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 Spring Semester of 2018 I am teaching MATH-2630 (Multivariable Calculus), MATH-2660 (Linear Algebra).

  • Office: 310A Goodwyn Hall
  • Office hour:
    • Mon,Wed 9:30am – 10:30am and 1:00pm – 2:00pm
    • Tue,Thu 1:00pm – 2:00pm
    • …and by appointment

I am also the advisor for the AUM Math Club.

News

I am one of the organizers for the Southern Regional Algebra Conference 2018 .

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)
    “A Product Formula for the Normalized Volume of Free Sums of Lattice Polytopes “
    [ arXiv ] [ ] [ ]
    Abstract: The free sum is a basic geometric operation among convex polytopes. This note focuses on the relationship between the normalized volume of the free sum and that of the summands. In particular, we show that the normalized volume of the free sum of full dimensional polytopes is precisely the product of the normalized volumes of the summands.
    @article{chen_product_2017,
    archivePrefix = {arXiv},
    arxivId = {1711.11130},
    author = {Chen, Tianran and Davis, Robert},
    eprint = {1711.11130},
    month = {nov},
    title = {{A Product Formula for the Normalized Volume of Free Sums of Lattice Polytopes}},
    url = {http://arxiv.org/abs/1711.11130},
    year = {2017}
    }
    
  • (with Robert Davis and Dhagash Mehta)
    “Counting equilibria of the Kuramoto model using birationally invariant intersection index “
    [ arXiv ] [ ] [ ]
    Abstract: Synchronization in networks of interconnected oscillators is a fascinating phenomenon that appear naturally in many independent fields of science and engineering. A substantial amount of work has been devoted to understanding all possible synchronization configurations on a given network. In this setting, a key problem is to determine the total number of such configurations. Through an algebraic formulation, for tree and cycle graphs, we provide an upper bound on this number using the birationally invariant intersection index of a system of rational functions on a toric variety.
    @article{chen_counting_2017,
    archivePrefix = {arXiv},
    arxivId = {1708.09246},
    author = {Chen, Tianran and Davis, Robert and Mehta, Dhagash},
    eprint = {1708.09246},
    month = {aug},
    title = {{Counting equilibria of the Kuramoto model using birationally invariant intersection index}},
    url = {http://arxiv.org/abs/1708.09246},
    year = {2017}
    }
    
  • “Unmixing the mixed volume computation”
    [ arXiv ] [ ] [ ]
    Abstract: Computing mixed volume of convex polytopes is an important problem in computational algebraic geometry. This paper establishes sufficient conditions under which the mixed volume of several convex polytopes exactly equals the normalized volume of the convex hull of their union. Under these conditions the problem of computing mixed volume of several polytopes can be transformed into a volume computation problem for a single polytope in the same dimension. We demonstrate through problems from real world applications that substantial reduction in computational costs can be achieved via this transformation in situations where the convex hull of the union of the polytopes has less complex geometry than the original polytopes. We also discuss the important implications of this result in the polyhedral homotopy method for solving polynomial systems.
    @article{chen_unmixing_2017,
    author = {Chen, Tianran},
    journal = {arXiv:1703.01684 [math]},
    keywords = {52B55- 65H10- 65H20,Mathematics - Algebraic Geometry},
    archivePrefix = {arXiv},
    arxivId = {1703.01684},
    month = {mar},
    title = {{Unmixing the mixed volume computation}},
    url = {http://arxiv.org/abs/1703.01684},
    year = {2017}
    }
    
  • (with Dhagash Mehta and Matthew Niemerg)
    “A network topology dependent upper bound on the number of equilibria of the Kuramoto model “
    [ arXiv ] [ ] [ ]
    Abstract: We begin with formulating the stationary equations of the Kuramoto model as a system of polynomial equations in a novel way. Then, based on an algebraic geometric root count, we give a prescription of computing an upper bound on the number of equilibria of the Kuramoto model for the most general case, i.e., defined on an arbitrary graph and having generic values of natural frequencies and inhomogeneous couplings. We demonstrate with computational experiments utilizing the numerical polynomial homotopy continuation method that our bound is tight for the number of complex equilibria for the Kuramoto model in the most general case. We then discuss the implications or our results in relation to finding all the real equilibria of the Kuramoto model.
    @article{chen_network_2016,
    archivePrefix = {arXiv},
    arxivId = {1603.05905},
    author = {Chen, Tianran and Mehta, Dhagash and Niemerg, Matthew},
    eprint = {1603.05905},
    month = {mar},
    title = {{A Network Topology Dependent Upper Bound on the Number of Equilibria of the Kuramoto Model}},
    url = {http://arxiv.org/abs/1603.05905},
    year = {2016}
    }
    
  • (with Dhagash Mehta)
    “An index-resolved fixed-point homotopy and potential energy landscapes”
    [ arXiv ] [ ] [ ]
    Abstract: Stationary points (SPs) of the potential energy landscapes can be classified by their Morse index, i.e., the number of negative eigenvalues of the Hessian evaluated at the SPs. In understanding chemical clusters through their potential energy landscapes, only SPs of a particular Morse index are needed. We propose a modification of the "fixed-point homotopy" method which can be used to directly target stationary points of a specified Morse index. We demonstrate the effectiveness of our approach by applying it to the Lennard-Jones clusters.
    @article{chen_index_2015,
    annote = {Comment: 7 pages, 2 figures},
    author = {Chen, Tianran and Mehta, Dhagash},
    journal = {arXiv:1504.06622 [cond-mat]},
    month = {apr},
    title = {An index-resolved fixed-point homotopy and potential energy landscapes},
    url = {http://arxiv.org/abs/1504.06622},
    year = {2015}
    }
    

Publications

  1. (with Christian Knoll, Dhagash Mehta, 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{knoll_fixed_2017,
    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 = {{Fixed Points of Belief Propagation - An Analysis via Polynomial Homotopy Continuation}},
    url = {http://ieeexplore.ieee.org/document/8027142/},
    year = {2017}
    }
    
  2. (with Dhagash Mehta)
    "On the Network Topology Dependent Solution Count of the Algebraic Load Flow Equations"
    IEEE Transactions on Power Systems
    [ link ] [ pdf ] [ ] [ ]

    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.
    @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 = {On the Network Topology Dependent Solution Count of the Algebraic Load Flow Equations},
    url = {http://ieeexplore.ieee.org/document/7971956/},
    year = {2017}
    }
    
  3. (with Tsung-Lin Lee and Tien-Yien Li)
    "Mixed cell computation in Hom4PS-3"
    Journal of Symbolic Computation (2017), pp. 516-534
    [ link ] [ ] [ ]

    Abstract: This article presents recent efforts in improving the efficiency and scalability of the mixed cell computation step in the context of the Polyhedral Homotopy method. Solving systems of polynomial equations is an important problem in applied mathematics. The Polyhedral Homotopy method is an important numerical method for this task. In this method, a necessary preprocessing step, known as the "mixed cell computation" problem has been the main bottleneck in the parallel efficiency and scalability. This article presents recent remarkable improvements in the parallel scalability of the algorithm that are applicable to a wide range of hardware architectures including multi-core systems, NUMA systems, computer clusters, and GPUs devices.
    @article{chen_mixed_2017,
    author = {Chen, Tianran and Lee, Tsung-Lin and Li, Tien-Yien},
    doi = {10.1016/j.jsc.2016.07.017},
    issn = {0747-7171},
    journal = {Journal of Symbolic Computation},
    month = {mar},
    pages = {516--534},
    series = {SI: Numerical Algebraic Geometry},
    title = {{Mixed cell computation in Hom4PS-3}},
    url = {http://www.sciencedirect.com/science/article/pii/S0747717116300542},
    volume = {79, Part 3},
    year = {2017}
    }
    
  4. (with Dhagash Mehta)
    "Parallel degree computation for binomial systems"
    Journal of Symbolic Computation (2017), pp. 535-558
    [ link ] [ ] [ ]

    Abstract: Solution sets of systems of binomial equations are of great interest in applied mathematics. For both theoretic and applied purposes, the degree of a solution set (its maximum number of isolated intersections with an affine space of complementary dimension) often plays an important role in understanding its geometric structure. This paper proposes a specialized parallel algorithm for computing the degree on GPUs that takes advantage of the massively parallel nature of GPU devices. The preliminary implementation shows remarkable efficiency and scalability when compared to the closest CPU-based counterpart. As a case study, the algorithm is applied to the master space problem of N = 1 gauge theories. The GPU-based implementation achieves nearly 30 fold speedup over its CPU-only counterpart enabling the discovery of previously unknown results.
    @article{chen_parallel_2017,
    author = {Chen, Tianran and Mehta, Dhagash},
    doi = {10.1016/j.jsc.2016.07.018},
    issn = {0747-7171},
    journal = {Journal of Symbolic Computation},
    keywords = {Algebraic Geometry,BKK root-count,Binomial systems,Folder - Unmixing,GPU computing,Supersymmetric gauge theories,homotopy continuation},
    month = {mar},
    pages = {535--558},
    series = {SI: Numerical Algebraic Geometry},
    title = {{Parallel degree computation for binomial systems}},
    url = {http://www.sciencedirect.com/science/article/pii/S0747717116300554},
    volume = {79, Part 3},
    year = {2017}
    }
    
  5. (with Dhagash Mehta, 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
    [ link ] [ ] [ ]

    Abstract: The comment notes that the Newton homotopy (NH) and Newton trajectory (NT) methods are related. By describing recent implementations of the NH method, we clarify the similarities and differences between the two approaches. The possible synergy between NH, NT and other flow methods could suggest further developments in mathematics and chemistry.
    @article{mehta_response_2015,
    author = {Mehta, Dhagash and Chen, Tianran and Morgan, John W. R. and Wales, David J.},
    doi = {10.1063/1.4939011},
    issn = {0021-9606, 1089-7690},
    journal = {The Journal of Chemical Physics},
    keywords = {Bifurcations,Difference equations,Flow instabilities,Flow simulations,Folder - LogEval,Folder - Unmixing,Singularity theory},
    month = {dec},
    number = {24},
    pages = {247102},
    title = {{Response to “Comment on ‘Exploring the potential energy landscape of the Thomson problem via Newton homotopies”' [J. Chem. Phys. 143, 247101 (2015)]}},
    url = {http://scitation.aip.org/content/aip/journal/jcp/143/24/10.1063/1.4939011 http://scitation.aip.org/deliver/fulltext/aip/journal/jcp/143/24/1.4939011.pdf;jsessionid=23ev3700uj07.x-aip-live-03?itemId=/content/aip/journal/jcp/143/24/10.1063/1.4939011{\&}mimeTyp},
    volume = {143},
    year = {2015}
    }
    
  6. (with 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 ] [ ] [ ]

    Abstract: This survey provides an overview of the homotopy continuation method for solving systems of nonlinear and polynomial equations including its theoretical background, numerical stability analysis, and implementation issues.
    @article{chen_homotopy_2015,
    author = {Chen, Tianran and Li, Tien-Yien},
    doi = {10.4310/CIS.2015.v15.n2.a1},
    issn = {15267555, 21634548},
    journal = {Commun. Inf. Syst.},
    number = {2},
    pages = {119--307},
    title = {Homotopy continuation method for solving systems of nonlinear and polynomial equations},
    volume = {15},
    year = {2015}
    }
    
  7. (with Dhagash Mehta, 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 ] [ ] [ ]

    Abstract: Locating the stationary points of a real-valued multivariate potential energy function is an important problem in many areas of science. This task generally amounts to solving simultaneous nonlinear systems of equations. While there are several numerical methods that can find many or all stationary points, they each exhibit characteristic problems. Moreover, traditional methods tend to perform poorly near degenerate stationary points with additional zero Hessian eigenvalues. We propose an efficient and robust implementation of the Newton homotopy method, which is capable of quickly sampling a large number of stationary points of a wide range of indices, as well as degenerate stationary points. We demonstrate our approach by applying it to the Thomson problem. We also briefly discuss a possible connection between the present work and Smale's 7th problem.
    @article{mehta_exploring_2015,
    author = {Mehta, Dhagash and Chen, Tianran and Morgan, John W. R. and Wales, David J.},
    doi = {10.1063/1.4921163},
    issn = {0021-9606, 1089-7690},
    journal = {The Journal of Chemical Physics},
    keywords = {Eigenvalues,Folder - Finished - NewtonHomotopyReply,Folder - Finished - Solving survey,Folder - LogEval,Folder - Power flow,Folder - Unmixing,Newton Raphson method,Nonlinear dynamics,Numerical solutions,Potential energy surfaces},
    month = {may},
    number = {19},
    pages = {194113},
    title = {{Exploring the potential energy landscape of the Thomson problem via Newton homotopies}},
    url = {http://scitation.aip.org/content/aip/journal/jcp/142/19/10.1063/1.4921163},
    volume = {142},
    year = {2015}
    }
    
  8. (with Tien-Yien Li and Xiaoshen Wang)
    "Theoretical aspects of mixed volume computation via mixed subdivision"
    Communications in Information and Systems (2014)
    [ link ] [ pdf ] [ ] [ ]

    Abstract: We analyze the computation of mixed volume of tuples of polytopes via fine mixed subdivisions. This method expresses the mixed volume as a sum of easily computable standard volumes of polytopes called mixed cells. Mixed cells play a critically important role in polyhedral homotopy continuation method, which in turn is a particularly efficient numerical method for the solution of systems of polynomial equations. We provide a complete and self-contained account of the underlying computational convexity techniques, assuming no background in algebraic geometry.
    @article{chen_theoretical_2014,
    author = {Chen, Tianran and Li, Tien-Yien and Wang, Xiaoshen},
    doi = {10.4310/CIS.2014.v14.n4.a1},
    issn = {15267555, 21634548},
    journal = {Communications in Information and Systems},
    number = {4},
    pages = {213--242},
    title = {Theoretical aspects of mixed volume computation via mixed subdivision},
    url = {http://www.intlpress.com/site/pub/pages/journals/items/cis/content/vols/0014/0004/a001/},
    volume = {14},
    year = {2014}
    }
    
  9. (with Dhagash Mehta, Jonathan Hauenstein, and David Wales)
    "Newton homotopies for sampling stationary points of potential energy landscapes"
    The Journal of Chemical Physics 141 (12), 121104 (2014)
    [ link ] [ arXiv ] [ ] [ ]

    Abstract: One of the most challenging and frequently arising problems in many areas of science is to find solutions of a system of multivariate nonlinear equations. There are several numerical methods that can find many (or all if the system is small enough) solutions but they all exhibit characteristic problems. Moreover, traditional methods can break down if the system contains singular solutions. Here, we propose an efficient implementation of Newton homotopies, which can sample a large number of the stationary points of complicated many-body potentials. We demonstrate how the procedure works by applying it to the nearest-neighbor ϕ4 model and atomic clusters.
    @article{mehta_newton_2014,
    author = {Mehta, Dhagash and Chen, Tianran and Hauenstein, Jonathan D and Wales, David J},
    doi = {10.1063/1.4896657},
    issn = {0021-9606, 1089-7690},
    journal = {The Journal of Chemical Physics},
    keywords = {Atomic and molecular clusters,Biomolecular structure,Nonlinear differential equations,Numerical modeling,Numerical solutions,real homotopy},
    month = {sep},
    number = {12},
    pages = {121104},
    shorttitle = {Communication},
    title = {Newton homotopies for sampling stationary points of potential energy landscapes},
    url = {http://scitation.aip.org/content/aip/journal/jcp/141/12/10.1063/1.4896657},
    volume = {141},
    year = {2014}
    }
    
  10. (with Tien-Yien Li)
    "Solutions to systems of binomial equations"
    Annales Mathematicae Silesianae 28:7–34 (2014)
    [ link ] [ ] [ ]

    Abstract: This paper provides an overview on the geometry of solutions to binomial equations (equations that have exactly two terms) as well as the numerical methods for solving such systems.
    @article{chen_solutions_2014,
    author = {Chen, Tianran and Li, Tien-Yien},
    journal = {Annales Mathematicae Silesianae},
    pages = {7--34},
    title = {{Solutions to systems of binomial equations}},
    volume = {28},
    year = {2014}
    }
    
  11. (with 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 ] [ ] [ ]

    Abstract: Homotopy continuation methods have been proved to be an efficient and reliable class of numerical methods for solving systems of polynomial equations which occur frequently in various fields of mathematics, science, and engineering. Based on the successful software package Hom4PS-2.0 for solving such polynomial systems, Hom4PS-3 has a new fully modular design which allows it to be easily extended. It implements many different numerical homotopy methods including the Polyhedral Homotopy continuation method. Furthermore, it is capable of carrying out computation in parallel on a wide range of hardware architectures including multi-core systems, computer clusters, distributed environments, and GPUs with great efficiency and scalability. Designed to be user-friendly, it includes interfaces to a variety of existing mathematical software and programming languages such as Python, Ruby, Octave, Sage and Matlab.
    @incollection{chen_hom4ps-3,
    author = {Chen, Tianran and Lee, Tsung-Lin and Li, Tien-Yien},
    booktitle = {Mathematical Software – ICMS 2014},
    editor = {Hong, Hoon and Yap, Chee},
    isbn = {978-3-662-44198-5 978-3-662-44199-2},
    keywords = {Algorithm Analysis and Problem Complexity,Discrete Mathematics in Computer Science,Math Applications in Computer Science,Numeric Computing,Software Engineering/Programming and Operating Sy,Theory of Computation,binomial system,homotopy continuation,polyhedral homotopy,polynomial systems},
    month = {jan},
    number = {8592},
    pages = {183--190},
    publisher = {Springer Berlin Heidelberg},
    series = {Lecture Notes in Computer Science},
    shorttitle = {Hom4PS-3},
    title = {{Hom4PS-3: A Parallel Numerical Solver for Systems of Polynomial Equations Based on Polyhedral Homotopy Continuation Methods}},
    url = {http://link.springer.com/chapter/10.1007/978-3-662-44199-2{\_}30},
    year = {2014}
    }
    
  12. (with Tsung-Lin Lee and Tien-Yien Li)
    "Mixed volume computation in parallel"
    Taiwanese Journal of Mathematics 18(1):93–114 (2014)
    [ link ] [ ] [ ]

    Abstract: Efficient algorithms for computing mixed volumes, via the computation of mixed cells, have been implemented in DEMiCs and MixedVol-2.0. While the approaches in those two packages are somewhat different, they follow the same theme and are both highly serial. To fit the need for the parallel computing, a reformulation of the algorithms is inevitable. This article proposes a reformulation of the algorithm for the mixed volume computation rooted from algorithms in graph theory, making it much more fine-grained and scalable. The resulting parallel algorithm can be readily adapted to both distributed and shared memory computing systems. Illustrated by the numerical results on several different architectures, the speedups of our parallel algorithms for the mixed volume computation are remarkable.
    @article{chen_mixed_2014,
    author = {Chen, Tianran and Lee, Tsung-Lin and Li, Tien-Yien},
    journal = {Taiwanese Journal of Mathematics},
    keywords = {bkk,mixed cells,mixed volume},
    number = {1},
    pages = {93--114},
    title = {Mixed volume computation in parallel},
    volume = {18},
    year = {2014}
    }
    
  13. (with Tien-Yien Li)
    "Spherical projective path tracking for homotopy continuation methods"
    Communications in Information and Systems 12(3):195-220 (2012)
    [ link ] [ ] [ ]

    Abstract: Solving systems of polynomial equations is an important problem in mathematics with a wide range of applications in many fields. The homotopy continuation method is a large class of reliable and efficient numerical methods for solving systems of polynomial equations. An essential component in the homotopy continuation method is the path tracking algorithm for tracking smooth paths of one real dimension. In this regard, “divergent paths” pose a tough challenge as the tracking of such paths is generally impossible. The existence of such paths is, in part, caused by \(\mathbb{C}^n\), the space in which homotopy methods usually operate, being non-compact. A well known remedy is to operate inside the complex projective space \(\mathbb{CP}^n\) instead. Path tracking inside \(\mathbb{CP}^n\) is the focus of this article. Taking the Riemannian geometry of \(\mathbb{CP}^n\) into account, we derive the basic algorithm for projective path tracking using the sphere, \(S^{2n+1}\), as the model of computation. Remarkable results from numerical experiments using this method are presented.
    @article{chen_spherical_2012,
    author = {Chen, Tianran and Li, Tien-Yien},
    journal = {Communication in Information and Systems},
    number = {3},
    pages = {195--220},
    title = {Spherical projective path tracking for homotopy continuation methods},
    volume = {12},
    year = {2012}
    }
    

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.