ICTS

This talk is divided into two parts, the first part will be on the approximation of the bilinear form of a matrix function, especially for large symmetric positive definite matrices. The optimal estimate of this bilinear form has different applications including the error estimate and stopping criteria for conjugate gradient method. Further, we will see how this bilinear form is related to Riemann-Stieltjes integral and its quadrature rule is crucial in estimating the energy norm of algebraic error. The second part of the talk is the application of first part in numerical homogenization using Fourier Galerkin method and its fast linear solvers. Two different discretization methods are considered with their computational advantages. The discretization error is measured using duality argument which has been compared with the algebraic error in order to have a bound on the total error during the entire numerical approximations.