Combinatorial Multigrid: Advanced Preconditioners For Ill-Conditioned Linear Systems

Publication Source: IEEE High Performance Extreme Computing Conference (HPEC) 2019, Waltham, MA

The Combinatorial Multigrid (CMG) technique is a practical and adaptable solver and combinatorial preconditioner for solving certain classes of large, sparse systems of linear equations. CMG is similar to Algebraic Multigrid (AMG) but replaces large groupings of fine-level variables with a single coarse-level one, resulting in simple and fast interpolation schemes. These schemes further provide control over the refinement strategies at different levels of the solver hierarchy depending on the condition number of the system being solved [1]. While many pre-existing solvers may be able to solve large, sparse systems with relatively low complexity, inversion may require O(n2) space; whereas, if we know that a linear operator has ~n = O(n) nonzero elements, we desire to use O(n) space in order to reduce communication as much as possible. Being able to invert sparse linear systems of equations, asymptotically as fast as the values can be read from memory, has been identified by the Defense Advanced Research Projects Agency (DARPA) and the Department of Energy (DOE) as increasingly necessary for scalable solvers and energy-efficient algorithms [2], [3] in scientific computing. Further, as industry and government agencies move towards exascale, fast solvers and communicationavoidance will be more necessary [4], [5]. In this paper, we present an optimized implementation of the Combinatorial Multigrid in C using Petsc and analyze the solution of various systems using the CMG approach as a preconditioner on much larger problems than have been presented thus far. We compare the number of iterations, setup times and solution times against other popular preconditioners for such systems, including Incomplete Cholesky and a Multigrid approach in Petsc against common problems, further exhibiting superior performance by the CMG. 1 2 Index Terms—combinatorial algorithms, spectral support solver, linear systems, fast solvers, preconditioners, multigrid, graph laplacian, benchmarking, iterative solvers
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Systems and Methods for Solving Unrestricted Incremental Constraint Problems

Publication Source: Patent US10402747B2

We present the architecture of a high-performance constraint solver R-Solve that extends the gains made in SAT performance over the past fifteen years on static decision problems to problems that require on-the-fly adaptation, solution space exploration and optimization. R-Solve facilitates collaborative parallel solving and provides an efficient system for unrestricted incremental solving via Smart Repair. R-Solve can address problems in dynamic planning and constrained optimization involving complex logical and arithmetic constraints.
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Polyhedral Tensor Schedulers

Publication Source: 2019 International Conference on High Performance Computing & Simulation (HPCS), Dublin, Ireland

Compiler optimizations based on the polyhedral model are able to automatically parallelize and optimize loopbased code. We acknowledge that while polyhedral techniques can represent a broad set of program transformations, important classes of programs could be parallelized just as well using less general but more tractable techniques. We apply this general idea to the polyhedral scheduling phase, which is one of the typical performance bottlenecks of polyhedral compilation. We focus on a class of programs in which enough parallelism is already exposed in the source program, and which includes Deep Learning layers and combinations thereof, as well as multilinear algebra kernels. We call these programs ”tensor codes”, and consequently call ”tensor schedulers” the tractable polyhedral scheduling techniques presented here. The general idea is that we can significantly speed up polyhedral scheduling by restricting the set of transformations considered. As an extra benefit, having a small search space allows us to introduce non-linear cost models, which fills a gap in polyhedral cost models.

Systems and Methods for Multiresolution Parsing

Publication Source: Patent US10313361B2

A multiresolution parser (MRP) can selectively extract one or more information units from a dataset based on the available processing capacity and/or the arrival rate of the dataset. Should any of these parameters change, the MRP can adaptively change the information units to be extracted such that the benefit or value of the extracted information is maximized while minimizing the cost of extraction. This tradeoff is facilitated, at least in part, by an analysis of the spectral energy of the datasets expected to be processed by the MRP. The MRP can also determine its state after a processing iteration and use that state information in subsequent iterations to minimize the required computations in such subsequent iterations, so as to improve processing efficiency.
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On the Bottleneck Structure of Positive Linear Programming

Publication Source: 2019 SIAM Workshop on Network Science

Positive linear programming (PLP), also known as packing and covering linear programs, is an important class of problems frequently found in fields such as network science, operations research, or economics. In this work we demonstrate that all PLP problems can be represented using a network structure, revealing new key insights that lead to new polynomial-time algorithms.  
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