Research in Inverse Problem and Numerical PDE

Question

I am taking a Thesis-based Master degree now and I am going to choose my supervisor soon. I plan to take a PHD degree after graduation, so if possible, I wish my PHD research area could be an extension of my Master Thesis ( so that the researching experience during my Master wouldn't be wasted).

Currently I am interested in the research areas of two professors. Professor A focuses on Numerical Solution on PDE. Professor B conducted research on Numerical PDE several years ago (He was the supervisor of Professor A's Master Thesis) but it seems that he has changed his research field to Inverse Problems in the last 5 years.

Because I am not quite familiar with Inverse Problem, currently I am more interested in Numerical PDE. But since Professor B has richer researching experience, I think his recommendation may be more helpful for my PHD application.

So may I ask what is the basic knowledge for research on Inverse Problem? Does it share the same research technique with researching on Numerical PDE? Is it possible for me to change from Numerical PDE to Inverse Problem without encountering many difficulties ( in PHD application and researching)?

Thanks for your help. Any comment would be highly appreciated.

Answer 1

If you plan to continue your PhD with the same adviser at the same university, then it seems to me your plan to keep the topics related is a good one. But, since it sounds like you want to do your Masters degree and then find a new (better?) place with a new adviser, I'd suggest that you're likely to find it hard to keep the exact topic going.

This doesn't mean that your Masters degree research will have been wasted, though! If you plan carefully, the background you develop while working on your Masters degree will dovetail nicely with your PhD topic. One prepares you for the other.

You've given a good example in your question, in fact. Numerical solution of PDEs is a part of the numerical solution of inverse problems. You can't to do the latter without the former. You could do a Masters with the PDE adviser and then move on to a PhD in inverse problems if that was interesting to you.

Answer 2

Forward modeling alone is of little use. 'Most' problems, both in engineering and science are really inverse problems. No matter how cool you forward models are unless you can use it to optimize something or estimate some parameters of interest (that have scientific implications) it will be of little use in applied sciences. So yes PDE constrained optimization is a good area.

However you also need to learn how to solve forward models (numerically I presume) and it does require some basic skills. So you can do you MS in numerical solution of PDE's (hopefully write a code) and then later pursue a PhD in inverse problems (and apply your code) to solve interesting problems and publish papers.

It is natural for people who do forward modeling to move into inverse modeling later on and it might very well be that Prof A follows the steps taken by Prof B. In fact there are many such examples, specially in the HPC community. A few decades back computers weren't powerful enough to solve forward problems, let alone inverse problems but now solution of inverse problems is routine, both in academia and industry.