Ohio University
Martin's Picture

Martin J. Mohlenkamp

Associate Professor
Department of Mathematics
College of Arts & Sciences
Ohio University

See my mathematical geneology and news about me.

Contact Information


Teaching

Courses

Resources

Wavelet Materials
I have organized some wavelet materials for a short course I taught in 2004.
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.

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

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
(University of Colorado APPM preprint #476 August 2, 2001; Abstract and final journal version.)
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
(University of Colorado APPM preprint #519, February 2004; (preprint).)
Musings on Multilinear Fitting
Martin J. Mohlenkamp
Linear Algebra and its Applications, 438(2): 834-852, 2013.
(final version; preprint.)

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.
(Copyright 2008 American Institute of Physics. This article may be found at http://link.aip.org/link/?JMP/49/032107. It 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.
(Copyright 2010 American Institute of Physics. This article may be found at http://link.aip.org/link/?JMP/51/022112. It 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
(link; 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.
(Copyright 2013 American Institute of Physics. This article may be found at http://link.aip.org/link/?JMP/54/062105. It 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

Regression is the art of building a function that approximately matches the data, and gives a reasonable value at new data locations. In this work we build a regression method that scales linearly with the dimension, and so can be used in high dimensions.

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).
(link; 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)
(link; reprint.)

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.
(preprint; An earlier version is available as: An Identity for Sine of the Sum of Several Variables. Martin J. Mohlenkamp and Lucas Monzon. University of Colorado APPM preprint #480, October 24, 2001.)

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.
(ID jcph.1999.6215; University of Colorado APPM preprint #392, August 12, 1998.)

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.
(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.


Resources


Martin J. Mohlenkamp
Last modified: Mon Aug 25 16:55:15 EDT 2014