Bornes pour fonctions de matrices et applications aux méthodes de calcul de structures électroniques

par Paola Boito (XLIM-DMI)

jeudi 20 janv. 2011, 10:30 → 11:30 Europe/Paris

XR.203 (Bâtiment XLIM)

Linear scaling methods, used in quantum chemistry and solid state physics for computation of electronic structures, rely on the phenomenon of localization: for certain choices of the basis functions, the Hamiltonian matrix that describes a physical system is banded or sparse. The associated density matrix, which is a function of the Hamiltonian, is expected to exhibit an off-diagonal decay behavior, which may allow to replace the matrix with a banded approximation. Our goal is to give a rigorous mathematical description and proof of this property. By means of polynomial approximation and matrix analysis techniques, we formulate asymptotic bounds on the entries of the density matrix (as well as more general matrix functions). Such results allow to bound the error due to sparse/banded approximation. Applications of bounds for matrix functions are not limited to computational physics. The talk will present examples of how our approach can be used to estimate matrix functions that describe the connectivity properties of networks. This is joint work with Michele Benzi and Nader Razouk.