On the Ill-Posed Nature of Inverse Problems: An Exactly Solvable Paradigm Inverse Neutron Diffusion Problem Illustrating the Solution's Non-Computability - Paper 48
In general, a physical system and/or the result of an indirect experimental measurement are modeled mathematically in terms of: (a) A system of linear and/or nonlinear equations that relate the system's independent variables and parameters to the system's state (i.e., dependent) variables; (b) Probability distributions, moments thereof, inequality and/or equality constraints that define the range of variations of the system's parameters; (c) One or several quantities, customarily referred to as system responses (or objective functions, or indices of performance), which are computed using the mathematical model. The 'direct problem' solves the 'parameter-to-output' mapping that describes the 'cause-to-effect' relationship in the respective physical process. In practice, the model parameters and/or responses are experimentally measured quantities and are therefore subject to uncertainties. Many measurement problems are 'inverse' to the direct problem in that they seek to determine the properties of the medium (i.e., various cross sections), or the properties of the source, and/or the size of the medium on its boundaries from measurements of quantities that depend on the neutron flux (or from direct measurements of the neutron flux). In general, two problems are called inverses of one another if the formulation of each involves all or part of the solution of the other. Some authors further group such inverse problems into 'invasive', when the interior flux is accessible for measurements as opposed to 'non-invasive', in which only fluxes on the boundary of (or exterior to) the medium can be measured. Examples of inverse radiative transfer problems have been reviewed by McCormick, while examples of inverse source problems for neutron transport have been provided by Sanchez and McCormick. More recently, Bledsoe et al., have presented two methods (the Levenberg-Marquard, and 'differential evolution method') for addressing numerically inverse transport problems. However, although these works (as well as other works cited therein) mentioned the detrimental effects of 'small perturbations in the input data', none of the works published thus far on inverse problems for particle transport or diffusion actually showed explicitly the effects that uncertainties in the measured responses (e.g., detector measurements) would actually have on the 'inverse' deduction (from such measurements) of the underlying medium's properties, sources, etc. This work highlights explicitly the nature of the difficulties encountered in so-called 'inverse problems' by presenting a paradigm inverse neutron diffusion problem admitting an exact solution, in which the 'inverse problem' is to determine the medium's diffusion coefficient from flux measurements. It will be shown explicitly (analytically) that this is not possible to achieve in a unique manner, even in the absence of measurement uncertainties, unless additional error-free information is provided about the quantity to be determined (in this case, the medium's diffusion coefficient). It if further shown that, even if this additional error-free information about the diffusion coefficient to be determined is available, the unavoidable experimental measurements errors, albeit it vanishingly small, in the measured flux render the 'exact solution' useless for computational purposes, since the exact solution turns out to be a divergent series. As discussed in this paper's ENDNOTES section, the paradigm problem presented in this work also underscores the need for 'regularization', which is a qualifier indicating that the inverse problem is to be solved in some approximate manner. (authors)