Parallelizing dense matrix computations to distributed memory architectures is a well-studied subject and generally considered to be among the best understood domains of parallel computing. Two packages, developed in the mid 1990s, still enjoy regular use: ScaLAPACK and PLAPACK. With the advent of many-core architectures, which may very well take the shape of distributed memory architectures within a single processor, these packages must be revisited since the traditional MPI-based approaches will likely need to be extended. Thus, this is a good time to review lessons learned since the introduction of these two packages and to propose a simple yet effective alternative. Preliminary performance results show the new solution achieves competitive, if not superior, performance on large clusters.

/pdf/elemental-a-new-framework-for-distributed-memory-dense-1cewzjr7ed.pdf

Elemental: A New Framework for Distributed Memory Dense Matrix Computations

We present a new parallel implementation of a divide and conquer algorithm for computing the spectral decomposition of a symmetric tridiagonal matrix on distributed memory architectures. The implementation we develop differs from other implementations in that we use a two-dimensional block cyclic distribution of the data, we use the Lowner theorem approach to compute orthogonal eigenvectors, and we introduce permutations before the back transformation of each rank-one update in order to make good use of deflation. This algorithm yields the first scalable, portable, and numerically stable parallel divide and conquer eigensolver. Numerical results confirm the effectiveness of our algorithm. We compare performance of the algorithm with that of the QR algorithm and of bisection followed by inverse iteration on an IBM SP2 and a cluster of Pentium PIIs.

/pdf/a-parallel-divide-and-conquer-algorithm-for-the-symmetric-52q6xzuegu.pdf

A Parallel Divide and Conquer Algorithm for the Symmetric Eigenvalue Problem on Distributed Memory Architectures

We present a new parallel algorithm for the dense symmetric eigenvalue/eigenvector problem that is based upon the tridiagonal eigensolver, Algorithm $\mbox{\sf MR}^3$, recently developed by Dhillon and Parlett. Algorithm $\mbox{\sf MR}^3$ has a complexity of O(n2) operations for computing all eigenvalues and eigenvectors of a symmetric tridiagonal problem. Moreover the algorithm requires only O(n) extra workspace and can be adapted to compute any subset of k eigenpairs in O(nk) time. In contrast, all earlier stable parallel algorithms for the tridiagonal eigenproblem require O(n3) operations in the worst case, while some implementations, such as divide and conquer, have an extra O(n2) memory requirement. The proposed parallel algorithm balances the workload equally among the processors by traversing a matrix-dependent representation tree which captures the sequence of computations performed by Algorithm $\mbox{\sf MR}^3$. The resulting implementation allows problems of very large size to be solved efficiently---the largest dense eigenproblem solved in-core on a 256 processor machine with 2 GBytes of memory per processor is for a matrix of size 128,000 $\times$ 128,000, which required about 8 hours of CPU time. We present comparisons with other eigensolvers and results on matrices that arise in the applications of computational quantum chemistry and finite element modeling of automobile bodies.

/pdf/a-parallel-eigensolver-for-dense-symmetric-matrices-based-on-4t2cyw5r9x.pdf

A Parallel Eigensolver for Dense Symmetric Matrices Based on Multiple Relatively Robust Representations

Determinantal Point Processes (DPPs) were introduced by Macchi as a model for repulsive (fermionic) particle distributions. But their recent popularization is largely due to their usefulness for encouraging diversity in the final stage of a recommender system. 
The standard sampling scheme for finite DPPs is a spectral decomposition followed by an equivalent of a randomly diagonally-pivoted Cholesky factorization of an orthogonal projection, which is only applicable to Hermitian kernels and has an expensive setup cost. Researchers have begun to connect DPP sampling to $LDL^H$ factorizations as a means of avoiding the initial spectral decomposition, but existing approaches have only outperformed the spectral decomposition approach in special circumstances, where the number of kept modes is a small percentage of the ground set size. 
This article proves that trivial modifications of $LU$ and $LDL^H$ factorizations yield efficient direct sampling schemes for non-Hermitian and Hermitian DPP kernels, respectively. Further, it is experimentally shown that even dynamically-scheduled, shared-memory parallelizations of high-performance dense and sparse-direct factorizations can be trivially modified to yield DPP sampling schemes with essentially identical performance. 
The software developed as part of this research, Catamari, this https URL, is released under the Mozilla Public License v2.0. It contains header-only, C++14 plus OpenMP 4.0 implementations of dense and sparse-direct, Hermitian and non-Hermitian DPP samplers.

High-performance sampling of generic Determinantal Point Processes

Determinantal point processes (DPPs) were introduced by Macchi (Macchi 1975 Adv. Appl. Probab. 7, 83122) as a model for repulsive (fermionic) particle distributions. But their recent popularization...

https://royalsocietypublishing.org/doi/pdf/10.1098/rsta.2019.0059

High-performance sampling of generic determinantal point processes

We describe a parallel algorithm for finding the eigenvalues and eigenvectors of a dense symmetric matrix, with an emphasis on the dense linear algebra operations. We follow the traditional three-step process: reduce to tridiagonal form, solve the tridiagonal problem, then backtransform the result. Since the different steps have different algorithmic characteristics, this problem serves as a perfect vehicle for exploring some issues associated with parallel linear algebra calculations. In particular, we examine the effects of matrix distribution and blocking on the computational performance of tridiagonalization and backtransformation. Through experiments on an Intel Paragon, we demonstrate that block storage of the matrix is not necessary for a highly efficient block algorithm. The performance of our approach compares very favorably with that of the corresponding ScaLAPACK routines.

Toward an Efficient Parallel Eigensolver for Dense Symmetric Matrices

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Toward an Efficient Parallel Eigensolver for Dense Symmetric Matrices