Solving large scale linear programming

The interior point method (IPM) is now well established as a competitive technique for solving very large scale linear programming problems. The leading variant of the interior point method is the primal dual - predictor corrector algorithm due to Mehrotra. The main computational steps of this algorithm are the repeated calculation and solution of a large sparse positive definite system of equations. We describe an implementation of the predictor corrector IPM algorithm on MasPar, a massively parallel SIMD computer. At the heart of the implemen-tation is a parallel Cholesky factorization algorithm for sparse matrices. Our implementation uses a new scheme of mapping the matrix onto the processor grid of the MasPar, that results in a more efficient Cholesky factorization than previously suggested schemes. The IPM implementation uses the parallel unit of MasPar to speed up the factorization and other computationally intensive parts of the IPM. An impor-tant part of this implementation is the judicious division of data and computation between the front-end computer, that runs the main IPM algorithm, and the par-allel unit. Performanc

An efficient null space inexact Newton method for hydraulic simulation of water distribution networks

Null space Newton algorithms are efficient in solving the nonlinear equations arising in hydraulic analysis of water distribution networks. In this article, we propose and evaluate an inexact Newton method that relies on partial updates of the network pipes' frictional headloss computations to solve the linear systems more efficiently and with numerical reliability. The update set parameters are studied to propose appropriate values. Different null space basis generation schemes are analysed to choose methods for sparse and well-conditioned null space bases resulting in a smaller update set. The Newton steps are computed in the null space by solving sparse, symmetric positive definite systems with sparse Cholesky factorizations. By using the constant structure of the null space system matrices, a single symbolic factorization in the Cholesky decomposition is used multiple times, reducing the computational cost of linear solves. The algorithms and analyses are validated using medium to large-scale water network models.Comment: 15 pages, 9 figures, Preprint extension of Abraham and Stoianov, 2015 (https://dx.doi.org/10.1061/(ASCE)HY.1943-7900.0001089), September 2015. Includes extended exposition, additional case studies and new simulations and analysi

Accelerating SPICE Model-Evaluation using FPGAs

Single-FPGA spatial implementations can provide an order of magnitude speedup over sequential microprocessor implementations for data-parallel, floating-point computation in SPICE model-evaluation. Model-evaluation is a key component of the SPICE circuit simulator and it is characterized by large irregular floating-point compute graphs. We show how to exploit the parallelism available in these graphs on single-FPGA designs with a low-overhead VLIW-scheduled architecture. Our architecture uses spatial floating-point operators coupled to local high-bandwidth memories and interconnected by a time-shared network. We retime operation inputs in the model-evaluation to allow independent scheduling of computation and communication. With this approach, we demonstrate speedups of 2–18× over a dual-core 3GHz Intel Xeon 5160 when using a Xilinx Virtex 5 LX330T for a variety of SPICE device models

Percolation-based compiling for evaluation of parallelism and hardware design trade-offs

This thesis investigates parallelism and hardware design trade-offs of parallel and pipelined architectures. To explore these trade-offs we developed a retargetable compiler based on a set of powerful code transformations called Percolation Scheduling (PS) that map programs with real-time constraints and/or massive time requirements onto synchronous, parallel, high-performance or semi-custom architectures.High-performance is achieved through extraction of application inherent fine-grain parallelism and the use of a suitable architecture. Exploiting fine-grain parallelism is a critical part of exploiting all of the parallelism available in a given program, particularly since highly irregular forms of parallelism are often not visible at coarser levels and since the use of low-level parallelism has a multiplicative effect on the overall performance.To extract substantial parallelism from both the hardware and the compiler, we use a clean, highly parallel VLIW-like architecture that is synchronous, has multiple functional units and has a single program counter. The use of a hazard-free and homogeneous architecture does not result only in a better VLSI design but also considerably increases the compiler's ability to produce better code. To further enhance parallelism we modified the uni-cycle VLIW model and extended the transformations such that pipelined units that provide extra parallelism are used.Another approach presented is of resource constrained scheduling (RCS). Since the RCS problem is known to be NP-hard, in practice it may be solved only by a heuristic approach. We argue that using the heuristic after extraction of the unlimited-resources schedule may yield better results than if the heuristic has been applied at the beginning of the scheduling process.Through a series of benchmarks we evaluate hardware design trade-offs and show that speed-ups on average of one order of magnitude are feasible with sufficient functional units. However, when resources are limited we show that the number of functional units needed may be optimized for a particular suite of application programs

A Lossy, Synchronization-Free, Race-Full, But Still Acceptably Accurate Parallel Space-Subdivision Tree Construction Algorithm

We present a new synchronization-free space-subdivision tree construction algorithm. Despite data races, this algorithm produces trees that are consistent enough for the client Barnes-Hut center of mass and force computation phases to use successfully. Our performance results show that eliminating synchronization improves the performance of the parallel algorithm by approximately 20%. End-to-end accuracy results show that the resulting partial data structure corruption has a neglible effect on the overall accuracy of the Barnes-Hut N-body simulation. We note that many data structure manipulation algorithms use many of the same basic operations (linked data structure updates and array insertions) as our tree construction algorithm. We therefore anticipate that the basic principles the we develop in this paper may effectively guide future efforts in this area

A New Method for Efficient Parallel Solution of Large Linear Systems on a SIMD Processor.

This dissertation proposes a new technique for efficient parallel solution of very large linear systems of equations on a SIMD processor. The model problem used to investigate both the efficiency and applicability of the technique was of a regular structure with semi-bandwidth $\beta,$ and resulted from approximation of a second order, two-dimensional elliptic equation on a regular domain under the Dirichlet and periodic boundary conditions. With only slight modifications, chiefly to properly account for the mathematical effects of varying bandwidths, the technique can be extended to encompass solution of any regular, banded systems. The computational model used was the MasPar MP-X (model 1208B), a massively parallel processor hostnamed hurricane and housed in the Concurrent Computing Laboratory of the Physics/Astronomy department, Louisiana State University. The maximum bandwidth which caused the problem\u27s size to fit the nyproc $\times$ nxproc machine array exactly, was determined. This as well as smaller sizes were used in four experiments to evaluate the efficiency of the new technique. Four benchmark algorithms, two direct--Gauss elimination (GE), Orthogonal factorization--and two iterative--symmetric over-relaxation (SOR) ($\omega$ = 2), the conjugate gradient method (CG)--were used to test the efficiency of the new approach based upon three evaluation metrics--deviations of results of computations, measured as average absolute errors, from the exact solution, the cpu times, and the mega flop rates of executions. All the benchmarks, except the GE, were implemented in parallel. In all evaluation categories, the new approach outperformed the benchmarks and very much so when N $\gg$ p, p being the number of processors and N the problem size. At the maximum system\u27s size, the new method was about 2.19 more accurate, and about 1.7 times faster than the benchmarks. But when the system size was a lot smaller than the machine\u27s size, the new approach\u27s performance deteriorated precipitously, and, in fact, in this circumstance, its performance was worse than that of GE, the serial code. Hence, this technique is recommended for solution of linear systems with regular structures on array processors when the problem\u27s size is large in relation to the processor\u27s size