 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.
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.
Uptodate 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.
Research
General Interests
 Fast Algorithms: How to get a computer to give you the (right)
answer as quickly as possible.
 Numerical Analysis: How to adapt a continuous problem (from
physics, for example) into something a computer can solve
(preferably with a fast algorithm).
 How to do Numerical Analysis in high dimension.
 Applied Mathematics: How to bring the power of Mathematics to bear
on problems from other fields (often using Numerical Analysis).
 (Computational) Harmonic Analysis: How to represent the world
efficiently in terms of waves (and wavelets).
 Mathematics: How to see the beautiful structures all around us.
Students
 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):1024610251, 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):21332159,
2005.
doi: 10.1137/040604959 (preprint)
 Musings on Multilinear Fitting
 Martin J. Mohlenkamp
Linear Algebra and its Applications, 438(2): 834852, 2013.
doi: 10.1016/j.laa.2011.04.019 (preprint.)
 The Optimization Landscape for Fitting a Rank2 Tensor with a Rank1 Tensor
 Xue Gong, Martin J. Mohlenkamp, and Todd R. Young.
SIAM Journal on Applied Dynamical Systems, 17(2): 14321477, 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): 12931333, 2019.
doi: 10.1137/18M1181389
(local copy, supplementary material)
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 multiyear project with many students to adapt
the general sumofseparable 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, CW. Shu, N. Yan, editors.
World Scientific, 2008.
(preprint)
 A CenterofMass Principle for the Multiparticle Schrodinger
Equation.
 Martin J. Mohlenkamp.
Journal of Mathematical Physics, 51(2):022112115, 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 Interelectron Cusp using a Geminal Layer on
an Unconstrained Sum of Slater Determinants.
 Martin J. Mohlenkamp
SIAM Journal on Applied Mathematics, 72(6):17421771, 2012. doi: 10.1137/110823900
(reprint)
 Function Space Requirements for the SingleElectron Functions
within the Multiparticle Schrodinger Equation

Martin J. Mohlenkamp
Journal of Mathematical Physics, 54(6):062105134, 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
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): 18401857
(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): 33813401 (2011).
doi: 10.1137/100805959 (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):6569, 2005.
doi: 10.1007/BF02985795
(preprint)
Spectral Projectors
 Fast Spectral Projection Algorithms for DensityMatrix
Computations.
 Gregory Beylkin, Nicholas Coult, Martin J. Mohlenkamp.
Journal of Computational Physics, 152(1):3254, 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):159184, 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
Last modified: Thu Jun 20 17:08:14 UTC 2019