Florin Dobrian

Additive Schwarz-based fully coupled implicit methods for resistive Hall magnetohydrodynamic problems

A parallel, fully coupled, nonlinearly implicit Newton-Krylov-Schwarz algorithm is proposed for the numerical simulation of a magnetic reconnection problem described by a system of resistive Hall magnetohydrodynamics equations in slab geometry. A key component of the algorithm is a restricted additive Schwarz preconditioner defined for problems with doubly periodic boundary conditions. We show numerically that with such a preconditioned nonlinearly implicit method the time step size is no longer constrained by the CFL number or the convergence of the Newton solver. We report the parallel performance of the algorithm and software on machines with thousands of processors.

The design of sparse direct solvers using object-oriented techniques

We describe our experience in designing object-oriented software for sparse direct solvers. We discuss Spindle, a library of sparse matrix ordering codes, and OBLIO, a package that implements the factorization and triangular solution steps of a direct solver. We discuss the goals of our design: managing complexity, simplicity of interface, flexibility, extensibility, safety, and efficiency. High performance is obtained by carefully implementing the computationally intensive kernels and by making several tradeoffs to balance the conflicting demands of efficiency and good software design. Some of the missteps that we made in the course of this work are also described.

Distributed-Memory Parallel Algorithms for Matching and Coloring

We discuss the design and implementation of new highly-scalable distributed-memory parallel algorithms for two prototypical graph problems, edge-weighted matching and distance-1 vertex coloring. Graph algorithms in general have low concurrency, poor data locality, and high ratio of data access to computation costs, making it challenging to achieve scalability on massively parallel machines. We overcome this challenge by employing a variety of techniques, including speculation and iteration, optimized communication, and randomization. We present preliminary results on weak and strong scalability studies conducted on an IBM Blue Gene/P machine employing up to tens of thousands of processors. The results show that the algorithms hold strong potential for computing at petascale.

The Design of I/O-Efficient Sparse Direct Solvers

We consider two problems related to I/O: First, find the minimum primary memory size required to factor a sparse, symmetric matrix when permitted to read and write the data exactly once. Second, find the minimum data traffic between core and external memory when permitted to read and write the data many times. These problems are likely to be intractable in general, but we prove upper and lower bounds on these quantities for several model problems with useful sparsity (i.e., whose computational graphs have small separators). We provide fast algorithms for computing these quantities through simulation for irregular problems. The choice of factorization algorithms (left-looking, right-looking, multifrontal), orderings (nested dissection or minimum degree), and blocking techniques (1- or 2- dimensional blocks) can change the memory size and traffic by orders of magnitude. Explicitly moving the data (files managed by the program) improves performance significantly over implicit data movement (pages managed by the operating system). Thus this work guides us in designing a software library that implements an external memory sparse solver.

Object-Oriented Design for Sparse Direct Solvers

We discuss the object-oriented design of a software package for solving sparse, symmetric systems of equations (positive definite and indefinite) by direct methods. At the highest layers, we decouple data structure classes from algorithmic classes for flexibility. We describe the important structural and algorithmic classes in our design, and discuss the trade-offs we made for high performance. The kernels at the lower layers were optimized by hand. Our results show no performance loss from our object-oriented design, while providing flexibility, ease of use, and extensibility over solvers using procedural design.

Oblio: design and performance

We discuss Oblio, our library for solving sparse symmetric linear systems of equations by direct methods. The code was implemented with two goals in mind: efficiency and flexibility. These were achieved through careful design, combining good algorithmic techniques with modern software engineering. Here we describe the major design issues and we illustrate the performance of the library.

Computing maximum matching in parallel on bipartite graphs: worth the effort?

We discuss parallel algorithms for computing maximum matchings in bipartite graphs on multithreaded computers, reporting for the first time, good speedups for the maximum cardinality matching problem. Experiments with serial matching algorithms have shown that their performance is sensitive to the order in which vertices are processed. In the execution of a multithreaded parallel algorithm for matching, variability in the order in which different threads process vertices is unavoidable. This sensitivity raises the possibility that different execution orderings might adversely affect the performance of parallel matching algorithms. In this paper, we answer this question by showing that good speedups are attainable by careful design of algorithms tuned to the characteristics of multithreaded architectures and the structure of the input graphs. We discuss preliminary results from parallel implementations of two key algorithms (Hopcroft-Karp and Pothen-Fan) and their variants on three multithreaded platforms (Cray XMT, AMD Opteron and Intel Nehalem) using a carefully chosen test set from real-world applications as well as synthetical graphs.

Domain-decomposed Fully Coupled Implicit Methods for a Magnetohydrodynamics Problem

1 University of Colorado at Boulder, Department of Computer Science, 430 UCB, Boulder, CO 80309, USA, serguei.ovtchinnikov@colorado.edu 2 Old Dominion University, Department of Computer Science, 3300 ECS Building, Norfolk, VA 23529, USA, dobrian@cs.odu.edu 3 University of Colorado at Boulder, Department of Computer Science, 430 UCB, Boulder, CO 80309, USA, cai@cs.colorado.edu 4 Columbia University, Department of Applied Physics & Applied Mathematics, 500 W. 120th St., New York, NY 10027, USA, david.keyes@columbia.edu