Parallel accelerated cyclic reduction preconditioner for three-dimensional elliptic PDEs with variable coefficients

We present a robust and scalable preconditioner for the solution of large-scale linear systems that arise from the discretization of elliptic PDEs amenable to rank compression. The preconditioner is based on hierarchical low-rank approximations and the cyclic reduction method. The setup and application phases of the preconditioner achieve log-linear complexity in memory footprint and number of operations, and numerical experiments exhibit good weak and strong scalability at large processor counts in a distributed memory environment. Numerical experiments with linear systems that feature symmetry and nonsymmetry, definiteness and indefiniteness, constant and variable coefficients demonstrate the preconditioner applicability and robustness. Furthermore, it is possible to control the number of iterations via the accuracy threshold of the hierarchical matrix approximations and their arithmetic operations, and the tuning of the admissibility condition parameter. Together, these parameters allow for optimization of the memory requirements and performance of the preconditioner.Comment: 24 pages, Elsevier Journal of Computational and Applied Mathematics, Dec 201

HURP/HURBA: Zero-configuration hierarchical Up/Down routing and bridging architecture for Ethernet backbones and campus networks

Ethernet switched networks do not scale appropriately due to limitations inherent to the spanning tree protocol. Ethernet architectures based on routing over a virtual topology in which turns are prohibited offer improved performance over spanning tree, although in some cases suffer from excessive computational complexity. Up/Down routing is a turn prohibition algorithm with low computational complexity. In this paper we propose HURBA, a new layer-two architecture that improves Up/Down routing performance due to an optimization based on the use of hierarchical addressing, while preserving the computational complexity of Up/Down. The resulting architecture requires zero-configuration, uses the same frame format as Ethernet, allows upgrades by software update, and is compatible with 802.1D bridges by means of encapsulation. HURP protocol builds automatically a core with the interconnected HURP routing bridges and the standard bridges get connected to the edges in standard spanning trees. Simulations show that the performance of HURP, evaluated over various combinations of network topology and size, is close to the one of shortest path, is consistently better than that of Up/Down, and is equal or better than Turn Prohibition, with the advantage of having a lower complexity.En prens

On a New Type of Information Processing for Efficient Management of Complex Systems

It is a challenge to manage complex systems efficiently without confronting NP-hard problems. To address the situation we suggest to use self-organization processes of prime integer relations for information processing. Self-organization processes of prime integer relations define correlation structures of a complex system and can be equivalently represented by transformations of two-dimensional geometrical patterns determining the dynamics of the system and revealing its structural complexity. Computational experiments raise the possibility of an optimality condition of complex systems presenting the structural complexity of a system as a key to its optimization. From this perspective the optimization of a system could be all about the control of the structural complexity of the system to make it consistent with the structural complexity of the problem. The experiments also indicate that the performance of a complex system may behave as a concave function of the structural complexity. Therefore, once the structural complexity could be controlled as a single entity, the optimization of a complex system would be potentially reduced to a one-dimensional concave optimization irrespective of the number of variables involved its description. This might open a way to a new type of information processing for efficient management of complex systems.Comment: 5 pages, 2 figures, to be presented at the International Conference on Complex Systems, Boston, October 28 - November 2, 200

Flexible Multi-layer Sparse Approximations of Matrices and Applications

The computational cost of many signal processing and machine learning techniques is often dominated by the cost of applying certain linear operators to high-dimensional vectors. This paper introduces an algorithm aimed at reducing the complexity of applying linear operators in high dimension by approximately factorizing the corresponding matrix into few sparse factors. The approach relies on recent advances in non-convex optimization. It is first explained and analyzed in details and then demonstrated experimentally on various problems including dictionary learning for image denoising, and the approximation of large matrices arising in inverse problems