Determination of the spatial distribution of a physical parameter from the distribution of another physical variable— a differential inverse problem

We propose a new type of inverse problem, in which the spatial distribution of the relative value of a physical parameter can be determined only from the distribution of another physical variable in the region of interest if there is no source in that region. The inverse problem proposed here has two features different from conventional remote probing type problems. One is that the physical variable data are given throughout the region of interest although the number of data variables is insufficient to determine directly the physical parameter of interest, while only remote data are given in a conventional problem. The other is the mathematical structure of the inverse problem: this new inverse problem yields a spatial differential equation on the target parameter, while the conventional problem becomes an integral equation on the target parameter. To show the nature of the problem, we formulated an illustrative inverse problem on the steady-state electric current field, in which the spatial distribution of the relative conductivity can be determined from the distribution of either the potential or the current density. We give examples of computational results obtained when using the simulated potential data and the current density data. This proposed formalism for a new type of inverse problem could provide a new approach and open up new measurement schemes in many fields of science and engineering.