# Hi, I am Tianran Chen

I am an Assistant Professor at Auburn University at Montgomery in the Department of Mathematics and Computer Science. My main research interests are numerical analysis and scientific computing as well as their applications in physics, chemistry, biology, and engineering. My current research focuses on numerical algebraic geometry. You can find out more about my research here. For more information, please see my CV here (PDF version).

In the Fall Semester of 2017 I am teaching MATH-2630 (Multivariable Calculus), MATH-4320 (Modern Algebra II). MATH-4400 (Math Models and Simulations).

• Office: 310A Goodwyn Hall
• Office hour:
• Mon,Wed 2pm – 3pm
• Tue,Thu 12:30pm – 3pm
• …and by appointment

I am also the advisor for the AUM Math Club.

## Research Interests

• Numerical analysis, scientific computing, high performance computing
• Application of numerical methods in physics, chemistry, and engineering
• Systems of polynomial equations
• Homotopy continuation methods
• Numerical algebraic geometry

## Preprints

• (with Robert Davis and Dhagash Mehta)
[ arXiv ] “Counting equilibria of the Kuramoto model using birationally invariant intersection index”

• [ arXiv ] “Unmixing the mixed volume computation”

• (with Dhagash Mehta and Matthew Niemerg)
[ arXiv ] “A network topology dependent upper bound on the number of equilibria of the Kuramoto model”

• (with Dhagash Mehta)
[ arXiv ] “An index-resolved fixed-point homotopy and potential energy landscapes”

## Publications

• Christian Knoll, Dhagash Mehta, Tianran Chen, and Franz Pernkopf
“Fixed points of belief propagation – An analysis via polynomial homotopy continuation”.
IEEE Transactions on Pattern Analysis and Machine Intelligence
[ link ] [ arXiv ] [ ] [ ]
Abstract: Belief propagation (BP) is an iterative method to perform approximate inference on arbitrary graphical models. Whether BP converges and if the solution is a unique fixed point depends on both the structure and the parametrization of the model. To understand this dependence it is interesting to find all fixed points.
        
@article{Knoll2017,
author = {Knoll, Christian and Mehta, Dhagash and Chen, Tianran and Pernkopf, Franz},
doi = {10.1109/TPAMI.2017.2749575},
issn = {0162-8828},
journal = {IEEE Transactions on Pattern Analysis and Machine Intelligence},
pages = {1--1},
title = ,
url = {http://ieeexplore.ieee.org/document/8027142/},
year = {2017}
}


• Tianran Chen and Dhagash Mehta.
“On the Network Topology Dependent Solution Count of the Algebraic Load Flow Equations”.
IEEE Transactions on Power Systems
[ pdf ] [ link ] [ ] [ ]

Abstract: Active research activity in power systems areas has focused on developing computational methods to solve load flow equations where a key question is the maximum number of solutions. Though several upper bounds exist, recent studies have hinted that much sharper upper bounds that depend the topology of underlying power networks may exist. This paper provides a significant refinement of these observations. We also develop a geometric construction called adjacency polytope which accurately captures the topology of a power network and is immensely useful in the computation of the solution bound. Finally we highlight the significant implications of the development of such solution bounds in numerically solving load flow equations.development of such solution bounds in numerically solving load flow equations.
        
@article{chen_network_2017,
author = {Chen, Tianran and Mehta, Dhagash},
doi = {10.1109/TPWRS.2017.2724030},
issn = {0885-8950},
journal = {IEEE Transactions on Power Systems},
pages = {1--1},
title = ,
url = {http://ieeexplore.ieee.org/document/7971956/},
year = {2017}
}


• Tianran Chen, Tsung-Lin Lee, and Tien-Yien Li.
“Mixed cell computation in Hom4PS-3”.
Journal of Symbolic Computation (2017), pp. 516-534

• Tianran Chen and Dhagash Mehta.
“Parallel degree computation for binomial systems”.
Journal of Symbolic Computation (2017), pp. 535-558

• Dhagash Mehta, Tianran Chen, John Morgan, and David Wales.
Response to “Comment on ‘Exploring the potential energy landscape of the Thomson problem via Newton homotopies’”.
The Journal of Chemical Physics 143, 247102 (2015)
[ link ] [ bibtex ].

• Tianran Chen and Tien-Yien Li.
“Homotopy continuation method for solving systems of nonlinear and polynomial equations”.
Communications in Information and Systems 15(2):119–307 (2015).
[ link ] [ bibtex ].

• Dhagash Mehta, Tianran Chen, David Wales, and John Morgan.
“Exploring the potential energy landscape of the Thomson problem via Newton homotopies”.
The Journal of Chemical Physics 142 194113 (2015)
[ link ] [ bibtex ].

• Tianran Chen, Tien-Yien Li, and Xiaoshen Wang.
“Theoretical aspects of mixed volume computation via mixed subdivision”.
Communications in Information and Systems (2014)
[ pdf ] [ link ] [ bibtex ].

• Dhagash Mehta, Tianran Chen, Jonathan Hauenstein, and David Wales.
“Newton homotopies for sampling stationary points of potential energy landscapes”.
The Journal of Chemical Physics 141 (12), 121104 (2014)
(Selected for a Journal of Chemical Physics Editors’ Choice for 2014)
[ arXiv ] [ link ] [ bibtex ].

• Tianran Chen and Tien-Yien Li.
“Solutions to systems of binomial equations”.
Annales Mathematicae Silesianae 28:7–34 (2014).
[ bibtex ].

• Tianran Chen, Tsung-Lin Lee, and Tien-Yien Li.
“Hom4PS-3: A Parallel Numerical Solver for Systems of Polynomial Equations Based on Polyhedral Homotopy Continuation Methods” (Extended abstract).
Mathematical Software – ICMS 2014 – 4th International Congress, Seoul, South Korea, August 5-9, 2014. Proceedings 8592:183–190
[ link ] [ bibtex ].

• Tianran Chen, Tsung-Lin Lee, and Tien-Yien Li.
“Mixed volume computation in parallel”.
Taiwanese Journal of Mathematics 18(1):93–114 (2014).
[ link ] [ bibtex ].

• Tianran Chen and Tien-Yien Li.
“Spherical projective path tracking for homotopy continuation methods”.
Communications in Information and Systems 12(3):195-220 (2012)
[ link ] [ bibtex ].

## Scientific Software

• Hom4PS-3: A parallel numerical solver for systems of polynomial equations based on the Polyhedral Homotopy Method.
• MixedVol-3: A parallel software package for computing mixed volume, BKK bound, and fine mixed subdivisions (now a part of Hom4PS-3).
• libtropicana: A library designed to find regular simplicial subdivision of lattice convex polytopes and also compute normalized volume as a byproduct. It is written completely in C++ with optional interface for leveraging spBLAS (Sparse BLAS) routines. It is open source software. Users may freely distribute its source under the terms of the LGPLv3 license.