.. _research:
==========
Research
==========
I come from a theoretical physics and applied mathematics background, with a
strong interest in building better tools for scientific computing. Here is a
brief description of topics I am currently interested in, with links to
relevant publications.
Neuroscience and scientific computing
=====================================
Briefly stated, today my work revolves at the intersection of algorithmic
development for neuroscience data analysis and scientific computing. I am
interested in mathematical methods that let us understand the structure of
brain function in a principled way from the data we can acquire today
non-invasively, and in the implementation of these ideas with open-source,
high-quality, well-documented and pleasant to use computational tools.
The main technique we use at the U.C. Berkeley `Brain Imaging Center`_ is
functional Magnetic Resonance Imaging (fMRI), but I am in general more
interested in the structure that we can extract from the data that in the
particular technique being used, as I'm convinced that we'll learn much more
about the system by combining multimodality data in a principled way.
I am collaborating with members of `Mark D'Esposito's lab`_ on the analysis of
several different types of resting state data, and in the process we are
developing new software tools that will be made available as part of the
open-source NiPy_ project. I also collaborate with `Michael Silver's lab`_,
and currently `Ariel Rokem`_ is leading the effort on the development of a
library for the analysis of time-series data. This is the nitime_ project,
another component of nipy. The following is a preprint of our recent paper on
this topic from the SciPy'09 conference (you can also see the slides_ of
Ariel's talk):
.. admonition:: Paper
A. Rokem, M. Trumpis, F. Pérez, `Nitime: time-series analysis for
neuroimaging data <../papers/nitime09_ro-tr-pe.pdf>`_, Proceedings of the
8th Python in Science Conference (SciPy 2009). `Bibtex
<../papers/nitime09_ro-tr-pe.bib>`_.
.. _slides: http://conference.scipy.org/static/wiki/rokem_timeseries.pdf
These computational projects are part of my interest in the development of
modern, high-quality tools for scientific computing that take advantage of the
best practices from the open source revolution in software development and
bring them to computational research.
In the last couple of decades, but increasingly so in the last few years, all
scientific disciplines have been tossed into the deep end of the computational
pool, often while not being too well prepared for the dip. Up until the end of
the XX century, scientific computing was mostly the domain of physicists,
chemists and engineers. Fortran and C dominated the scene, and for the most
part scientific computing was synonymous with high-end numerical analysis. But
now, all disciplines are drowning in quantitative data and need to develop a
culture of scientific computing. Fortunately the revolution in open source
(made possible by the opening of the Internet in the late 90's) has created a
lot of freely available tools, and a culture of how to use them and develop
them, that is causing lasting changes in scientific computing.
I believe that as discipline-focused scientists we have much to learn from
these developments, and I am trying to do my part in advancing the state of the
art on this problem, which I believe to be one of major importance. Since late
2001, I have been actively involved in developing tools of this kind based on
the Python_ programming language. I started the IPython_ project to build an
interactive environment I could use for my everyday work, inspired by the tools
I was accustomed to as a physicist, especially Mathematica and IDL. I've also
contributed in various forms to NumPy_, SciPy_, matplotlib_ and Mayavi_, all
tools that combined give us today an amazing array of computational
functionality based on the excellent qualities of Python as a programming
language. Here are two publications on this topic:
.. admonition:: Papers
F. Pérez, H-P. Langtangen and R. LeVeque, `Python for Scientific Computing
at CSE09 <../papers/py4science09_siam_news.pdf>`_, SIAM News, 42:5, 4
(2009). `Bibtex entry`_. (An `html version`_ is available at the SIAM
site).
F. Pérez and B. E. Granger. `IPython: a System for Interactive Scientific
Computing <../papers/ipython07_pe-gr_cise.pdf>`_, Comput. Sci. Eng. 9:3, 21-23
(2007).
You can find `more details `_ on the use of Python for scientific
computing and my work on this front.
.. _Brain Imaging Center: bic_
.. _Mark D'Esposito's lab: despolab_
.. _Michael Silver's lab: silverlab_
.. _Ariel Rokem: arokem_
.. _html version: siam_news_cse09_
.. _bibtex entry: ../papers/py4science09_siam_news.bib
.. _fast_algorithms:
New algorithms for PDEs and the Schrödinger equation
====================================================
As a post-doc in the `Applied Mathematics`_ department at the University of
Colorado (Boulder) I worked with prof. `Gregory Beylkin`_ on the development
of novel algorithms for the solution of partial differential equations (PDEs)
in multiple dimensions. These methods use a multiresolution approach to
decompose functions and operators and a set of approximations from the familiy
of methods broadly referred to as "tensor decompositions" that G. Belkyin
developed in collaboration with Lucas Monzón and prof. `Martin Mohlenkamp`_, to
address the "curse of dimensionality". In this formulation, PDE problems are
handled in an integral formulation (which has a number of advantages) since we
now have an efficient way of applying Green's functions.
The long-term goal of this effort is the development of accurate and efficient
methods to solve the many-electron Schrödinger equation, and we made some
theoretical and numerical progress on this front as well.
.. _Gregory Beylkin: beylkin_
.. _Applied Mathematics: amath_cu_
.. _Martin Mohlenkamp: http://www.math.ohiou.edu/~mjm
.. admonition:: Papers
G. Beylkin, M. J. Mohlenkamp and F. Pérez, `Approximating a Wavefunction as
an Unconstrained Sum of Slater Determinants
<../papers/freeslater08_be-mo-pe.pdf>`_, Journal of Mathematical Physics, 49,
(2008).
G. Beylkin, V. Cheruvu and F. Pérez, `Fast adaptive algorithms in the
non-standard form for multidimensional problems
<../papers/nsform08_be-ch-pe.pdf>`_, Applied and Computational Harmonic
Analysis, 24 (2008) 354--377.
G. Beylkin, M. J. Mohlenkamp and F. Pérez, `Preliminary results on
approximating a wavefunction as an unconstrained sum of Slater determinants
<../papers/freeslater07_be-mo-pe.pdf>`_, Proc. Appl. Math. Mech., 7, (2007).
The attic
=========
The following is an archive of projects I have worked on in the past and I am
no longer actively pursuing. It turns out, however, that my current interest
in the structure of neuroimaging datasets ties in closely with the problems in
semi-classical chaos listed here, via questions on the highly excited
eigenstates of the Laplacian operator. So I may end up returning to at least
some of these topics that I actually hold very dear.
.. toctree::
Old projects:
.. include:: links.txt