Algebraic Algorithms

  • Polynomial system solving, sparse/toric elimination theory, structured matrices. Resultants reduce the problem of solving a non-linear system of equations to linear algebra. Sparse elimination exploits the structure of polynomials. Newton polytopes model the sparseness of polynomials and provide the bridge between algebra and combinatorial geometry. Some related Algebraic software.

  • Real solving, Real algebraic numbers. Fast solvers based on Continued Fractions and Sturm sequences, implemented in C++ (library SYNAPS, now Mathemagix). MAPLE software SLV for operations on real algebraic numbers (comparison, sign determination), real solving of a polynomial equation and of bivariate systems. The software handles arbitrary inputs, and outputs all real solutions in isolating interval representation.

  • Resultant formulae. Optimal sparse resultant matrices for unmixed and certain classes of mixed multihomogeneous polynomial systems, by means of complexes. (cf. arXiv:0904.4064). MAPLE software MHRES constructs these matrices. An efficient representation of complexes is implemented and used to output Sylvester, Bezout-type or even hybrid resultant matrices free of extraneous factors. Macaulay-type formulae for sparse resultants of generalized unmixed polynomial systems, by means of a single lifting function of the input Newton polytopes (cf. article).

  • Software development and Benchmarking. In Mathemagix, we developed module Realroot, solvers for univariate polynomials (e.g. continued fractions), for systems of polynomials (subdivision methods); interval-Newton method. Polytopal computations for algebraic manipulation: normals, faces, ridges, extreme or interior points, Minkowski sums and Mixed volumes; connection to polynomial representation to exploit toric elimination theory (module Polytopix).
    Experiments on state-of-the-art black-box univariate real solvers (isolating real roots over the integers). To the best of our knowledge, this is the largest number of tests for univariate real solving up to date.


Geometric Algorithms

  • Nonlinear computational geometry and geometric design. Voronoi diagrams of circles and ellipses (IMA nugget), using the incremental algorithm of CGAL. Arrangements of curved objects (talk). Computer-aided Geometric design on modeler "Axel" (video of A.Mantzaflaris' talk).

  • Implicitization of parametric hypersurfaces: Prediction of the implicit support and computation of a matrix kernel for exploiting sparseness. Approximate implicitization by means of numerical linear algebra (Τ.Kalinka's PhD Thesis) (webpage).

  • Convex geometry in general dimension. Convex hull and symbolic perturbation, mixed volume (software). Regular fine mixed subdivisions of Minkowski sums, regular triangulations, secondary polytopes, resultant polytopes (software). Pictured is an illustration of Cayley's trick.

  • Approximate geometric algorithms. Nearest neighbors and data structures for searching (survey). Randomized kd-trees, and specific methods for exploiting the structure of the input dataset.
    Minkowski decomposition, pictured (webpage): optimal algorithms with a fixed-size summand, approximation of general (NP-hard) problem.
    Approximation algorithms for visibility.

  • Python projects and CGAL-Python bindings, including visibility tools: webpage. The goal is to provide a user-friendly interface, especially for educational purposes, as well as visibility tools for algorithms and objects from CGAL, and to develop geometric algorithms in Python.

Applications

  • Molecular conformations in Structural bioinformatics. Enumeration of all possible conformations of (small) molecules/proteins under some geometric constrains. The problem is formulated in algebraic terms as a system of polynomial equations which is then solved using sparse elimination and matrix methods. Utilizing data from NMR and RDC, reduces the problem to linear algebra.

  • Structure of Transmembrane proteins. Geometric modelling of β-barrels and detection of the transmembrane region of a β-barrel transmembrane protein. The geometric modelling algorithm combines a non-linear least square minimization method and a genetic algorithm in order to find the characteristics (axis, radius) of a shape with axial symmetry which best models a β-barrel. The transmembrane region is detected by profiling the external residues of the β-barrel along its axis in terms of hydrophobicity and existence of aromatic and charged residues. TbB-Tool implements these algorithms.

  • Lakes. "Lakes" is a program for the prediction of protein binding sites. It analyzes the solvent and its contacts with proteins (obtained by X-ray crystallography) and defines clusters of water molecules, which mark potentially exposed interaction and binding sites of the protein. You can download the program here. Contact person: Dr Thanassis Tartas (in the photo black is the protein, colored are the Oxygen atoms of the clusters, with red being the largest cluster).

  • Robot kinematics. Robust calibration of parallel robots (by applying elimination techniques), forward kinematics of Stewart/Gough platforms (by exploiting the structure of the underlying polynomial system), and design of robotic platforms for medical applications such as physiotherapy.

  • Geometric graphs. Graph embedding in Euclidean spaces, rigid graphs and rigidity theory. Geometric graphs, distributed routing in graphs, and applications to sensor networks.