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Sofja Kovalevskaja project

Direct and inverse problems of algebra and analysis

Research team

Former members

  • Anders Jensen (commutative algebra, symbolic computations); currently a postdoc at the Mathematical Institute in Goettingen.
  • Nenad Moraca (linear algebra, combinatorics, probability); currently teaching in Novyi Sad, Serbia.
  • Zhiqiang Xu (approximation theory, splines, combinatorics); currently an assistant professor at the Academy of Mathematics and Systems of the Chinese Academy of Sciences.

Topics

Accurate and efficient linear algebra and related algorithms

Computational complexity and numerical stability of sequential and parallel linear algebra algorithms, stability of multivariate polynomial evaluation in floating point arithmetics.

Publications:

  • Olga Holtz and Noam Shomron, Computational complexity and numerical stability of linear problems, Jun 2009.    arXiv version.
  • Grey Ballard, James Demmel, Olga Holtz and Oded Schwartz, Minimizing communication in linear algebra, May 2009.    arXiv version.
  • Grey Ballard, James Demmel, Olga Holtz and Oded Schwartz, Communication-optimal parallel and sequential Cholesky decomposition, Feb 2009.    arXiv version.
  • James Demmel, Ioana Dumitriu, Olga Holtz and Plamen Koev, Accurate and efficient expression evaluation and linear algebra, Acta Numerica, 17 (2008), 87--145.    arXiv version.
  • James Demmel, Ioana Dumitriu and Olga Holtz, Fast linear algebra is stable, Numer. Math., 108 (2007), 59-91.     arXiv version.
  • James Demmel, Ioana Dumitriu, Olga Holtz and Robert Kleinberg, Fast matrix multiplication is stable, Numer. Math., 106 (2007), no. 2, 199-224.     arXiv version
  • James Demmel, Ioana Dumitriu and Olga Holtz, Toward accurate polynomial evaluation in rounded arithmetic, Foundations of Computational Mathematics: Santander 2005 (L. Pardo et al, eds.) Cambridge University Press, 2006, pp. 36-105.     arXiv version

Direct and inverse problems of core linear algebra

The inverse eigenvalue problem for nonnegative matrices and its variants, eigenvalue localization of special matrix classes, Markov chains, other related problems and probability and combinatorics.
Publications:

  • Gautam Bharali and Olga Holtz, Functions preserving nonnegativity of matrices, SIMAX, 30 (2008), no.1, 84-101.     arXiv version
  • Nenad Moraca, Bounds for norms of the matrix inverse and the smallest singular value, Linear Alg. Appl., 429 (2008), no.10, 2589--2601.
  • Olga Holtz and Bernd Sturmfels, Hyperdeterminantal relations among symmetric principal minors, J. Algebra, 316 (2007), no.2, 634-648.     arXiv version

Root localization and asymptotics of polynomials

Matrix criteria for stability and root location, applications to probability and theoretical computer science, connections with scalar- and matrix-valued orthogonal polynomials, including their asymptotics.

Publications:

  • Olga Holtz and Mikhail Tyaglov, Structured matrices, continued fractions, and root localization of polynomials, Dec 2009.    arXiv version.
  • Mikhail Tyaglov, On the number of real critical points of logarithmic derivatives and the Hawaii conjecture, Feb 2009.    arXiv version.
  • Diego Dominici, Polynomial solutions of nonlinear integral equations, Mar 2008.    arXiv version.
  • Yury Barkovsky (translated by Olga Holtz and Mikhail Tyaglov), Lectures on the Routh-Hurwitz problem, Feb 2008.    arXiv version.
  • Diego Dominici, Asymptotic analysis of the Bell polynomials by the ray method, Sep 2007.    arXiv version.
  • Diego Dominici, Fisher information of orthogonal polynomials I, Aug 2007.    arXiv version.

The algebra of box splines and zonotopes

Connections between polynomial spaces, polynomial ideas, constant-coefficient PDEs, multivariate difference equations, multivariate polynomial interpolation, tilings and integer points in zonotopes, combinatorics of hyperplane arrangements, and various counting functions on graphs.

Publications:

  • Olga Holtz, Amos Ron and Zhiqiang Xu, Hierarchical zonotopal spaces, Oct 2009.    arXiv version.
  • Bernd Sturmfels and Zhiqiang Xu, Sagbi bases of Cox-Nagata rings, Mar 2008.    arXiv version.
  • Olga Holtz and Amos Ron, Zonotopal algebra, Aug 2007.    arXiv version.

Interplay between approximation theory and other fields

Connections between approximation theory, classical geometry and analysis, algorithms, and probability theory.

Publications:

  • Olga Holtz, Fedor Nazarov and Yuval Peres, New coins from old, smoothly, Aug 2008.    arXiv version.
  • Shaowei Lin, Bernd Sturmfels and Zhiqiang Xu, Marginal likelihood integrals for mixtures of independence models, May 2008.    arXiv version.

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