Homotopy, Polynomial Equations, Gross Boundary Data, and Small Helmholtz Systems
✍ Scribed by M.E. Everett
- Book ID
- 102584929
- Publisher
- Elsevier Science
- Year
- 1996
- Tongue
- English
- Weight
- 340 KB
- Volume
- 124
- Category
- Article
- ISSN
- 0021-9991
No coin nor oath required. For personal study only.
✦ Synopsis
Inverse problems with boundary measurements also netic induction, and groundwater flow. The objective is to determine have a medical application. In a technique known as impea spatially varying coefficient in a partial differential equation from dance tomography [8], current is injected into the human incomplete knowledge of the dependent variable and its normal body and voltages are measured. Like the analogous geogradient at the boundary. Equivalent 2D discrete inverse problems physical data, the voltages are sensitive to the distribution based on the Helmholtz or modified Helmholtz equation reduce to systems of polynomial equations indicating that there are only a of electrical conductivity within the body, which is an indifinite number of exact solutions, excluding certain pathological cator of blood flow and the health of organs. Finally, there cases. A homotopy procedure decides whether real, positive soluis a close relationship between the above problems and tions exist and, if so, generates the entire list. The computational that of determining from spectral data the inhomogeneous complexity of the algorithm scales as M M/2 , where M is the number mass distribution of a vibrating string or membrane. This of model parameters to be found. Measurement errors are accommodated by oversampling the boundary data at additional frequen-is the classic inverse problem that once prompted the mathcies. For test Helmholtz and modified Helmholtz inverse problems ematician Mark Kac to ask [2] ''Can one hear the shape based on (i) perfect and (ii) noisy data I generate the full list of of a drum?'' exact solutions. The homotopy approach applies to large scale, In this paper I show that a 2D discrete Helmholtz inverse multidimensional geophysical inverse problems but at present is problem with enough boundary measurements reduces to practical only for small systems, up to M ϭ 9. Recent advances in homotopy theory should, however, reduce the complexity, making a well-determined system of polynomial equations. Such larger problems tractable in the future.