ABINIT is a package whose main program allows one to find the total energy, charge density, electronic structure and many other properties of systems made of electrons and nuclei, (molecules and periodic solids) within Density Functional Theory (DFT), Many-Body Perturbation Theory (GW approximation and Bethe-Salpeter equation) and Dynmical Mean Field Theory (DMFT). ABINIT also allows to optimize the geometry according to the DFT forces and stresses, to perform molecular dynamics simulations using these forces, and to generate dynamical matrices, Born effective charges and dielectric tensors. The present paper aims to describe the new capabilities of ABINIT that have been developed since 2009. It covers both physical and technical developments inside the ABINIT code, as well as developments provided within the ABINIT package. The developments are described with relevant references, input variables, tests and tutorials.

We propose a non-intrusive reduced-order modeling method based on the notion of space-timeparameter proper orthogonal decomposition for approximating the solution of non-linear parametrized time-dependent partial differential equations. A two-level proper orthogonal decomposition method is introduced for constructing spatial and temporal basis functions with special properties such that the reduced-order model satisfies the boundary and initial conditions by construction. A radial basis function approximation method is used to estimate the undetermined coefficients in the reduced-order model without resorting to Galerkin projection. This nonintrusive approach enables the application of our approach to general problems with complicated nonlinearity terms. Numerical studies are presented for the parametrized Burgers' equation and a parametrized convection-reaction-diffusion problem. We demonstrate that our approach leads to reduced-order models that accurately capture the behavior of the field variables as a function of the spatial coordinates, the parameter vector and time.

SUMMARYThis paper presents a methodology for constructing low-order surrogate models of finite element/finite volume discrete solutions of parameterized steady-state partial differential equations. The construction of proper orthogonal decomposition modes in both physical space and parameter space allows us to represent high-dimensional discrete solutions using only a few coefficients. An incremental greedy approach is developed for efficiently tackling problems with high-dimensional parameter spaces. For numerical experiments and validation, several non-linear steady-state convection-diffusion-reaction problems are considered: first in one spatial dimension with two parameters, and then in two spatial dimensions with two and five parameters. In the two-dimensional spatial case with two parameters, it is shown that a 7×7 coefficient matrix is sufficient to accurately reproduce the expected solution, while in the five parameters problem, a 13×6 coefficient matrix is shown to reproduce the solution with sufficient accuracy. The proposed methodology is expected to find applications to parameter variation studies, uncertainty analysis, inverse problems and optimal design.