Inverse Problems web site

Do you have an Inverse Problem?

Do you have an Inverse Problem?

By Bob Marlow, Loughborough University

Did you ever play with a thing called a "Puzzle Box" in your
physics class? It was a wooden box with 4 or more sockets on top,
and inside the box the sockets were connected by resistors,
capacitors and the like. You were armed with ammeters, voltmeters
etc, but you weren't allowed to look inside the box (unless you
got so curious you had to!). So you had to establish exactly what,
if anything, connected the sockets up. It could be very frustrating.

This was my first serious introduction to "Inverse Problems". a subject
coming more and more into prominence in the last ten years (although
its origins could be said to date back to the 1930's!).

It's curious that these little puzzles were given to us to do only
once, as a bit of 'lateral thinking', or light relief from the main
wodge of theory we had to absorb. Because in a lot of everyday
professional applications of course, we don't always have exact
measurements to put into specific formalae. However, there is
an area of mathematics and physics where exactly these sorts of
problems are dealt with - Inverse Problems.

They 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 three streams join to form a river,
and we know that three 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 what the pollutants are
in the river, and we have to establish which factory is putting what
into which 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.

For another, simpler example:
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 in the field of mathematics,
and it is in that area where most research takes place.
Prof Yaroslav Kurylev, of Loughborough University writes:-

"Inverse problems are natural for many applications. For many centuries people are searching for hiding places by tapping walls and analysing the echo. This is a particular case of an inverse problem. Generally, inverse problems are those of finding some characteristics of a medium from knowledge of some fields interacting with the medium. These fields (or some of their characteristics) are usually measured outside the medium, for instance, on its boundary.
Mathematically, a problem may be formulated in the following way: assume the behaviour of a field is described by a differential equation Pu = f where f is a source and coefficients of P reflect properties of the medium. Assume we are able to measure u outside M, where M is a region occupied by the medium. What can we say about coefficients of P (and sometimes f)?
The name "inverse" has come from the fact that it is traditional for mathematics to consider the problem of finding u in the case when P and f are given. Then the problem of determination of P from u may be regarded as "inverse" to the one described above."

What are the Applications of Inverse Problems?

The table below shows roughly the practice/theory grid:-

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 taken only on the outside.
Algorithm development.   Genetic programming. Letting software evolve rather than using a pre-determined algorithm.

Bob Marlow is a system developer, with experience
of mathematical modelling in fluid dynamics, geophysics
and Kalman filtering techniques.
He is currently involved with encryption issues and
providing network support, and is studying Inverse
problems as a PhD subject at Loughborough University.

For more information on Inverse Problems, visit:-

http://www.inverse-problems.com