unknown parameters are highly correlated, or one of the parameters does not have a measurable effect on the response variables, matrix A may become singular or near-singular. In such a case we have a so called ill-posed problem and matrix A is ill-conditioned. 

A. N. Tikhonov, V. Y. Arsenin 1977, Solutions of Ill-posed problems, John Wiley Sons, New York. 

Other authors concentrated their attention on the evaluation of the time constants. It is well known that multiexponential fit is an ill-posed problem, and its solution, obtained with typical non-linear optimization... 

We note in passing that many more spectra will be accumulated compared to the observable species present. This represents an over-determined problem, one of many qualities present in real physical ill-posed problems. 

A. B. Bakushinski, A. U. Goncharskii, Ill-posed Problems Theory and Application. Kluwer, Dordrecht 1994. 

We repeat the argument that, even if we are going to obtain numerical solutions to a problem, we still need to pose a proper mathematical problem for the computer program to solve. A problem involving two equations with three unknowns cannot be solved with the most sophisticated computer problem. Even worse, with an ill-posed problem, the computer may give you answers, but they wiU be nonsense. 

In general, problems having solutions that vary radically or discon-tinuously for small input changes are said to be ill-posed. Deconvolution is an example of such a problem. Tikhonov was one of the earliest workers to deal with ill-posed problems in a mathematically precise way. He developed the approach of regularization (Tikhonov, 1963 Tikhonov and Arsenin, 1977) that has been applied to deconvolution by a number of workers. See, for example, papers by Abbiss et al (1983), Chambless and Broadway (1981), Nashed (1981), and Bertero et al. (1978). Some of the methods that we have previously described fall within the context of regularization (e.g., the method of Phillips and Twomey, discussed in Section V of Chapter 3). Amplitude bounds, such as positivity, are frequently used as key elements of regularization methods. 

Vol. 3. Ill-Posed Problems for Integrodifferential Equations in Mechanics and Electromagnetic Theory Frederick Bloom... 

Different data interpretation models have been applied simple dissociation constants (Langford and Khan, 1975), discrete multi-component models (Lavigne et al., 1987 Plankey and Patterson, 1987 Sojo and de Haan, 1991 Langford and Gutzman, 1992), discrete kinetic spectra (Cabaniss, 1990), continuous kinetic spectra (Olson and Shuman, 1983 Nederlof et al., 1994) and log normal distribution (Rate et al., 1992 1993). It should be noted that for heterogeneous systems, analysis of rate constant distributions is a mathematically ill-posed problem and slight perturbations in the input experimental data can yield artefactual information (Stanley et al., 1994). 

At this point it has to be stressed that the minimization of the functional (4) is an ill-posed problem [2], This is due to the fact that the number of normal frequencies n for a given polyatomic molecule is less than the number of independent adjustable force-constants, whichis given by n(n+1)/2. The situation is even worse since the number m of vibrations accessible by spectroscopic techniques is smaller than the total number of normal vibrations. It is obvious that additional restrictions have to be applied on the set of force-constants in order to obtain a well defined molecular force field. 

Tikhonov AN, Leonov AS, Yagola AG (1998) Nonlinear Ill-Posed Problems. Chapman and Hall, London... 

Tikhonov, A. N. Arsenin, V. Y. "Solutions of Ill-Posed Problems" John, F., Translator Wiley New York, 1977. 

Hansen, PC., Truncated singular value decomposition solutions to discrete ill-posed problems with ill-determined numerical rank, SIAM J. Sci. Stat. Comput., 11, 503-519, 1990. 

Hansen, P.C., Rank-Deficient and Discrete Ill-Posed Problems Numerical Aspects of Linear Inversion, SIAM, Philadelphia, 1998. 

Kilmer, M.E. and O Leary, D.P., Choosing regularization parameters in iterative methods for ill-posed problems, SIAM J. Matrix Anal. Appl., 22, 1204—1221, 2001. 

Tikhonov, A.N. and Goncharsky, A.V., Solutions of Ill-Posed Problems, Winston Sons, Washington, D.C., 1977. 

Finding the source location and the time history of the solute in ground-water can be categorized as a problem of time inversion. This means that we have to solve the governing equations backward in time. Modeling contaminant transport using reverse time is an ill-posed problem since the process, being dispersive is irreversible. Because of this ill-posedness, the problems have discontinuous dependence on data and are sensitive to the errors in data. 

As for the non-uniqueness of the solution, there is no method that can bypass this inherent problem. In inverse problems, one of the common practices to overcome the stability and non-uniqueness criteria is to make assumptions about the nature of the unknown function so that the finite amount of data in observations is sufficient to determine that function. This can be achieved by converting the ill-posed problem to a properly posed one by stabilization or regularization methods. In the case of groundwater pollution source identification, most of the time we have additional information such as potential release sites and chemical fingerprints of the plume that can help us in the task at hand. 

Ames KA, Clark GW, Epperson JF, Oppenheimer SF (1998) A comparison of regularizations for an ill-posed problem. Math Comput 67 1451-1471... 

Beck JV, Blackwell B, St Clair Jr CR (1985) Inverse heat conduction ill-posed problems. Wiley, New York... 

Tikhonov AN, Arsenin VY (1977) Solutions of ill-posed problems. VH Winston, Washington, DC... 

By virtue of the relationship (14) between frequency and time representation it must be clearly understood that the G(x) calculation does not provide in itself anything more than another way of describing the dynamic behavior of dielectrics in time domain [12]. Moreover, such a calculation is a mathematically ill-posed problem [13,14], which leads to additional mathematical difficulties. 

The integral equation (145) presents a classic example of an ill-posed problem, by which one means that the solution i/dx) does not depend continuously on the data function R(X). In the above formulation of the problem, R(X) is known only for X Xj (j = 1,2,..., m) and the data are given with known errors AR/Xj). With these inadequate data, it is extremely difficult, in general, to solve Eq. (145) (see e.g. Ref. 329). One possible approach is to apply the statistical regularization method (STREG) [330]. 

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