Homotopy, Polynomial Equations, Gross Boundary Data, and Small Helmholtz Systems
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Inverse problems of the boundary measurement type appear in several geophysical contexts including DC resistivity, electromagnetic induction, and groundwater flow. The objective is to determine a spatially varying coefficient in a partial differential equation from incomplete knowledge of the dependent variable and its normal gradient at the boundary. Equivalent 2D discrete inverse problems based on the Helmholtz or modified Helmholtz equation reduce to systems of polynomial equations indicating that there are only a finite number of exact solutions, excluding certain pathological cases. A homotopy procedure decides whether real, positive solutions exist and, if so, generates the entire list. The computational complexity of the algorithm scales as MM/2, where M is the number of model parameters to be found. Measurement errors are accommodated by oversampling the boundary data at additional frequencies. For test Helmholtz and modified Helmholtz inverse problems based on (i) perfect and (ii) noisy data I generate the full list of exact solutions. The homotopy approach applies to large scale, multidimensional geophysical inverse problems but at present is practical only for small systems, up to M = 9. Recent advances in homotopy theory should, however, reduce the complexity, making larger problems tractable in the future. 1996 Academic Press, Inc.
