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A hybrid MPI+OpenMP parallelization strategy has been implemented into the density functional theory code FLEUR. Based on the full-potential linearized augmented plane-wave (FLAPW) method, FLEUR is a well-established all-electron code specialized on the simulation of materials properties of crystalline bulk solids and surfaces with significant electronic and magnetic complexity. Developed in over 30 years the Fortran implementation included two layers of MPI-based distributed memory parallelization that serves as a reference for our work. The revised code version shows superior performance, improved scalability and thereby opens the path to exploit current and future high performance computing architectures efficiently. Multiple threads per MPI process can be utilized by interfacing with optimized linear algebra subroutines from the BLAS and LAPACK libraries as well as in code sections with explicit OpenMP statements. We demonstrate that the additional multithreading helps to avoid the communication induced scalability limit of the pure-MPI version and simultaneously boosts the single node-performance on current multi-core systems. This enables FLEUR calculations for unit cells with over 1000 atoms to simulate extended defects, surfaces and disordered solids.

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The work is devoted to integro-differential equations related to stochastic processes. We study the relationship between differential equations with random perturbations - stochastic differential equations (SDEs) - and deterministic equations for the probabilistic characteristics of processes determined by random perturbations. The resulting deterministic pseudodifferential equations are investigated by semigroup methods and Fourier transform methods.