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   Inverse Problems Forum

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Inverse Problems are a mathematical topic related to environmental science, water pollution and they would also be of interest to any oil company. They can also arise when considering a medical question, or analysing medical equipment, for example, a medical specialist might want to know about tomography and scanning.

What are Inverse Problems?

Well, roughly speaking, Inverse Problems could be described as problems where the answer is known, but not the question. Or where the results, or consequences are known, but not the cause. For example, if 3 streams join to form a river, and we know that 3 factories are putting known amounts of pollutant into the streams, then we can calculate the resultant pollutant in the river. This would be the forward, or classical problem. But a more likely problem is that we only know the pollutant in the river, and we have to establish which factory is putting what into what stream. This is the inverse problem.

Another example is in Medical Imaging: If the exact properties of some internal organ were known, then on doing a scan, i.e. Targetting that area with radiation or ultrasound, the resultant reflection/ attenuation map would be known. That would be the forward problem. But it is nearly always the properties of the internal organ that we are trying to find, and ideally without invasive surgery. Thus we have to solve an inverse problem.

As another, simpler example, If you studied Science or maths at school, you may be familiar with the property known as 'Specific Heat Capacity' of a liquid. If you know this, you can calculate, for instance, how long it will take to boil a given quantity of water. Now suppose you know how long a liquid takes to boil, can you tell what liquid it is? You can see the problem has become more complex, especially if more than one liquid shares the same specific heat capacity. It is probably for that reason that you don't usually come across inverse problems in traditional education. This is not a drawback of education as such, it is just that these problems are not easily 'packaged' into a 'set question and answer problem.

Academically, this topic belongs mostly under mathematics, and it is in that area where most research takes place. See
 Prof Kurylev's definition. Also see the example on the right  (bottom) from the mathematical world. 

Theory vs Practice. The chart below shows what theory is being developed for each practical problem. See links on left for the actual resources needed. If you have still not found the resource, please use the form below to request more info.

Geophysical prospecting. General Acoustic problems	Inverse Scattering theory: Given a signal and an unknown obstacle, what does the obstacle look like?
Industrial processing where direct measurement is difficult, particularily in the case of fluid flows.	Inverse problems in fluids: fiinding fluid flows from only the boundary measurements.
Medical diagnosis, scanners	Tomography, electrical imaging. Finding out something about the inside of a body from measurements only taken on the outside.
Algorithm development.	Genetic programming. Letting software evolve rather than using a pre-determined algorithm.

SIAM have published "Inverse Problem Theory and Methods for Model Parameter Estimation" by Albert Tarantola. This can be downloaded free in pdf  form here.

Other books:-

Professor Yaroslav Kurylev, of Loughborough University, Professor Alexander Katchalov, of the Steklov Institute, St. Petersburg and Dr Matti Lassas, of the Rolf Nevanlinna Institute,Helsinki wrote a book on:-

"Inverse Boundary Spectral Problems"

The Gelfand Inverse Problem of the reconstruction of a Riemann manifold and a 2nd order Elliptic Linear Operator on it, from it's Boundary Spectral Data, and applications of this technique for more Spectral Classes of Operators.

Published by Chapman/CRC Press.

    
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So, as t tends to infinity, we can see that u(x,t) tends to zero, i..e. the heat or dispersant decays.  More