Intersections, ideals, and inversion
Techniques from computational algebra provide a framework for treating large classes of inverse problems. In particular, the discretization of many types of integral equations and of partial differential equations with undetermined coefficients lead to systems of polynomial equations. The structure...
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| Format: | Report |
| Language: | English |
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Lawrence Berkeley National Laboratory
1998
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| Online Access: | https://doi.org/10.2172/6448 https://digital.library.unt.edu/ark:/67531/metadc690439/ |
| Summary: | Techniques from computational algebra provide a framework for treating large classes of inverse problems. In particular, the discretization of many types of integral equations and of partial differential equations with undetermined coefficients lead to systems of polynomial equations. The structure of the solution set of such equations may be examined using algebraic techniques. For example, the existence and dimensionality of the solution set may be determined. Furthermore, it is possible to bound the total number of solutions. The approach is illustrated by a numerical application to the inverse problem associated with the Helmholtz equation. The algebraic methods are used in the inversion of a set of transverse electric (TE) mode magnetotelluric data from Antarctica. The existence of solutions is demonstrated and the number of solutions is found to be finite, bounded from above at 50. The best fitting structure is dominantly onedimensional with a low crustal resistivity of about 2 ohm-m. Such a low value is compatible with studies suggesting lower surface wave velocities than found in typical stable cratons. |
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