We present a computational framework for high-performance tensor contractions on GPUs. High-performance is difficult to obtain using existing libraries, especially for many independent contractions where each contraction is very small, e.g., sub-vector\/warp in size. However, using our framework to batch contractions plus application-specifics, we demonstrate close to peak performance results. In particular, to accelerate large scale tensor-formulated high-order finite element method (FEM) simulations, which is the main focus and motivation for this work, we represent contractions as tensor index reordering plus matrix-matrix multiplications (GEMMs). This is a key factor to achieve algorithmically many-fold acceleration (vs. not using it) due to possible reuse of data loaded in fast memory. In addition to using this context knowledge, we design tensor data-structures, tensor algebra interfaces, and new tensor contraction algorithms and implementations to achieve 90+% of a theoretically derived peak on GPUs. On a K40c GPU for contractions resulting in GEMMs on square matrices of size 8 for example, we are 2.8x faster than CUBLAS, and 8.5x faster than MKL on 16 cores of Intel Xeon ES-2670 (Sandy Bridge) 2.60GHz CPUs. Finally, we apply autotuning and code generation techniques to simplify tuning and provide an architecture-aware, user-friendly interface.<\/p>\n","protected":false},"excerpt":{"rendered":"
We present a computational framework for high-performance tensor contractions on GPUs. High-performance is difficult to obtain using existing libraries, especially for many independent contractions where each contraction is very small, e.g., sub-vector\/warp in size. However, using our framework to batch contractions plus application-specifics, we demonstrate close to peak performance results. In particular, to accelerate large […]<\/p>\n","protected":false},"author":351,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":"","jetpack_publicize_message":"","jetpack_publicize_feature_enabled":true,"jetpack_social_post_already_shared":true,"jetpack_social_options":{"image_generator_settings":{"template":"highway","default_image_id":0,"font":"","enabled":false},"version":2}},"categories":[36,11,89,3],"tags":[1787,215,1782,14,1037,212,37,324,20,252,1543],"class_list":["post-15361","post","type-post","status-publish","format-standard","hentry","category-algorithms","category-computer-science","category-nvidia-cuda","category-paper","tag-algorithms","tag-code-generation","tag-computer-science","tag-cuda","tag-fem","tag-finite-element-method","tag-linear-algebra","tag-matrix-multiplication","tag-nvidia","tag-openmp","tag-tesla-k40"],"views":5768,"jetpack_publicize_connections":[],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/hgpu.org\/index.php?rest_route=\/wp\/v2\/posts\/15361","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/hgpu.org\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/hgpu.org\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/hgpu.org\/index.php?rest_route=\/wp\/v2\/users\/351"}],"replies":[{"embeddable":true,"href":"https:\/\/hgpu.org\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=15361"}],"version-history":[{"count":0,"href":"https:\/\/hgpu.org\/index.php?rest_route=\/wp\/v2\/posts\/15361\/revisions"}],"wp:attachment":[{"href":"https:\/\/hgpu.org\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=15361"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/hgpu.org\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=15361"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/hgpu.org\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=15361"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}