G. Parravicini et al., MINIMAL A-PRIORI ASSIGNMENT IN A DIRECT METHOD FOR DETERMINING PHENOMENOLOGICAL COEFFICIENTS UNIQUELY, Inverse problems, 11(3), 1995, pp. 611-629
We identify the coefficients of the transport equation in N dimensions
grad c . grad h + c Delta h = d partial derivative h/partial derivati
ve t + f by solving a differential system of the form grad c + ca = b.
The assignment of c at one point only yields a unique solution, found
by integration along arbitrary paths. This arbitrariness guarantees a
good control of the error, notwithstanding the ill-posedness of the p
roblem. For N = 2, the hypotheses allowing for this identification are
satisfied when one knows two stationary potentials with non-overlappi
ng equipotential lines and a third nonstationary one-this last needed
only for determining d. The theory is applied to a numerical synthetic
example, for various grid sizes or for noisy data. Notwithstanding th
e minimal a priori information required for the coefficients, we are a
ble to compute these at a large number of nodes with good precision. F
or the sake of completeness, we give other results on identification.