Ioana Dumitriu Awarded National Science Foundation CAREER Grant

Ioana Dumitriu has received a CAREER grant from the National Science Foundation (NSF). The NSF awards these prestigious grants to “junior faculty who exemplify the role of teacher-scholars through outstanding research, excellent education and the integration of education and research within the context of the mission of their organizations.” The following article gives a look into Ioana’s contributions to the areas of random matrix theory, numerical linear algebra and numerical analysis.

Ioana Dumitriu

Ioana Dumitriu’s research career could be described as a journey through discrete and applied mathematics. During her undergraduate years at New York University, she worked on a variety of combinatorial and probabilistic problems; her PhD thesis was on random matrix theory; during her postdoctoral stay at University of California, Berkeley, she started working on numerical analysis and numerical linear algebra; and recently, as an Assistant Professor at University of Washington, her work has been focused on random structures and scientific computing.

Some of Ioana’s most important research contributions are in random matrix theory, a field of mathematics with a two-fold origin in nuclear physics (with works authored by Eugene Wigner and Freeman Dyson) and statistics (started by John Wishart and L. K. Hua). Random matrix theory brings together tools from fields as varied as probability, combinatorics, special functions, linear algebra, and perturbation theory, to answer questions formulated in quantum and statistical physics, number theory, wireless communications, finance, and scientific and numerical computing. The basic problems in this field deal with analyzing eigenvalue and eigenvector statistics of (mostly large) matrices whose entries are random variables.

In classical random matrix ensembles (Gaussian, Wishart, Jacobi, Circular) the parameter β, which describes the eigenvalue distributions, has traditionally taken only three values: 1, 2, and 4, corresponding to entries distributed over the real, complex, and quaternion fields. There is, however, no intrinsic need for this restriction from the distributional viewpoint, and in fact, interpolating β-ensembles of eigenvalues have been studied since the re-discovery of the Selberg integral in the 1970s. Studying such ensembles, even in the absence of a matrix model when β is not 1, 2, or 4, has provided tremendous insight into eigenvalue statistics, as well as a way to construct a unified theory for the previously distinct cases.

Ioana’s work in random matrix theory has been in the area of β-ensembles; her first major contribution, which was part of her PhD thesis, was to construct real, symmetric, tridiagonal matrix models for all β-Gaussian (β-Hermite) and β-Wishart (β-Laguerre) ensembles. These matrix models have sparked a flurry of research in the area, as well as lead to the subsequent discovery of β-analogues for other ensembles (Jacobi, anti-symmetric Gaussian). Among the many results obtained through use of the new models are a functional central limit theorem for the eigenvalues of large random matrices and a very surprising connection between β-Hermite and β-Laguerre ensembles and stochastic Schrödinger operators (discovered by Jose Ramirez, Brian Ryder, and Balint Virag). Ioana’s thesis received an Honorable Mention for the 2004 Householder Prize, a triennial award for the best PhD thesis in numerical analysis.

Figure 1: Distributions of a random eigenvalue from the n = 4 ensembles with β = 2, 4, 6, 8, 10. The “bumps” increase with β. As β grows large, the distribution approaches the average of 4 Dirac delta distributions.

Another component of Ioana’s work in random matrices has been the development of software for computing eigenvalue statistics of β-ensembles, both numerically and symbolically. This has been an important computational achievement in this area, both for its use in research and as an educational tool (in classes taught at MIT, Berkeley, and UC Davis). It was used to create Figure 1.

The second major area of research in which Ioana has made important contributions is numerical linear algebra and numerical analysis. During her postdoctoral studies in Berkeley, together with James Demmel and Olga Holtz, Ioana initiated a wide-scope study of accurate evaluation of multivariate polynomials; accuracy here means that the computational error is allowed to be only a (small) fraction of the true answer, as opposed to some absolute tolerance bound. This type of relative accuracy is particularly useful in evaluating the polynomials close to their zero sets, and it is thus of considerable interest in computational geometry and numerical linear algebra. In her work, Ioana has found necessary and (sometimes) sufficient conditions for the existence of accurate algorithms for generic polynomials; the resulting paper was awarded the Leslie Fox Prize for Numerical Analysis in 2007. Subsequently, Ioana has worked on the stability of fast linear algebra algorithms, showing that, using any fast matrix multiplication algorithm, one can construct a stable matrix multiplication algorithm that runs just as fast. The stable, fast algorithm can then be used to construct fast and stable algorithms for virtually all linear algebra computations.

Ioana’s current research involves using randomization as a means to speed up and stabilize numerical algorithms, as well as constructing algorithms that minimize communication between the various layers of memory in computer architecture. She is also continuing to work on eigenvalue statistics of β-ensembles as well as on eigenvector statistics of adjacency matrices of random graphs.