Simone Brugiapaglia: The Future of Computational Mathematics

Simone is a PIMS sponsored postdoctoral fellow based in the Department of Mathematics at Simon Fraser University (SFU).

In many ways, Simone considers himself a computational mathematician — indeed, all of his publications have involved, to some extent, computer experiments. As his time as a prolific PIMS postdoc at SFU comes to a close, Simone is excited about the next chapter of his career as an Assistant Professor of Mathematics at Concordia University, where he will work alongside artificial intelligence and deep learning experts.

PIMS sat down with Simone to talk about work, music, and his career now and in the future; this is an extract of the full interview.

1. Why did you write your PhD on numerical methods for partial differential equations (PDEs)

I got my Master’s Degree at the University of Pisa. My Master’s thesis was about numerical linear algebra. Although I really enjoyed this topic from a mathematical viewpoint, I felt the need to work on a problem more directly linked with applications. That is the main reason why I decided to apply for a PhD in “Mathematical Models and Methods in Engineering” at MOX Laboratory of Politecnico di Milano and to work on numerical methods for PDEs. It was a great opportunity to see Mathematics “in action” in the real world.

My PhD on the Numerical solution for PDEs is an essential part of computer simulation. Engineers use this on a daily basis. For example, if you’re designing a car or an aeroplane, instead of going to the wind tunnel, you can create a total simulation experiment on a computer, so you can save a lot of money. There’s a strong motivation for people in the private sector to implement simulations by finding the PDE they need to simulate their particular problem and then solve the PDE, which is just not possible with pen and paper, you need huge computational power.

2. Are there any real-world implications/uses for your research? If so, what are they?

Yes, there are plenty of them!

I decided in my first years of university that I liked the idea of seeing that what I do matters to someone; to work on problems and algorithms that people can potentially use in the real world.

It’s really exciting to see how mathematicians can create such a huge impact in the real world.

A lot of my research activity revolves around the compressive sensing paradigm, a very powerful principle used in signal processing to acquire a signal by using the minimal amount of measurements. Compressive sensing can be applied to acquire measurements more efficiently in medical imaging (for example, MRI or X-Ray CT), in image or video acquisition, or in seismic imaging.

Comparison between different acquisition strategies in MRI. The quality of the reconstructed image can be substantially improved if the measurements are carefully chosen according to advanced mathematical principles. Figure from [8] (see bibliography below).

For example, I study the algorithms used to choose the measurements that the MRI does and how the MRI reconstructs the image once the measurements are given.

Regarding my research on numerical methods for PDEs, the main goal is to produce computer simulations of physical phenomena usually related to mechanics, fluid dynamics or electromagnetism. The applications are countless. For example, this type of computer simulation is crucial in aerospace engineering, to design the shape of an airplane in the most efficient way.

3. If you were given an unlimited research budget, what would you like to work on and why?
 
Interesting question! With an unlimited budget, I would not be particularly worried about the research topic. I would rather invest to improve the computational aspect of my research.

It would be extremely cool to have access to a virtually unlimited amount of computing power and to have many collaborators expert in coding who could exploit this computational power in the best way. In this way, I could instantly try out a new idea for a new algorithm or check if a conjecture that I have in mind is reasonable or not. And I could do this in an effortless way, avoiding time-consuming things like debugging or running multiple simulations, which may take a long time on my laptop. This would allow me to focus mainly on the mathematical aspect of my research.

For example, through Compute Canada we can wait in line to do big simulations on a cluster; however, it requires IT skills to adapt the code to run on big machines. It’s a long-term goal to have IT people helping me and collaborating with more data scientists, combining my theoretical skills with their practical implementation skills.

4. What is your favourite aspect of teaching?
 
I really love how teaching helps me learn and understand concepts in a fresh and deeper way. Very often, while preparing a lecture, while teaching, or while answering a student’s question, I get new insights and opportunities to expand my knowledge. This is particularly true during office hours (which I love!).

Interaction with students is a great way to help them and me understand each other’s way of thinking about a problem, a definition, a theorem, or a concept. We learn from each other and have new insights.

5. You’re an active speaker, giving numerous talks in various locations. What do you enjoy about giving these talks?
 
 Communicating my research to other people forces me to explain it in a clear and compelling way. To give a nice talk, you need to optimize the narrative of your presentation, stress the motivation of your research, convey the main intuitions behind your work. This process helps me understand the problems I am working on at a deeper level and gives me new ideas, which may lead to further research goals.

A great aspect of giving talks is the Q&A after the presentations. This moment is extremely valuable and can lead to surprising ideas, future research projects, and new collaborations.

6. Why did you decide to become the President of the Simon Fraser University Postdoctoral Association?
 
My first year of postdocs was not easy. My wife and I just moved to Canada, we had to start a new life from scratch. I was also looking for new research directions for my postdoc. All of these aspects created pressure and stress. Moreover, the life of a postdoc (especially in Math), can be very isolated. This experience, which is very common among postdocs, made me realize how important it is to have a nice and healthy postdoc community around you.

This is why I decided to get directly involved with the SFU PDA in 2017 as a vice president of Policy and as its president in 2018. Serving the SFU PDA is my way to help SFU postdocs find a welcoming, lively, and supporting community around them. I am really honoured to have this opportunity.

7. How does music, your drumming, and being in a band compliment your work as a mathematician?
 
Usually, people think that music and Math go nicely together because rhythm and harmony are based on proportions, subdivision, and numbers. Although this is certainly true, I feel that this is not the main aspect that attracts me towards music and, in particular, to drumming. I really see drumming as a way to express the most irrational and intuitive side of me.

Math is the queen of rationality — it is pure mind. On the other hand, drums are the most physical instrument. They are pure vibration! In addition, a great aspect of music is the connection it generates. Some of the greatest friends I have are musicians and playing together is a great way to cultivate a relationship while having a lot of fun!

8. How has PIMS helped your career?

I got the PIMS fellowship, which allowed me to have a nice postdoc experience, which I’m very grateful for. PIMS is a really good group with many talented mathematicians, and this was a great way to meet people, not only at SFU but also UBC and other parts of Canada.

I have particularly fond memories of the Postdoctoral Training Centre in Stochastics (PTCS) at the Banff International Research Station. It featured really high-quality math — a very nice experience for me.

I also co-organized a PIMS mini-symposium about sparse recovery, learning, and neural networks, and it was really exciting to work with my advisor at SFU, Ben Adcock, but also some professors at UBC (Yaniv Plan and Ozgur Yilmaz) who are part of the PIMS community. It’s great to have this around you.

9. What’s next for you?
 
 It looks like Math will still be an important part of my life in the years to come. I have just accepted an offer for a tenure-track position at the Assistant Professor level in the Department of Mathematics and Statistics at Concordia University. I am very excited about starting this new chapter of my life. My wife and I are really looking forward to moving to Montreal!

I’m also working on a project, in collaboration with another mathematician and an engineer, based on the work of my PhD thesis. This involves combining compressive sensing with isogeometric analysis, to simulate the stress on industrial components. Once a designer has the Computer Aided Design File (CAD) of a particular component it can be combined with the PDE solver. The collaborators in Italy are working on this big software called Geo-PDEs and we’re combining their isogeometirc analysis software with the CORSING method proposed in my PhD thesis. Plug in your CAD file, choose which PDE you want to solve on that domain and you solve it.


Contact Simone

Address: Department of Mathematics, Simon Fraser University, 8888 University Drive Burnaby, BC V5A 1S6

E-mail: simone_brugiapaglia@sfu.ca

Website: sites.google.com/view/paglia


Publications

1. S.B., S. Micheletti, F. Nobile, S. Perotto. Wavelet-Fourier CORSING techniques for multi-dimensional advection-diffusion-reaction equations. Submitted, 2018. [arXiv:1812.09403]

2. B. Adcock and S.B. Sparse approximation of multivariate functions from small datasets via weighted orthogonal matching pursuit. Submitted, 2018. [arXiv:1810.11115]

3. S.B. A compressive spectral collocation method for the diffusion equation under the restricted isometry property. Submitted, 2018. [arXiv:1807.06606]

4. B. Adcock, C. Boyer, S.B. On oracle-type local recovery guarantees in compressed sensing. Submitted, 2018. [arXiv:1806.03789]

5. B. Adcock, A. Bao, S.B. Correcting for unknown errors in sparse high-dimensional function approximation. Submitted, 2017. [arXiv:1711.07622]

6. S.B., B. Adcock. Robustness to unknown error in sparse regularization. IEEE Transactions on Information Theory, 64(10), 2018. [arXiv:1705.10299]

7. B. Adcock, S.B., C.G. Webster. Compressed sensing approaches for polynomial approximation of high- dimensional functions. In “Compressed Sensing and its Applications”, Springer series “Applied and Numerical Harmonic Analysis”, 2018. [arXiv:1703.06987]

8. S.B., B. Adcock, R. K. Archibald. Recovery guarantees for compressed sensing with unknown errors. Proceedings of the 12th International Conference “Sampling Theory and Applications” (SampTA). Tallinn, Estonia, 2017. [arXiv:1702.04424]

9. S.B., F. Nobile, S. Perotto, S. Micheletti. A theoretical study of COmpRessed SolvING for advection-diffusion-reaction problems. Mathematics of Computation, 87 (309), pp. 1–38, 2018

10. S.B., S. Perotto, S. Micheletti. Compressed solving: a numerical approximation technique for PDEs based on compressed sensing. Computers & Mathematics with Applications, 70, pp. 1306–1335, 2015.

11. S.B., L. Gemignani. On the simultaneous refinement of the zeros of H-palindromic polynomials. Journal of Computational and Applied Mathematics, 272, pp. 293–303, 2014.


Fellowships and awards

  1. 2017 Awarded a spotlight talk prize at SFU Postdoc Research Day 2017.

2. 2016–18 “Postdoctoral Training Center in Stochastics” Fellowship from the Pacific Institute for the Mathematical Sciences.

3. 2012–15 PhD scholarship from the Italian Ministry of Education, Universities and Research.

4. 2010–12 Scholarship for graduate students from Istituto Nazionale di Alta Matematica. (Ranked 2nd on a national level in Italy)

5. 2007–10 Scholarship for undergraduate students from Istituto Nazionale di Alta Matematica. (Ranked 25th on a national level in Italy)