Daniel M. Dunlavy
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Current Research
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Homotopy Optimization Methods and Protein Structure Prediction
My current research involves the development of homotopy methods for
minimizing potential energy functions associated with proteins and
protein models. This work has the focus of my doctoral studies and has
been been funded by a grant from the National Library of Medicine
(NLM) at the National Institutes of Health (NIH)
in which I am the principal investigator (Grant 5F37LM008162-02,
2 years, $91,450).
The goal of this work has been to produce an global optimization
algorithm for finding a global minimizer for potential energy
functions for pairs of homologous, or sequentially related,
proteins. The method developed, Homotopy Optimization using
Perturbations and Ensembles (HOPE), shares features with comparative
modeling, smoothing methods, and simulated annealing, all of which
have been used by various researchers in the past to solve to predict
the native state of proteins, or simplified models of proteins and
clusters of atoms.
Publications and Presentations:
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A Homotopy Optimization Method for Protein Structure Prediction
Preprint, February 2005.
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Homotopy Optimization Methods and Protein Structure Prediction
Dissertation Prospectus, AMSC Program, University of Maryland, February 2005.
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A Homotopy Method for Predicting Low Energy Conformations of Polypeptides
Contributed talk, SIAM Conference on the Life Sciences, July 2004.
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A Homotopy Method for Predicting the Lowest Energy Conformations of Proteins
Ph.D. Candidacy Prospectus, April 2003.
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A Homotopy Method for Predicting the State of Minimal Energy for Chains of Charged Particles
Winner's Talk, Spotlight on Graduate Research, Department of Mathematics,
University of Maryland, February 2002.
Past Research
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Document Summarization
Under the direction of Dianne O'Leary of the University of Maryland
and John Conroy of the Center for Computing Sciences, I developed the
QCS system for querying, clustering, and summarizing documents with
the goal of improving the efficiency of scientific literature searches
via the internet. QCS combines previously developed components for
solving each of the three retrieval/extraction tasks into a single
application. Documents are represented using a vector space model,
where the importance of the m terms appearing in the n
documents is represented using an m-by-n term-document
matrix. Querying is accomplished using latent semantic indexing (LSI),
a method which uses the the singular value decomposition (SVD) of the
term-document matrix to assign the relevance of each document for a
given query. Clustering of the documents retrieved during the querying
phase is accomplished using the spherical k-means
algorithm. This algorithm differs from the classical k-means
algorithm in that the value of k, the number of clusters into
which to partition the documents, is allowed to take values from a
prescribed range rather than being fixed, thus removing the need to
choose a specific number of clusters that would be suitable for all
queries. Extract summaries are produced for each document using a
hidden Markov model (HMM) trained to predict which sentences in a
document best summarize that document. Using the summary sentences
from all of the documents in a cluster, a term-sentence matrix is then
produced for that cluster. A pivoted-QR method is then run on this
term-sentence matrix to produce a multiple-document summary for each
cluster, with redundant filtered out.
I also helped in modifying the summarization module used in the
current system. This summarization system was submitted for evaluation
in the 2003
Document Understanding Conference.
Details of the system can be found
here,
and a publicly accessible version can be found
here.
Publications and Presentations:
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Structure Preserving Eigensolvers
With
Niloufer Mackey
and D. Steve Mackey of Western Michigan University, I implemented and
tested algorithms for solving structured eigenvalue problems using
perplectic orthogonal (i.e., centrosymmetric orthogonal)
transformations. These algorithms are Jacobi-like iterative methods
for solving the complete eigenvalue problem for matrices with the
double structure of symmetry/skew-symmetry across both the diagonal as
well as anti-diagonal. Based on the direct solution of 4-by-4
subproblems constructed via quaternions, the algorithms calculate
structured orthogonal bases for the invariant subspaces of the
associated matrix. In addition to preserving structure, these methods
are inherently parallelizable, numerically stable, and show asymptotic
quadratic convergence.
Images and animations (Java, AVI, PPT, Flash, HTML) of the
symmetric persymmetric algorithm in action on matrices of sizes
n = 12, 16, and 32 can be found
here.
Images and animiations of the sweep pattern used in the current
implementation of the algorithms are also available there.
Publications and Presentations:
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Surrogate Models in Derivative-Free Optimization
During an internship in the
Computational Science and Mathematics Research (CSMR) Department
at
Sandia National Laboratories,
I worked with
Tamara Kolda
on creating surrogate models for use with the derivative-free
optimization software package
APPSPACK,
an optimization package employing an asynchronous parallel pattern
search method. The surrogate models were developed to help increase
the convergence rate by using points generated during the pattern
search. These points were used to generate a smooth approximation of
the objective function, and a trust region method was used to optimize
the approximation. When the evaluation of a function is
computationally expensive (e.g., the function is a simulation), the
approximation can be generated while waiting for function values to be
computed and may accelerate the determination of optimal
points.
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RF Communication Circuits
As part of a Mathematical Modeling
in Industry workshop for graduate students at the Institute for
Mathematics and Its Applications (IMA), I worked with Bob Melville of
Agere (then Lucent)
As part of the
Mathematical Modeling in Industry
workshop at the
IMA
, I worked with Robert Melville of Agere (then Lucent) and graduate students from several universities on
developing methods for determining the steady-state response of
radio-frequency (RF) integrated circuits. We developed iterative
methods using finite differences and harmonic balancing to find the
steady-state solution of nonlinear differential algebraic equations
(DAEs) arising from these circuits. The use of tensor products and
stride permutations helped reduce the storage and computational costs
associated with the Jacobian matrices. Several preconditioners were
also developed which significantly reduced the number of iterations
required for convergence.
Publications and Presentations: