Efficient Out-of-Core Algorithms for Linear Relaxation Using Blocking Covers

AbstractWhen a numerical computation fails to fit in the primary memory of a serial or parallel computer, a so-called “out-of-core” algorithm, which moves data between primary and secondary memories, must be used. In this paper, we study out-of-core algorithms for sparse linear relaxation problems in which each iteration of the algorithm updates the state of every vertex in a graph with a linear combination of the states of its neighbors. We give a general method that can save substantially on the I/O traffic for many problems. For example, our technique allows a computer withMwords of primary memory to performT=Ω(M1/5) cycles of a multigrid algorithm for a two-dimensional elliptic solver over an n-point domain using onlyΘ(nT/M1/5) I/O transfers, as compared with the naive algorithm which requiresΩ(nT) I/O's. Our method depends on the existence of a “blocking” cover of the graph that underlies the linear relaxation. A blocking cover has the property that the subgraphs forming the cover have large diameters once a small number of vertices have been removed. The key idea in our method is to introduce a variable for each removed vertex for each time step of the algorithm. We maintain linear dependences among the removed vertices, thereby allowing each subgraph to be iteratively relaxed without external communication. We give a general theorem relating blocking covers to I/O-efficient relaxation schemes. We also give an automatic method for finding blocking covers for certain classes of graphs, including planar graphs andd-dimensional simplicial graphs with constant aspect ratio (i.e., graphs that arise from dividingd-space into “well-shaped” polyhedra). As a result, we can performTiterations of linear relaxation on anyn-vertex planar graph using onlyΘ(n+nTlgn/M1/4) I/O's or on anyn-noded-dimensional simplicial graph with constant aspect ratio using onlyΘ(n+nTlgn/MΩ(1/d)) I/O's

Quantitative performance modeling of scientific computations and creating locality in numerical algorithms

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 1995.Includes bibliographical references (p. 141-150) and index.by Sivan Avraham Toledo.Ph.D

A Maximum Satisfiability Based Approach to Bi-Objective Boolean Optimization

Many real-world problem settings give rise to NP-hard combinatorial optimization problems. This results in a need for non-trivial algorithmic approaches for finding optimal solutions to such problems. Many such approaches—ranging from probabilistic and meta-heuristic algorithms to declarative programming—have been presented for optimization problems with a single objective. Less work has been done on approaches for optimization problems with multiple objectives. We present BiOptSat, an exact declarative approach for finding so-called Pareto-optimal solutions to bi-objective optimization problems. A bi-objective optimization problem arises for example when learning interpretable classifiers and the size, as well as the classification error of the classifier should be taken into account as objectives. Using propositional logic as a declarative programming language, we seek to extend the progress and success in maximum satisfiability (MaxSAT) solving to two objectives. BiOptSat can be viewed as an instantiation of the lexicographic method and makes use of a single SAT solver that is preserved throughout the entire search procedure. It allows for solving three tasks for bi-objective optimization: finding a single Pareto-optimal solution, finding one representative solution for each Pareto point, and enumerating all Pareto-optimal solutions. We provide an open-source implementation of five variants of BiOptSat, building on different algorithms proposed for MaxSAT. Additionally, we empirically evaluate these five variants, comparing their runtime performance to that of three key competing algorithmic approaches. The empirical comparison in the contexts of learning interpretable decision rules and bi-objective set covering shows practical benefits of our approach. Furthermore, for the best-performing variant of BiOptSat, we study the effects of proposed refinements to determine their effectiveness

A non-hybrid method for the PDF equations of turbulent flows on unstructured grids

In probability density function (PDF) methods of turbulent flows, the joint PDF of several flow variables is computed by numerically integrating a system of stochastic differential equations for Lagrangian particles. A set of parallel algorithms is proposed to provide an efficient solution of the PDF transport equation, modeling the joint PDF of turbulent velocity, frequency and concentration of a passive scalar in geometrically complex configurations. An unstructured Eulerian grid is employed to extract Eulerian statistics, to solve for quantities represented at fixed locations of the domain (e.g. the mean pressure) and to track particles. All three aspects regarding the grid make use of the finite element method (FEM) employing the simplest linear FEM shape functions. To model the small-scale mixing of the transported scalar, the interaction by exchange with the conditional mean model is adopted. An adaptive algorithm that computes the velocity-conditioned scalar mean is proposed that homogenizes the statistical error over the sample space with no assumption on the shape of the underlying velocity PDF. Compared to other hybrid particle-in-cell approaches for the PDF equations, the current methodology is consistent without the need for consistency conditions. The algorithm is tested by computing the dispersion of passive scalars released from concentrated sources in two different turbulent flows: the fully developed turbulent channel flow and a street canyon (or cavity) flow. Algorithmic details on estimating conditional and unconditional statistics, particle tracking and particle-number control are presented in detail. Relevant aspects of performance and parallelism on cache-based shared memory machines are discussed.Comment: Accepted in Journal of Computational Physics, Feb. 20, 200