Numerical linear algebra

Acceleration strategies in numerical linear algebra

Electronic structure calculations are sources of infinitely many problems dealing with numerical linear algebra. Recently, I got interested with collaborators on acceleration techniques for gradient-type methods. The idea is that standard optimal step gradient descent methods for high-dimensional problems often suffer from the strange zig-zag effect, where the residuals live in a two dimensional subspace. Avoiding this effect allows to significantly accelerated the convergence of these methods by exploiting their link with power methods and the alignment of the residual with eigendirections.

Related works:

• J.P. Chehab, G. Kemlin, M. Raydan, Y. Saad. Eigenvector-based acceleration strategies for gradient-type methods. pdf

Lanczos based acceleration for minimizing quadratic functionals.

Domain decomposition methods

During my Master studies, I also worked on domain decomposition methods for the linearized Boussinesq equations in coastal oceanography. We showed that using transparent boundary conditions (which are not easy to compute...) helps a lot as transmission conditions in domain decomposition algorithms.

Related works:

• J. G. Caldas Steinstraesser, G. Kemlin, and A. Rousseau. A domain decomposition method for linearized Boussinesq-type equations. Journal of Mathematical Study, 52(3):320-340, 2019. doi - pdf