“Back to their source the holy rivers turn their tide. Order and the universe are being reversed…” (Euripides, Medea)
Most of the time, when people find out I am a mathematician, they are intrigued to hear that mathematics is not a static, dead science, but rather a rich and exciting area of research. Many are then curious to know what I am researching... “Inverse problems” is my answer, just to add to their bewilderment…
But what is an inverse problem? The definition is not straightforward even for a trained mathematician. To emphasise this difficulty, Charles W. Groetsch mentions a situation famous in the history of the United States Supreme Court, whereby in 1964, Justice Potter Stewart said he could not define ‘obscenity’, but he knew it when he saw it. The Supreme Court eventually ruled that obscenity is defined by ‘community standards’.
Similarly, mathematicians find it hard to describe an ‘inverse problem’, but most of them recognise one when they see it. The recognition stems from their familiarity with accepted scientific ‘standards’ for ‘direct’ problems that dominate the undergraduate training in mathematics, where students are provided with enough information to carry out a sequence of well-defined and stable computational steps leading to a unique solution.
One of the first ‘direct’ problems we learn about in school is multiplication: find the product of any two numbers. The corresponding ‘inverse’ problem is to find a pair of factors for a given number. Obviously, the problem of factorisation, like many other inverse problems, does not always have a unique solution. Moreover, by trying to impose uniqueness on the solution of this inverse problem, we open the door to the magical world of prime numbers.
From the mathematical point of view, the prevailing paradigm in direct problems can be described by the following: cause -> model -> effect. However, this is only one-third of the story, as there are two inverse problems corresponding to every direct problem. One is the causation problem: given a model and an effect, find the cause of the effect. The other is the model identification problem: given cause-effect information, identify the model.
Formally, an inverse problem is a mathematical framework that is used to obtain information about a physical object (cause) or theory (model) from observed measurements (effect). The solution to this problem is useful because it generally provides information about physical parameters that we cannot directly observe, and has been an important factor in the development of mathematics and science in the past 30 years.
The realm of applications of inverse problems is vast, including medical imaging techniques, geophysical explorations, computer vision, astronomy, non-destructive testing and non-invasive evaluation (see Sound Bites).
Most of the research is interdisciplinary and the community is constantly expanding. About 300 participants will attend the 9th ‘Inverse Problems: Modelling and Simulation’ international conference that will take place in Malta in May 21-25, 2018, at Paradise Bay Hotel.
Reference: Charles W. Groetsch, Inverse Problems. Activities for Undergraduates. The Mathematical Association of America, 1999.
Cristiana Sebu is a member of the international programme and organising committees of the abovementioned conference.
Did you know?
• In Book VII of his Republic, Plato discusses a situation in which human beings who are kept captive and chained in a cave since their childhood are faced with solving an inverse problem: reconstructing the cause (real world outside) of the effects (shadows projected on the back of the cave) of a distant fire (the model).
• Aristotle presented one argument for the Earth’s sphericity based on inverse theory: the shape is inferred from the observations of Earth’s curved shadow on the moon’s surface during eclipses.
• Eratosthenes (276-195? BC) could not measure the circumference of the Earth directly, but by a brilliant use of indirect reasoning he solved the inverse problem of determining the angle from the arc and arrived at an impressively accurate estimate of Earth’s girth.
• According to Mark Twain’s Adam’s Diary, Eve used to solve the Inverse Problem of Eden by shape recognition. “The new creature names everything that comes along, before I can get in a protest. And always that same pretext is offered – it looks like the thing.”
For more trivia see: www.um.edu.mt/think
Sound bites
• Prof. Cristiana Sebu from the University of Malta in collaboration with colleagues at University of Mainz, Germany, and Oxford Brookes University, UK, has been working on a new way to quickly and easily screen people for breast cancer in a non-invasive way.
Electrical Impedance Mammography (EIM) works by applying a very low electrical current to the breast tissue with a flat sensor. The sensor then reads back the voltage and current across the different areas of tissue, which are then used by an inversion algorithm to reconstruct the electrical properties (conductivity and permittivity) of breast tissue. Cancer tissue gives readings of around three to four times higher than that of normal tissue, making it easier to detect a change and spot its exact location. Benefits of EIM are its low cost and portability: tests are much cheaper and easier to conduct than the traditional mammography we use today. Prof. Sebu is even working on developing a bra that has the same technology, making it incredibly quick and easy to perform, which could revolutionise breast cancer diagnosis. This work was featured in a 3D hologram set-up for the festival Science in the City held in Valletta on September 29, 2017.
• On October 16, 2017, Phys.org reported that scientists from the Ligo-Virgo collaboration provided the first-ever glimpse of two neutron stars colliding. Einstein’s theory of relativity predicted that mergers of two such exotic bodies would create ripples in the fabric of space-time known as gravitational waves, as well as bright flashes of high-energy radiation called gamma ray bursts. Detectors witnessed both phenomena, 1.7 seconds apart, coming from the same spot in the Hydra constellation. The impact happened 130 million years ago but was so far away that the light and gravitational waves have only just reached us. The stars’ masses were 10-20 per cent bigger than the sun’s, but they were no larger than 30km across. One of the most challenging inverse problems is to determine the equation of state of matter at supranuclear densities via gravitational-wave detection [Monthly notices of the Royal Astronomical Society:320(3), 307–315, (2001)].
For more science news, listen to Radio Mocha on Radju Malta every Saturday at 11.05am.