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Advanced Math/underdetermined problem


Iím conducting a R&D project recently.  It belongs to an underdetermined problem (no. of equations < no. of variables).  Luckily, several tests can be conducted for the same system; nevertheless, each test is always an underdetermined problem.  My question is: how to reach the final solution based upon several tests?  Many thanks for your valuable suggestions

This sounds like it could be a tomography problem. Acoustic tomography is used in oceanography to determine the density and sound speed profiles over an extended area.

The basic idea in that application is to measure the travel times of acoustic (or seismic) pulses between two points located at opposite sides around a perimeter of a large area. The travel time between pairs depends on the integrated sound speed along the path between them. The values from many pairs around the perimeter can be inverted to provide estimates of the sound speed in the interior of the area.

This a non-trivial problem but there is lots of literature on it. General inverse theory, such as used in geophysics, would be a good place to look to see if your problem is similar to standard problems. Resolution versus accuracy is the trade-off you'll face, but it is a solved problem.

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randy patton


college mathematics, applied math, advanced calculus, complex analysis, linear and abstract algebra, probability theory, signal processing, undergraduate physics, physical oceanography


26 years as a professional scientist conducting academic quality research on mostly classified projects involving math/physics modeling and simulation, data analysis and signal processing, instrument development; often ocean related

J. Physical Oceanography, 1984 "A Numerical Model for Low-Frequency Equatorial Dynamics", with M. Cane

M.S. MIT Physical Oceanography, B.S. UC Berkeley Applied Math

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