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In this paper, we present a highly efficient approach for numerically solving the Black–Scholes equation in order to price European and American basket options. Therefore, hardware features of contemporary high performance computer architectures such as non-uniform memory access and hardware-threading are exploited by a hybrid parallelization using MPI and OpenMP which is able to drastically reduce the computing time. In this way, we achieve very good speed-ups and are able to price baskets with up to six underlyings. Our approach is based on a sparse grid discretization with finite elements and makes use of a sophisticated adaption. The resulting linear system is solved by a conjugate gradient method that uses a parallel operator for applying the system matrix implicitly. Since we exploit all levels of the operator's parallelism, we are able to benefit from the compute power of more than 100 cores. Several numerical examples as well as an analysis of the performance for different computer architectures are provided. [ABSTRACT FROM PUBLISHER]

Walls represent globally distributed, locally extensive, artificial ecosystems. Wall vegetation is still poorly known in the Mediterranean and Temperate regions of southeastern Europe. The aim of this study is to classify and describe chasmophytic vegetation of walls, covering southeastern Europe from Slovenia to North Macedonia. From a total 463 phytosociological relevés, we identify and describe 12 distinct species – poor to moderately rich vegetation units using TWINSPAN evaluated by NMDS, and indicator values. The vegetation units are included within three alliances from two macroclimate regions: (1) vegetation of cool-temperate Europe of the Cymbalario-Asplenion alliance, and (2) Mediterranean vegetation of the Galio valantiae-Parietarion judaicae and Artemisio arborescentis-Capparidion spinosae alliances. The southernmost limit of the Cymbalario-Asplenion was determined in Central Bosnia. The presence of the Artemisio arborescentis-Capparidion spinosae in the eastern Adriatic is highlighted. [ABSTRACT FROM AUTHOR]