Wouldn’t it be great to have all the answers? But what if you had none of the questions? Imagine getting a load of data, but not knowing exactly what conditions produced it – and your job is to figure that out. Some mathematicians do this every day. Their upside-down, inside-out world is the field of inverse problems, and it actually makes a lot of sense, when you think about the way many scientists and doctors work. Right now, this specialty is growing and finding ever more applications, in areas from medical imaging to weather forecasting and seismic studies. We learned more about how it works and how this area of research in mathematics is always bringing improvements on the practical side.
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Wouldn’t it be great to have all the answers? But what if you had none of the questions? Imagine getting a load of data, but not knowing exactly what conditions produced it – and your job is to figure that out. Some mathematicians do this every day. Their upside-down, inside-out world is the field of inverse problems, and it actually makes a lot of sense, when you think about the way many scientists and doctors work. In areas from medical imaging to seismic studies, we know what comes out of, say, the MRI scanner or the seismograph, and have to work backwards to the initial conditions that existed at the start.
Right now, this area of study is growing and finding ever more applications. For the past three months, researchers in inverse problems from the world over have been gathered at the Institut Henri Poincaré in Paris for an intensive quarter of study, discussion, and brainstorming. Two of its organizers, Jérôme Le Rousseau, a professor at the University of Orléans, and David Dos Santos Ferreira, himself a professor at the University of Lorraine, both in France, told us more about these backward-facing problems.
An MRI scan of the human head (Wikipedia CC-BY)
But before we go backwards, what does it mean to go forwards? With a complex subject of study like the human body or the Earth, if you already know the characteristics of a particular system, you can feed numbers into it—the starting parameters of your scenario—and see what comes out. If the defined system is an earthquake model, a mathematician can input the given conditions and simulate the quake they would produce.
Now imagine you don’t already know your system inside and out. Maybe it’s a patient who needs to be checked for internal tumors, or a region where geologists want to determine the location of magma below the Earth’s surface in order to assess the risk of volcanic activity. In either case, you need to examine the interior, but making a hole and shining a light inside is not a practical option. One approach is to send waves of energy through it and look at how they behave. The way they are knocked off their original course, slowed down or deflected, tells us about the material they have passed through. This time, the numbers you have to work with are the results recorded when these waves come out on the other side, rather than the starting point. So, what were the conditions that produced them? By answering that question, you can reconstruct the structure of the kidney that would scatter waves in the way detected by your MRI machine or the components of the earth beneath your feet. It’s a problem worked out in reverse – an inverse problem.
Screenshot from a presentation given by Prof. Samuli Siltanen during the introductory school of the IHP quarter on inverse problems: “Reconstruction methods for Ill-posed inverse problems - Part 1”
Always advancing with inverse problems
This area of mathematics is what makes such techniques possible, in all sorts of fields, like oil and gas exploration and weather forecasting. And inverse problem mathematicians continue working to improve them, making them more reliable, more precise or safer for patients. Consider medical imaging again. A major limit on the resolution that can be achieved in scans is patient safety: their exposure to the energetic waves must be kept below a certain limit. How can math get around this basic, physical constraint? “That’s an issue mathematicians will address anyway in their work,” explains Jérôme Le Rousseau. “They ask themselves, ‘If I reduce my data, do we lose resolution? Do we lose quality?’” If they succeed in adapting their methods to less data and can answer “no” to those questions, the consequence could be shorter exams to obtain the same result, meaning less x-ray exposure for patients.
An example of this principle at work can be found in the dentist’s chair. Affixing dental implants requires digging into the jaw, where there are nerves that you do not want touched, but they aren’t in the same place for everyone. Working alongside dentists and engineers, specialists of inverse problems helped achieve better imaging results for the localization of the nerves, while keeping x-ray exposure down.
Screenshot from Prof. Siltanen’s presentation “Reconstruction methods for Ill-posed inverse problems - Part 1”.
State-of-the-art applications aside, inverse problems are nothing new. “They’re as old as humanity. Indirect measurements were used by the Ancient Greeks to access ‘impossible-to-measure’ physical quantities. Eratosthenes measured the length of shadows to estimate the circumference of the Earth,” explains David Dos Santos Ferreira. “Anyone who says, ‘I don’t have access to the right measurement, but I’ll find the answer by other means.’ is using an inverse problems approach.” But today’s uses do make clear the intimate relationship that exists between downright hardcore mathematical research and applications that are perfectly familiar, even essential.
To continue this progress that inverse problems have made possible, Prof. Le Rousseau underlines both the importance of investing in research – and without expectation of immediate returns – and the critical need for scientific freedom. The latter is what makes creativity possible. This is a fundamental ingredient in research and innovation, he says, and the IHP, host of the quarter dedicated to inverse problems, is a renowned location for just such academic freedom. Prof. Dos Santos Ferreira highlights the enriching quality of exchange across disciplines, as when a group of geophysicists came to speak to the mathematicians assembled. “These interactions bring new ideas and methods. Our research is nourished by these other disciplines, all the while letting us remain mathematicians.”
Find out more:
Connect with the researchers of the inverse problems community and learn more about their work, on the IHP Polaris platform: https://ihp.mysciencework.com/
Watch a short introduction to inverse problems and the Institut Henri Poincaré program, with co-organizer Colin Guillarmou
Gunther Uhlmann’s talk, “Harry Potter’s Cloak”, on another application of inverse problems: metamaterials and cloaking