An Optimized and Scalable Eigensolver for Sequences of Eigenvalue Problems

In many scientific applications the solution of non-linear differential equations are obtained through the set-up and solution of a number of successive eigenproblems. These eigenproblems can be regarded as a sequence whenever the solution of one problem fosters the initialization of the next. In addition, in some eigenproblem sequences there is a connection between the solutions of adjacent eigenproblems. Whenever it is possible to unravel the existence of such a connection, the eigenproblem sequence is said to be correlated. When facing with a sequence of correlated eigenproblems the current strategy amounts to solving each eigenproblem in isolation. We propose a alternative approach which exploits such correlation through the use of an eigensolver based on subspace iteration and accelerated with Chebyshev polynomials (ChFSI). The resulting eigensolver is optimized by minimizing the number of matrix-vector multiplications and parallelized using the Elemental library framework. Numerical results show that ChFSI achieves excellent scalability and is competitive with current dense linear algebra parallel eigensolvers.

Towards an Efficient Use of the BLAS Library for Multilinear Tensor Contractions

Mathematical operators whose transformation rules constitute the building blocks of a multi-linear algebra are widely used in physics and engineering applications where they are very often represented as tensors. In the last century, thanks to the advances in tensor calculus, it was possible to uncover new research fields and make remarkable progress in the existing ones, from electromagnetism to the dynamics of fluids and from the mechanics of rigid bodies to quantum mechanics of many atoms. By now, the formal mathematical and geometrical properties of tensors are well defined and understood; conversely, in the context of scientific and high-performance computing, many tensor- related problems are still open. In this paper, we address the problem of efficiently computing contractions among two tensors of arbitrary dimension by using kernels from the highly optimized BLAS library. In particular, we establish precise conditions to determine if and when GEMM, the kernel for matrix products, can be used. Such conditions take into consideration both the nature of the operation and the storage scheme of the tensors, and induce a classification of the contractions into three groups. For each group, we provide a recipe to guide the users towards the most effective use of BLAS.

A Parallel and Scalable Iterative Solver for Sequences of Dense Eigenproblems Arising in FLAPW

In one of the most important methods in Density Functional Theory - the Full-Potential Linearized Augmented Plane Wave (FLAPW) method - dense generalized eigenproblems are organized in long sequences. Moreover each eigenproblem is strongly correlated to the next one in the sequence. We propose a novel approach which exploits such correlation through the use of an eigensolver based on subspace iteration and accelerated with Chebyshev polynomials. The resulting solver, parallelized using the Elemental library framework, achieves excellent scalability and is competitive with current dense parallel eigensolvers.

Block Iterative Eigensolvers for Sequences of Correlated Eigenvalue Problems

In Density Functional Theory simulations based on the LAPW method, each self-consistent field cycle comprises dozens of large dense generalized eigenproblems. In contrast to real-space methods, eigenpairs solving for problems at distinct cycles have either been believed to be independent or at most very loosely connected. In a recent study [7], it was demonstrated that, contrary to belief, successive eigenproblems in a sequence are strongly correlated with one another. In particular, by monitoring the subspace angles between eigenvectors of successive eigenproblems, it was shown that these angles decrease noticeably after the first few iterations and become close to collinear. This last result suggests that we can manipulate the eigenvectors, solving for a specific eigenproblem in a sequence, as an approximate solution for the following eigenproblem. In this work we present results that are in line with this intuition. We provide numerical examples where opportunely selected block iterative eigensolvers benefit from the reuse of eigenvectors by achieving a substantial speed-up. The results presented will eventually open the way to a widespread use of block iterative eigensolvers in ab initio electronic structure codes based on the LAPW approach.