Ill-posed inverse problem equals loads of fun?
Unfortunately two negatives don’t make one positive in the case of inverse problems. However inverse problems turned out to be very important in applied mathematics when I met Samuli Siltanen, the professor of industrial mathematics at the University of Helsinki.
Professor Samuli Siltanen. Photo by Jan Jansson.
I went to the department of mathematics and statistics at the University of Helsinki to find out what inverse problems in mathematics mean. I had done my googling before I met Samuli Siltanen. It somehow seemed to me that inverse problems like ´what are the factors of six´, instead of asking ´what does multiplying two by three give´ aren’t the kind of problems that a professor of industrial mathematics would go over. Indeed, it turned out that usually inverse problems are a bit more complicated.
MY Campers getting to know modern mathematical methods
When he has time, Samuli Siltanen promotes studying mathematics in schools. He was delighted to be able to join the Millennium Youth Camp team, and guide the MY Campers to work on an inverse problem. For the MY Camp, the problem was made a bit simple but it still is inverse and closely related to the real problems of applied mathematics. In this example a 2D object of unknown inner structure is X-rayed four times. The resolution of the picture to be created is 49 pixels. It is a bit like solving a sudoku with the exception that there may be any numbers in the grid which is 7 by 7 and the sum of the numbers in each row and column isn’t the same but they are given by the x-ray measurement.
Since sudokus have several descriptive restrictions and contain a lot of hints, one can just try out all the possibilities. You could probably find some correct answer to the MY Campers’ problem by just trying out different numbers, but it wouldn’t do when working with real-life objects. To be able to do some industrially applicable mathematics one has to be able to work with mathematical calculation software to solve the extremely complicated equations that one gets from problems like this. The MY Campers are also working with computers trying to write a program to solve the problem. With modern maths blackboard just isn’t enough anymore.
To model complex physical situations one has to know some programming as well. To model physical situations at all one also has to know some physics. Siltanen points out that with applied mathematics and real-life problems the mathematician needs to have a general good knowledge of several areas of mathematics. Theoretical mathematicians concentrate in one area of mathematics but in applied mathematics one can use any appropriate way to attack a problem.
Mathematics seems to require a bit of creativity every now and then.
Real-life inverse problems are solved with even less information
Siltanen quite doesn’t think that asking for the factors of six grasps the magnitude of inverse problems. The question is a pair to an easier direct problem which is true for all inverse problems. It does have multiple correct answers (including 1*6 and 2*3) too but it isn’t as sensitive to measurement errors as real-life inverse problems are. To explain this further the professor of industrial mathematics takes a plastic model of human jaw with teeth and for a short moment I think I might have taken a wrong turn and ended up in the biology department after all.
A more typical inverse problem arises in the situation of taking an X-ray at the dentist’s. Siltanen explains that if one is missing a tooth and it needs to be replaced with an implant, it is important to know how deep the dentist can drill to attach the implant without damaging any nerves for example. X-rays pass through tissues and bone but are also partially absorbed. The denser the material the less X-rays pass to the other side. This gives us an idea about the total density in the path that the ray has passed. It doesn’t tell where exactly the density was located.
For an X-ray image to give adequate information to the dentist, it needs to be quite sharp. To create a 3D picture one has to know location of everything within 0,1 mm. However it isn’t safe to take very many doses of radiation to get a tooth implant. In the typical case there is some milliard (109) 0,1*0,1*0,1 mm cubes of material that need to be characterised in a jaw. If you take just five pictures with a typical X-ray you get five times a megapixel or two of data. This only gives some millions (106) pieces of information. When there is less information than variables, possibility of several correct answers arises. The answer is not unambiguous anymore but ambiguous.
Siltanen has actually worked on this problem with some 20 other people for 8 years. They solved it and made mathematical technology serve a real purpose. I can see why it took some time.
Jan Jansson is a chemistry teacher that spends his little free time doing molecular gastronomic experiments.