Parallel Shift-Invert Spectrum Slicing on Distributed Architectures with GPU Accelerators

The solution of large scale eigenvalue problems (EVP) is often the computational bottleneck for many scientific and engineering applications. Traditional eigensolvers, such as direct (e.g. ScaLAPACK) and Krylov subspace (e.g. Lanczos) methods, have struggled in achieving high scalability on large computing resources due to communication and synchronization bottlenecks which are inherent in their implementation. This includes a difficulty in developing well-performing ports of these algorithms to architectures which rely on the use of accelerators, such as graphics processing units (GPU), for the majority of their floating point operations. Recently, there has been significant research into the development of eigensolvers based on spectrum slicing, in particular shift-invert spectrum slicing, to alleviate the communication and synchronization bottlenecks of traditional eigensolvers. In general, spectrum slicing trades the global EVP for many smaller, independent EVPs which may be combined to assemble some desired subset of the entire eigenspectrum. The result is a method which utilizes more floating point operations than traditional eigensolvers, but in a way which allows for the expression of massive concurrency leading to an overall improvement in time-to-solution on large computing resources. In this work, we will examine the performance of parallel shift-invert spectrum slicing on modern GPU clusters using state-of-the-art linear algebra software.