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Inverse Problems
Let's say you're a doctor, and you want to see the inside of someone using only ordinary, visible light. You might think this isn't possible because people don't transmit light, but it turns out that people do transmit light -- if you use a bright enough light. You can see this for yourself if you go into a dark room and cover a flashlight with your hand. You'll see some light coming through your hand, but you won't see a clear image of what's inside your hand.
The real problem is that the light inside your hand is scattered. By this I mean that it bounces around a lot at random inside the hand, so by the time it comes out of your hand, it doesn't carry information about any one part of your hand in particular. That's why if you try this flashlight experiment, you won't see much more than some blurry redness.
We can do better though, if we use the mathematical models of how light behaves in a scattering medium. We know that the light inside your hand behaves according to certain equations whose parameters are determined by what's inside your hand. So mathematically, we have the following question: If we know what the solutions of these equations look like on the outside of the hand, can we determine what the parameters of the equations are inside the hand? It turns out that we can do this, and the resulting procedure is called optical tomography.
This is a classic example of an inverse problem -- a problem in which we measure the solutions to an equation and try to determine the parameters of the equation. There are many more examples. For example, you could try to build a 3-D image of the inside of a person by sending X-rays through at different angles: this is what happens in a CT scan. Or you could try to apply electrical current to the boundary of an object and measure the resulting voltages: this is called electrical impedance tomography. Or you could try to image the inside of the earth by measuring the pressure waves from earthquakes around the world: seismic tomography. All of these cases can be described by a mathematical problem in which we try to determine the parameters of an equation by making some sort of measurement on its solutions.
I study the mathematics behind these kinds of problems. For more information on what I've worked on in the past, you could return here and read some of my papers. But those definitely have lots of PDEs in them.