Christian Lubich, Ivan Oseledets, and Bart Vandereycken received the SIAM Outstanding Paper Prize at the 2018 SIAM Annual Meeting. They were recognized for their paper, “Time Integration of Tensor Trains,” SIAM Journal on Numerical Analysis, Volume 53, Issue 2 (2015).
The SIAM Outstanding Paper Prize is awarded to the authors of the three most outstanding papers, in the opinion of the selection committee, published in SIAM journals in the three calendar years preceding the year before the award year. Priority is given to papers that bring a fresh look at an existing field or that open up new areas of applied mathematics.
Christian Lubich has been Professor of Numerical Mathematics at the University of Tübingen since 1994. He received his doctoral degree from the University of Innsbruck in 1983. His research is focused on the numerical analysis of time-dependent problems.
Ivan Oseledets is currently an associate professor at Skolkovo Institute of Science and Technology (Skoltech), where he is a leader of the Scientific Computing Group. Before joining Skoltech in 2013, he earned his PhD in 2007 and his Doctor of Science in 2012 from the Institute of Numerical Mathematics of the Russian Academy of Sciences. His research focuses on the development of breakthrough numerical techniques (matrix and tensor methods) for solving a broad range of high-dimensional problems.
Bart Vandereycken is currently associate professor in the Section of Mathematics of the University of Geneva. He received his PhD from Katholieke Universiteit Leuven in 2010. His research is focused on numerical linear algebra and numerical optimization. He is particularly interested in geometric methods for low-rank matrices and tensors.
The authors collaborated on their answers to our questions.
Q: Why are you excited to receive the SIAM Outstanding Paper Prize?
A: We are very honored to receive this prize since it not only recognizes our work on the time integration of low-rank tensor trains but also that of our colleagues who have worked on this and closely related topics. In addition, we interpret this award as a broader acceptance of tensor techniques in the scientific community.
Q: Could you tell us a bit about the research that won you the prize?
A: Our paper combines two lines of research:
(1) Tensor trains, a very successful data-sparse format for approximating high-dimensional data or solutions to high-dimensional partial differential equations. They were independently studied in physics (under the name of matrix product states) and in mathematics starting with a paper by Ivan Oseledets in SIAM Journal on Scientific Computing in 2011.
(2) Dynamical low-rank approximation, which studies the low-rank approximation of time-dependent large matrices and tensors, which may be given time-dependent data or unknown solutions to time-dependent differential equations. This originated with a paper by Othmar Koch and Christian Lubich in SIAM Journal on Matrix Analysis and Applications in 2007 for the matrix case. This time-continuous approach was later extended to various classes of tensors in the same journal.
Our collaboration began after Christian and Ivan proposed a fully discrete time integrator for low-rank matrices based on a splitting of the projection to the tangent space of the low-rank manifold [BIT Numerical Mathematics 54 (2014)]. The extension to tensor trains was done in collaboration with Bart and focused on efficiently extending the matrix case to tensors. In our winning paper we also applied the method to a high-dimensional problem in computational chemistry. (In an immediate follow-up paper, our method was adopted by the quantum physics community for the simulation of spin systems [Haegeman et al (2016), Phys. Rev. B 94].)
The projector-splitting methods for low-rank tensors have remarkable properties: they are robust to small singular values in the approximation matrix (this is not the case for standard methods); they conserve energy and norm when applied to Hamiltonian systems (important in physical simulations); the necessary linear algebra to represent the approximation format can be computed very efficiently (despite the need to integrate many subproblems in very high dimension); and the subproblems are defined on flat subspaces so they can be integrated one dimension after the other in a high-dimensional problem (effectively exploiting that the low-rank manifolds are ruled surfaces). While a series of follow-up papers by us and others have studied these properties and extended the range of applications, many properties still remain mysterious. For example, changing the order of the integration steps in the splitting method destroys many of the beautiful and practical properties, but we lack a simple explanation why!
Ivan Oseledets (center) of Skolkovo Institute of Science and Technology and Bart Vandereycken (right) of the University of Geneva were awarded the SIAM Outstanding Paper Prize from SIAM President Nick Higham (left) at the 2018 SIAM Annual Meeting.
Q: What does your research mean to the public?
A: Our contribution on time integration with low-rank tensors is only one example of methods based on tensor techniques that have numerous applications in science and engineering. As a sign of the popularity of computations with tensors, take for example the 2009 overview paper "Tensor Decomposition and Applications" by Tamara. G. Kolda and Brett W. Bader, which was the second-most downloaded paper since the beginning of SIAM Review.
Computing with tensors is a very efficient and well-structured way to compute with high-dimensional data and unknowns --- and these are everywhere around us! Our paper is one in a series of papers that study the use of tensors for time-varying high-dimensional data and unknowns.
Q: What does participation in SIAM mean to you?
Bart: As a SIAM member, I regularly attend the SIAM conferences to keep in contact with colleagues and to get to know the latest research. In addition, the many excellent SIAM journals play an indispensable role in my academic research, both for publishing my own and for reading other people's work.
Christian: I highly appreciate the publications from SIAM, be it the SIAM journals or the SIAM books. They provide first-rate examples showing that publishing by a professional association in the service of the research community can be highly successful and even class-leading.
Ivan: SIAM journals have always been the sign of quality in applied mathematics, with exceptionally high publication standards. This mixture of ideas, applications, and, most importantly, people creates a foundation for a whole community. Without SIAM many new brilliant ideas would not be possible.