A scalable H-matrix approach for the solution of boundary integral equations on multi-GPU clusters

Abstract

In this work, we consider the solution of boundary integral equations by means of a scalable hierarchical matrix approach on clusters equipped with graphics hardware, i.e. graphics processing units (GPUs). To this end, we extend our existing single-GPU hierarchical matrix library hmglib such that it is able to scale on many GPUs and such that it can be coupled to arbitrary application codes. Using a model GPU implementation of a boundary element method (BEM) solver, we are able to achieve more than 67 percent relative parallel speed-up going from 128 to 1024 GPUs for a model geometry test case with 1.5 million unknowns and a real-world geometry test case with almost 1.2 million unknowns. On 1024 GPUs of the cluster Titan, it takes less than 6 minutes to solve the 1.5 million unknowns problem, with 5.7 minutes for the setup phase and 20 seconds for the iterative solver. To the best of the authors' knowledge, we here discuss the first fully GPU-based distributed-memory parallel hierarchical matrix Open Source library using the traditional H-matrix format and adaptive cross approximation with an application to BEM problems.

Acceleration of the space–time boundary element method using GPUs

“…The mass matrix M h represents the identity operator in the boundary integral equations ( 4) and (5) and in the corresponding variational formulations. It has block dimensions E t × E t with blocks of size E s × N s , the same as K h .…”

“…In one of the early works [24] authors use the on-the-fly approach to accelerate the boundary element method for the Helmholtz equation. More recently, the focus has been on acceleration of the fast BEM techniques, such as adaptive cross approximation [5,25] or fast multipole method [26]. An example of an open-source GPU-accelerated library of BEMbased solvers for the Laplace, Helmholtz, and Maxwell problems is Bempp-cl [2].…”

“…Due to the expected access pattern, we store the quadrature reference on the GPUs in the constant memory. We have a GPU-constant quadrature reference variable for each of the four available quadrature orders (1,2,4,5) and an associated boolean variable denoting if the quadrature reference of the corresponding order has already been initialized.…”