AMaSiS 2018 Workshop: Abstracts

FEAST v4.0: Tutorial and practices

Eric Polizzi

University of Massachusetts, Department of Electrical and Computer Engineering, Department of Mathematics and Statistics

FEAST is a general purpose eigenvalue solver which takes its inspiration from the density-matrix representation and contour integration technique in quantum mechanics [1]. The algorithm gathers key elements from complex analysis, numerical linear algebra and approximation theory, and it can be interpreted as a generalization of shift-and-invert iterations that uses multiple shifts in the complex plane leading to an optimal filter projector [2]. Once a given search domain is selected, FEAST’s main computational task consists of a numerical quadrature computation that involves solving independent linear systems along a complex contour. FEAST offers a set of appealing features including remarkable robustness with well-defined convergence rate, and the capability to exploit natural parallelism at three different levels: (i) search domains can be treated separately (no overlap), (ii) linear systems can be solved independently across the quadrature nodes of the complex contour, and (iii) each complex linear system with multiple right-hand-sides can be solved in parallel (using direct or iterative solvers). Parallel resources can be placed at all three levels simultaneously in order to achieve scalability and optimal use of the computing platform.

During these past few years, the FEAST public software package [3] for solving the eigenvalue problem has reached a significant international visibility with many hundreds of new downloads of its most recent version (v3.0) each year. Since the integration of FEAST v2.1 in Intel-MKL in 2013, the solver is now accessible to virtually everyone in the community. The library offers “black-box” reverse communication interfaces (RCI) which are both matrix format and linear system solver independent, and can then be fully customized by expert users to allow maximum flexibility for their applications. Consequently, the software package has been very well received by application developers.

In recent work (which includes the upcoming v4.0 of the software), the solver has been reimplemented to make use of residual inverse iterations. Although, the new filter form is mathematically equivalent to the original FEAST linear projector, it is numerically more efficient and more appealing in a number of new situations. We have demonstrated the effectiveness of the FEAST residual inverse iterations for addressing: (i) the inexact inner-outer iterative approach (IFEAST or FEAST without factorization), (ii) the mixed precision arithmetic iterative procedure, and (iii) the non-linear eigenvalue problem. In addition to all these new features, version v4.0 contains also the PFEAST package including three levels of MPI parallelism which can be integrated with distributed solvers such as cluster pardiso, MUMPS or any custom domain decomposition techniques.

In this tutorial, all the new capabilities of v4.0 package with be reviewed and discussed with practical examples.

Acknowledgments: This work is supported by NSF#1510010 and #1739423.