Martin J. Mohlenkamp

Professor and Chair
Department of Mathematics
College of Arts & Sciences
Ohio University

See my mathematical geneology, AMS author profile, and news about me.

Contact Information

  • Solid mail: Department of Mathematics / Morton Hall 321 / 1 Ohio University / Athens OH 45701 USA
  • E-mail: mohlenka@ohio.edu
  • Office: 321-C Morton Hall
  • Phone: (740) 593-1259
  • Fax: (740) 593-9805
  • For office hours check one of the courses I am currently teaching.
Martin's Picture

Teaching

Courses

Resources

Numerical Methods Textbook
Introduction to Numerical Methods and Matlab Programming for Engineers. By Todd Young & Martin J. Mohlenkamp
Good Problems
We have developed a method to gently teach mathematical writing.
    Good Problems: teaching mathematical writing
D. Bundy, E. Gibney, J. McColl, M. Mohlenkamp, K. Sandberg, B. Silverstein, P. Staab, and M. Tearle.
University of Colorado APPM preprint #466, August 15, 2001.
Up-to-date materials through a Student's Guide.
Sage Cells
For Calculus and Linear Algebra.
Wavelet Materials
I have organized some wavelet materials for a short course I taught in 2004.

Wavelet book cover Wavelets, Their Friends, and What They Can Do for You.
Martin J. Mohlenkamp and Maria Cristina Pereyra.
EMS Series of Lectures in Mathematics, June 2008.
(flyer; order in the Americas; order elsewhere)

2x2x2 random slice Tensor Rank Visualization Tool.
Martin J. Mohlenkamp.
First release November 2009.

Associate Legendre Function matrix libftsh: A Fast Transform for Spherical Harmonics.
Martin J. Mohlenkamp.
First release October 2000.

Research

General Interests

Students

Indupama Herath
PhD 2022. Multivariate Regression using Neural Networks and Sums of Separable Functions
Nathaniel McClatchey
PhD 2018. Tensors: An Adaptive Approximation Algorithm, Convergence in Direction, and Connectedness Properties
Ryan Botts
PhD 2010. Recovery and Analysis of Regulatory Networks from Expression Data Using Sums of Separable Functions

Projects and Publications

Numerical Analysis in High Dimensions

It is a common experience in numerical analysis to develop a very nice algorithm in dimension one or two, discover it is painfully slow in dimension three or above, and then give up and go work on other nice algorithms in dimension one or two. The cause of this is clear: computational costs grow exponentially with dimension, an effect called the Curse of Dimensionality. We have developed methods to bypass this curse by representing multivariate functions as sums of separable functions. I am now working with collaborators and students to better understand and improve the key approximation algorithms.

Numerical Operator Calculus in Higher Dimensions.
Gregory Beylkin and Martin J. Mohlenkamp.
Proceedings of the National Academy of Sciences, 99(16):10246-10251, August 6, 2002. doi:10.1073/pnas.112329799
Algorithms for Numerical Analysis in High Dimensions
Gregory Beylkin and Martin J. Mohlenkamp
SIAM Journal on Scientific Computing, 26(6):2133-2159, 2005. doi: 10.1137/040604959 (preprint)
Musings on Multilinear Fitting
Martin J. Mohlenkamp
Linear Algebra and its Applications, 438(2): 834-852, 2013. doi: 10.1016/j.laa.2011.04.019 (preprint.)
The Optimization Landscape for Fitting a Rank-2 Tensor with a Rank-1 Tensor
Xue Gong, Martin J. Mohlenkamp, and Todd R. Young.
SIAM Journal on Applied Dynamical Systems, 17(2): 1432-1477, 2018. doi:10.1137/17M112213X (local copy)
The Dynamics of Swamps in the Canonical Tensor Approximation Problem
Martin J. Mohlenkamp.
SIAM Journal on Applied Dynamical Systems, 18(3): 1293--1333, 2019. doi: 10.1137/18M1181389 (local copy, supplementary material)
Transient Dynamics of Block Coordinate Descent in a Valley
Martin J. Mohlenkamp, Todd Young, and Balázs Bárány.
International Journal of Numerical Analysis and Modeling, 17(4): 557--591, 2020. (Journal link)

The Multiparticle Schrodinger Equation

It is notoriously difficult to compute numerical solutions to this basic governing equation in quantum mechanics, in part because it is posed in high dimensions. I worked on a multi-year project with many students to adapt the general sum-of-separable function methods to this problem.

Approximating a Wavefunction as an Unconstrained Sum of Slater Determinants.
Gregory Beylkin, Martin J. Mohlenkamp, and Fernando Perez.
Journal of Mathematical Physics, 49(3):032107, 2008. doi: 10.1063/1.2873123
(Copyright 2008 American Institute of Physics. This article can also be downloaded here for personal use only; any other use requires prior permission of the author and the American Institute of Physics.)
Convergence of Green Iterations for Schrodinger Equations.
Martin J. Mohlenkamp and Todd Young.
in Recent Advances in Computational Science: Selected Papers from the International Workshop on Computational Sciences and Its Education. P. Jorgensen, X. Shen, C-W. Shu, N. Yan, editors. World Scientific, 2008.
(preprint)
A Center-of-Mass Principle for the Multiparticle Schrodinger Equation.
Martin J. Mohlenkamp.
Journal of Mathematical Physics, 51(2):022112-1--15, 2010. doi: 10.1063/1.3290747
(Copyright 2010 American Institute of Physics. This article can also be downloaded here for personal use only; any other use requires prior permission of the author and the American Institute of Physics.)
Capturing the Inter-electron Cusp using a Geminal Layer on an Unconstrained Sum of Slater Determinants.
Martin J. Mohlenkamp
SIAM Journal on Applied Mathematics, 72(6):1742-1771, 2012. doi: 10.1137/110823900 (reprint)
Function Space Requirements for the Single-Electron Functions within the Multiparticle Schrodinger Equation
Martin J. Mohlenkamp
Journal of Mathematical Physics, 54(6):062105-1--34, 2013. doi: 10.1063/1.4811396
(Copyright 2013 American Institute of Physics. This article can also be downloaded here for personal use only; any other use requires prior permission of the author and the American Institute of Physics.)

Multivariate Regression and Machine Learning

Regression is the art of building a function that approximately matches the data, and gives a reasonable value at new data locations.

Multivariate Regression and Machine Learning with Sums of Separable Functions.
Gregory Beylkin, Jochen Garcke, and Martin J. Mohlenkamp.
SIAM Journal on Scientific Computing, 31(3): 1840-1857 (2009). doi: 10.1137/070710524 (preprint)
Learning to Predict Physical Properties using Sums of Separable Functions.
Mayeul d'Avezac, Ryan Botts, Martin J. Mohlenkamp, and Alex Zunger
SIAM Journal on Scientific Computing, 33(6): 3381-3401 (2011). doi: 10.1137/100805959 (reprint)
Leveraging high-throughput screening data and conditional generative adversarial networks to advance predictive toxicology.
Adrian J. Green, Martin J. Mohlenkamp, Jhuma Das, Meenal Chaudhari, Lisa Truong, Robyn L. Tanguay, David M. Reif.
PLOS Computational Biology 17(7): e1009135, 2021. doi: 10.1371/journal.pcbi.1009135

Trigonometric Identities

Although it seems like there should be nothing new in trigonometry, we stumbled upon some rather cute identities for sine of the sum of several variables.

Trigonometric Identities and Sums of Separable Functions
Martin J. Mohlenkamp and Lucas Monzon
The Mathematical Intelligencer, 27(2):65--69, 2005. doi: 10.1007/BF02985795 (preprint)

Spectral Projectors

Fast Spectral Projection Algorithms for Density-Matrix Computations.
Gregory Beylkin, Nicholas Coult, Martin J. Mohlenkamp.
Journal of Computational Physics, 152(1):32-54, 10 June 1999. doi: jcph.1999.6215

Spherical Harmonics

My thesis was a Fast Transform for Spherical Harmonics. (Like an FFT, but for the sphere.) Completed in the spring of 1997 under the direction of R.R. Coifman at Yale University.
(Abstract, Thesis itself (.ps))

A Fast Transform for Spherical Harmonics
Martin J. Mohlenkamp
Journal of Fourier Analysis and Applications, 5(2/3):159-184, 1999. doi: 10.1007/BF01261607 (preprint)
libftsh
is a software library implementing the transform.

I have also created A User's Guide to Spherical Harmonics for those new to the area.


Martin J. Mohlenkamp