- 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:
- Vladimir Kostov, Boris Shapiro, Mikhail Tyaglov, Maximal univalent disks of real rational functions and Hermite-Biehler polynomials, May 2010. arXiv version.
- 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:
- Matthias Lenz, The f-vector of a matroid complex is log-concave, Jun 2011.
arXiv version.
- Matthias Lenz, Hierarchical zonotopal power ideals, Nov 2010.
arXiv version.
- 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.
ERC Starting Grant (2010-2015)