August 4, 2014 at 4:31 pm

Math Professors Get NSF Grant to Attack the Curse of Dimensionality

Dr. Martin J. Mohlenkamp, Associate Professor and Graduate Chair of Mathematics, and Dr. Todd Young, Professor of Mathematics, received an NSF grant for computational mathematics work.

The Curse of Dimensionality is that high-dimensional problems are much, much harder than low-dimensional problems. “Although in our everyday lives we perceive space as only three dimensional, electrons interact in a much higher dimensional space, with dimensionality of the number of electrons times three,” says Mohlenkamp.

By combining Mohlenkamp’s expertise in high-dimensional numerical analysis, Dr. Young’s expertise in dynamical systems, and the enthusiastic research efforts of Ohio University graduate and undergraduate students, this project strives to overcome a crucial bottleneck in a computational method for high-dimensional problems.

Their award on “Dynamical Systems on Tensor Approximations” is from the National Sciences Foundation’s Division of Mathematical Sciences.

Abstract: Functions of many variables arise in numerous mathematical, statistical, and scientific problems; a particularly notable example is the multiparticle Schrodinger equation in quantum mechanics. The effort required to compute in a straightforward way with such functions grows extremely rapidly as the number of variables increases, and soon becomes prohibitive. Mathematical methods have been developed that in some cases allow one to compute without this rapid growth, but crucial parts of the method are poorly understood and unreliable. This project seeks to understand and then fix these crucial parts. Students will be actively involved in the project and so learn mathematics and how to conduct mathematical research; they will also develop skills in writing, presenting seminars and posters, and software development and usage.

A mathematical study will be conducted on the approximation of tensors using sums of separable tensors and the approximation of multivariate functions using sums of separable functions. The objectives are to understand (1) how such approximations behave and (2) how such approximations can be effectively computed. The method is to consider iterative tensor approximation algorithms as dynamical systems to probe the set of sum-of-separable tensors and to understand the behavior of the algorithm within this set. The approximation of tensors by sums of separable tensors enables a promising computational paradigm for bypassing the curse of dimensionality when working with functions of many variables. This project addresses a bottleneck, in understanding and in computation, that prevents the computational paradigm from achieving its full potential.

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