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Register a new userA Shift Selection Strategy for Parallel Shift-Invert Spectrum Slicing in Symmetric Self-Consistent Eigenvalue Computation
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Added by
David Williams-Young
Publication date
2019
fields
Informatics Engineering
and research's language is
English
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The central importance of large scale eigenvalue problems in scientific computation necessitates the development of massively parallel algorithms for their solution. Recent advances in dense numerical linear algebra have enabled the routine treatment of eigenvalue problems with dimensions on the order of hundreds of thousands on the worlds largest supercomputers. In cases where dense treatments are not feasible, Krylov subspace methods offer an attractive alternative due to the fact that they do not require storage of the problem matrices. However, demonstration of scalability of either of these classes of eigenvalue algorithms on computing architectures capable of expressing massive parallelism is non-trivial due to communication requirements and serial bottlenecks, respectively. In this work, we introduce the SISLICE method: a parallel shift-invert algorithm for the solution of the symmetric self-consistent field (SCF) eigenvalue problem. The SISLICE method drastically reduces the communication requirement of current parallel shift-invert eigenvalue algorithms through various shift selection and migration techniques based on density of states estimation and k-means clustering, respectively. This work demonstrates the robustness and parallel performance of the SISLICE method on a representative set of SCF eigenvalue problems and outlines research directions which will be explored in future work.
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In this paper, a parallel structured divide-and-conquer (PSDC) eigensolver is proposed for symmetric tridiagonal matrices based on ScaLAPACK and a parallel structured matrix multiplication algorithm, called PSMMA. Computing the eigenvectors via matrix-matrix multiplications is the most computationally expensive part of the divide-and-conquer algorithm, and one of the matrices involved in such multiplications is a rank-structured Cauchy-like matrix. By exploiting this particular property, PSMMA constructs the local matrices by using generators of Cauchy-like matrices without any communication, and further reduces the computation costs by using a structured low-rank approximation algorithm. Thus, both the communication and computation costs are reduced. Experimental results show that both PSMMA and PSDC are highly scalable and scale to 4096 processes at least. PSDC has better scalability than PHDC that was proposed in [J. Comput. Appl. Math. 344 (2018) 512--520] and only scaled to 300 processes for the same matrices. Comparing with texttt{PDSTEDC} in ScaLAPACK, PSDC is always faster and achieves $1.4$x--$1.6$x speedup for some matrices with few deflations. PSDC is also comparable with ELPA, with PSDC being faster than ELPA when using few processes and a little slower when using many processes.
We offer a lattice-theoretic account of dynamic slicing for {pi}-calculus, building on prior work in the sequential setting. For any run of a concurrent program, we exhibit a Galois connection relating forward slices of the start configuration to backward slices of the end configuration. We prove that, up to lattice isomorphism, the same Galois connection arises for any causally equivalent execution, allowing an efficient concurrent implementation of slicing via a standard interleaving semantics. Our approach has been formalised in the dependently-typed language Agda.
The parallel full approximation scheme in space and time (PFASST) introduced by Emmett and Minion in 2012 is an iterative strategy for the temporal parallelization of ODEs and discretized PDEs. As the name suggests, PFASST is similar in spirit to a space-time FAS multigrid method performed over multiple time-steps in parallel. However, since the original focus of PFASST has been on the performance of the method in terms of time parallelism, the solution of any spatial system arising from the use of implicit or semi-implicit temporal methods within PFASST have simply been assumed to be solved to some desired accuracy completely at each sub-step and each iteration by some unspecified procedure. It hence is natural to investigate how iterative solvers in the spatial dimensions can be interwoven with the PFASST iterations and whether this strategy leads to a more efficient overall approach. This paper presents an initial investigation on the relative performance of different strategies for coupling PFASST iterations with multigrid methods for the implicit treatment of diffusion terms in PDEs. In particular, we compare full accuracy multigrid solves at each sub-step with a small fixed number of multigrid V-cycles. This reduces the cost of each PFASST iteration at the possible expense of a corresponding increase in the number of PFASST iterations needed for convergence. Parallel efficiency of the resulting methods is explored through numerical examples.
This paper continues to develop a fault tolerant extension of the sparse grid combination technique recently proposed in [B. Harding and M. Hegland, ANZIAM J., 54 (CTAC2012), pp. C394-C411]. The approach is novel for two reasons, first it provides several levels in which one can exploit parallelism leading towards massively parallel implementations, and second, it provides algorithm-based fault tolerance so that solutions can still be recovered if failures occur during computation. We present a generalisation of the combination technique from which the fault tolerant algorithm is a consequence. Using a model for the time between faults on each node of a high performance computer we provide bounds on the expected error for interpolation with this algorithm. Numerical experiments on the scalar advection PDE demonstrate that the algorithm is resilient to faults on a real application. It is observed that the trade-off of recovery time to decreased accuracy of the solution is suitably small. A comparison with traditional checkpoint-restart methods applied to the combination technique show that our approach is highly scalable with respect to the number of faults.
We study interacting massive spin-1 theories in de Sitter (dS) and anti-de Sitter (AdS) space that possess shift symmetries parametrized by (A)dS Killing vectors. We show how they emerge from the massless limit of massive spin-2 theories on (A)dS space. In the case of massive gravity, the corresponding spin-1 theory realizes a symmetry breaking pattern that takes two copies of the (A)dS isometry group down to a diagonal subgroup. By taking the flat space limit of this theory, we find a new symmetry of the decoupling limit of massive gravity in flat space. This symmetry acts on the vector modes, is parametrized by an antisymmetric tensor, and fixes the nonlinear structure of the scalar-vector sector of the decoupling limit.
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