• This web page describes an activity within the Department of Mathematics at Ohio University, but is not an official university web page.
• If you have difficulty accessing these materials due to visual impairment, please email me at mohlenka@ohio.edu; an alternative format may be available.

# Dynamical Systems on Tensor Approximations

### 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.

### Acknowledgments

This work is supported by the National Science Foundation under Grant No. 1418787 from August 1, 2014 to July 31, 2017, with principal investigator Martin J. Mohlenkamp and Co-Principal Investigator Todd Young. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.

## Goals

### Scientific Goal

The primary scientific goal is to understand why tensor approximations behave the way they do and then improve them.

### Educational Goal

By participating in this research project, students will both learn specific mathematical and computational skills and also develop an understanding of how science and research works. Since the students will range from advanced doctoral students to undergraduates, the desired outcome will vary with the individual. Students at all levels will develop the ability to express mathematical ideas, both in writing and orally.

### Process Goal

Both science and education are rather messy, imperfect processes. By keeping aware of the process we use when pursuing the scientific and educational goals, we will attempt to continually improve this process.

## Opportunities to Participate

This project has many facets, from purely theoretical mathematical considerations, to very concrete programming tasks. The interests of individual students will be accommodated whenever possible. There are several levels at which you can participate, from exploratory to full-time research.

### Exploratory Participation

Anyone who wishes to can join the project for one semester. You will learn more about the project, as well as develop valuable skills in mathematical writing, searching the literature, reading a research paper, presenting a seminar talk, and programming. I will learn about your strengths and weaknesses, and we both will learn how well you would fit into the research effort.

We will have regular group meetings where we will discuss what was accomplished on the project so far and what the next steps should be. As exploratory participants, for each group meeting you will have a small task to complete and will write a journal entry about what you did. Some sample things you might do in your exploratory term:

• Software:
• Create a design for a subroutine to compute something related to this project, and write about it.
• Implement the subroutine you designed, and test it.
• Writing:
• Locate and get copies of two papers referenced by this project.
• Read the two papers and write a paragraph summary of the topic of each. Choose your favorite.
• Present your paper. Critique the presentations of others.
• Starting Research:
• Read part of the latest draft of the paper about this project. Decide what is unclear, missing, etc. and help to improve it.
• Work on a simple case.

### Research Assistants

For graduate students in the Mathematics department, a limited number of part-time paid research assistantships are available. Typically, these will be for 5 hours per week and supplement a teaching assistantship; the student is then not allowed to take any additional work (e.g. an extra class as overload). Summer research assistantships will also be available. In general, a student must participate at the exploratory level for one semester before being considered for a paid assistantship.

For undergraduate students, paid research assistantships are available for 5-10 hours per week.

Each participant's research project will be determined to meet their interests and level. To help keep them on track and progressing toward the overall project's three goals, there are the following specific expectations:

• For the group meetings:
• Attend.
• Submit a journal of what they did since the last meeting, questions or difficulties they encountered, and what they plan to do next.
• Be prepared to informally present to the group what they have done.
• Maintain a summary document of what they have learned so far that term. This document contains research results and other information that should be preserved. At the end of the term this becomes their final report.
• Give a formal presentation at least once per semester.
• Assist in the direction of more junior participants.
• Give feedback and suggestions on how the group's research process could be improved.

# Student Participants

## Summer 2017

Nate McClatchey
PhD student in Mathematics; Mathematical Autobiography

## Spring 2017

Nate McClatchey
PhD student in Mathematics; Mathematical Autobiography

## Fall 2016

Yichao Li
MS student in Mathematics and PhD student in Computer Science; Mathematical Autobiography; Journal.
Nate McClatchey
PhD student in Mathematics; Mathematical Autobiography; Journal
Kevin Pomorski
MS student in Mathematics; Mathematical Autobiography; Journal

## Summer 2016

Xue Gong
PhD student in Mathematics; Mathematical Autobiography
Nate McClatchey
PhD student in Mathematics; Mathematical Autobiography
Kevin Pomorski
MS student in Mathematics; Mathematical Autobiography; Journal

## Spring 2016

Xue Gong
PhD student in Mathematics; Mathematical Autobiography; Journal
Nate McClatchey
PhD student in Mathematics; Mathematical Autobiography; Journal
Kevin Pomorski
MS student in Mathematics; Mathematical Autobiography

## Fall 2015

David Avornyo
MS student in Mathematics; Mathematical Autobiography, Journal
Xue Gong
PhD student in Mathematics; Mathematical Autobiography, Journal
Cesar Lopez Castillo
MS student in Mathematics; Mathematical Autobiography
Nate McClatchey
PhD student in Mathematics; Mathematical Autobiography; Journal
Ikenna Nwajagu
MS student in Mathematics; Mathematical Autobiography
Isaac Owusu-Mensah
PhD student in Mathematics; Mathematical Autobiography
Kevin Pomorski
MS student in Mathematics; Mathematical Autobiography, Journal
Yuanhang Zhang
MS student in Mathematics; Mathematical Autobiography

## Summer 2015

Xue Gong
PhD student in Mathematics
Nate McClatchey
PhD student in Mathematics; Mathematical Autobiography
Gregory Moses
PhD student in Mathematics; Mathematical Autobiography

## Spring 2015

S. Elaine Hale
Masters student in Mathematics; Mathematical Autobiography; Journal; Visualization tool
Samantha Hampton
Masters student in Mathematics; Mathematical Autobiography; Journal
Nate McClatchey
PhD student in Mathematics; Mathematical Autobiography; Journal
Gregory Moses
PhD student in Mathematics; Mathematical Autobiography