Below you can find the research topics that I have worked on in the past or I am working right now. Generally speaking, I am interested in parallel numerical algorithms for the solution of large-scale linear systems, eigenvalue problems, and problems in data analysis.

#### Numerical methods for the solution of linear systems

*Mean relative error of a Monte Carlo stochastic estimator to approximate the diagonal of the inverse of a model covariance matrix.*

**1.**V. Kalantzis, C. Bekas, A. Curioni, and E. Gallopoulos, "Accelerating Data Uncertainty Quantification By Solving Linear Systems with Multiple Right-Hand Sides,"

*Numerical Algorithms*, Vol. 62, No. 4, pp. 637-653.

#### Numerical methods for the solution of the symmetric eigenvalue problem

*The Newton Branch-hopping scheme to compute the two smallest eigenvalues of a 2D discretized Laplacian.*

**2.**V. Kalantzis, J. Kestyn, E. Polizzi, and Y. Saad, "Domain Decomposition Approaches for Accelerating Contour Integration Eigenvalue Solvers for Symmetric Eigenvalue Problems," Preprint, Dept. Computer Science and Engineering, University of Minnesota, Minneapolis, MN, 2016.

**1.**V. Kalantzis, R. Li, and Y. Saad, "Spectral Schur Complement Techniques for Symmetric Eigenvalue Problems".

*Electronic Transactions on Numerical Analysis*, Vol. 45, pp. 305-329, 2016.

#### Trace of matrix functions

*An example of fitting the approximate diagonal of the matrix inverse using PCHIP. Red: the true diagonal entries. Blue: the original approximation. Green: The fitted approximation.*

**1.**L. Wu, J. Laeuchli, V. Kalantzis, A. Stathopoulos, and E. Gallopoulos "Estimating the Trace of the Matrix Inverse by Interpolating from the Diagonal of an Approximate Inverse".

*Journal of Computational Physics*, Vol. 326, pp. 828-844, 2016.