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DOE OSTI2017

Parallel replica dynamics method for bistable stochastic reaction networks: Simulation and sensitivity analysis

Stochastic reaction networks that exhibit bistable behavior are common in systems biology, materials science, and catalysis. Sampling of stationary distributions is crucial for understanding and characterizing the long-time dynamics of bistable stochastic dynamical systems. However, simulations are often hindered by the insufficient sampling of rare transitions between the two metastable regions. In this paper, we apply the parallel replica method for a continuous time Markov chain in order to improve sampling of the stationary distribution in bistable stochastic reaction networks. The proposed method uses parallel computing to accelerate the sampling of rare transitions. Furthermore, it can be combined with the path-space information bounds for parametric sensitivity analysis. With the proposed methodology, we study three bistable biological networks: the Schlögl model, the genetic switch network, and the enzymatic futile cycle network. We demonstrate the algorithmic speedup achieved in these numerical benchmarks. More significant acceleration is expected when multi-core or graphics processing unit computer architectures and programming tools such as CUDA are employed

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2014

TH-E-18A-01: Developments in Monte Carlo Methods for Medical Imaging

Monte Carlo simulation methods are widely used in medical physics research and are starting to be implemented in clinical applications such as radiation therapy planning systems. Monte Carlo simulations offer the capability to accurately estimate quantities of interest that are challenging to measure experimentally while taking into account the realistic anatomy of an individual patient. Traditionally, practical application of Monte Carlo simulation codes in diagnostic imaging was limited by the need for large computational resources or long execution times. However, recent advancements in high-performance computing hardware, combined with a new generation of Monte Carlo simulation algorithms and novel postprocessing methods, are allowing for the computation of relevant imaging parameters of interest such as patient organ doses and scatter-to-primaryratios in radiographic projections in just a few seconds using affordable computational resources. Programmable Graphics Processing Units (GPUs), for example, provide a convenient, affordable platform for parallelized Monte Carlo executions that yield simulation times on the order of 10{sup 7} xray/ s. Even with GPU acceleration, however, Monte Carlo simulation times can be prohibitive for routine clinical practice. To reduce simulation times further, variance reduction techniques can be used to alter the probabilistic models underlying the x-ray tracking process, resulting in lower variance in the results without biasing the estimates. Other complementary strategies for further reductions in computation time are denoising of the Monte Carlo estimates and estimating (scoring) the quantity of interest at a sparse set of sampling locations (e.g. at a small number of detector pixels in a scatter simulation) followed by interpolation. Beyond reduction of the computational resources required for performing Monte Carlo simulations in medical imaging, the use of accurate representations of patient anatomy is crucial to the virtual generation of medical images and accurate estimation of radiation dose and other imaging parameters. For this, detailed computational phantoms of the patient anatomy must be utilized and implemented within the radiation transport code. Computational phantoms presently come in one of three format types, and in one of four morphometric categories. Format types include stylized (mathematical equation-based), voxel (segmented CT/MR images), and hybrid (NURBS and polygon mesh surfaces). Morphometric categories include reference (small library of phantoms by age at 50th height/weight percentile), patient-dependent (larger library of phantoms at various combinations of height/weight percentiles), patient-sculpted (phantoms altered to match the patient's unique outer body contour), and finally, patient-specific (an exact representation of the patient with respect to both body contour and internal anatomy). The existence and availability of these phantoms represents a very important advance for the simulation of realistic medical imaging applications using Monte Carlo methods. New Monte Carlo simulation codes need to be thoroughly validated before they can be used to perform novel research. Ideally, the validation process would involve comparison of results with those of an experimental measurement, but accurate replication of experimental conditions can be very challenging. It is very common to validate new Monte Carlo simulations by replicating previously published simulation results of similar experiments. This process, however, is commonly problematic due to the lack of sufficient information in the published reports of previous work so as to be able to replicate the simulation in detail. To aid in this process, the AAPM Task Group 195 prepared a report in which six different imaging research experiments commonly performed using Monte Carlo simulations are described and their results provided. The simulation conditions of all six cases are provided in full detail, with all necessary data on material composition, source, geometry, scoring and other parameters provided. The results of these simulations when performed with the four most common publicly available Monte Carlo packages are also provided in tabular form. The Task Group 195 Report will be useful for researchers needing to validate their Monte Carlo work, and for trainees needing to learn Monte Carlo simulation methods. In this symposium we will review the recent advancements in highperformance computing hardware enabling the reduction in computational resources needed for Monte Carlo simulations in medical imaging. We will review variance reduction techniques commonly applied in Monte Carlo simulations of medical imaging systems and present implementation strategies for efficient combination of these techniques with GPU acceleration. Trade-offs involved in Monte Carlo acceleration by means of denoising and “sparse sampling” will be discussed. A method for rapid scatter correction in cone-beam CT (<5 min/scan) will be presented as an illustration of the simulation speeds achievable with optimized Monte Carlo simulations. We will also discuss the development, availability, and capability of the various combinations of computational phantoms for Monte Carlo simulation of medical imaging systems. Finally, we will review some examples of experimental validation of Monte Carlo simulations and will present the AAPM Task Group 195 Report. Learning Objectives: Describe the advances in hardware available for performing Monte Carlo simulations in high performance computing environments. Explain variance reduction, denoising and sparse sampling techniques available for reduction of computational time needed for Monte Carlo simulations of medical imaging. List and compare the computational anthropomorphic phantoms currently available for more accurate assessment of medical imaging parameters in Monte Carlo simulations. Describe experimental methods used for validation of Monte Carlo simulations in medical imaging. Describe the AAPM Task Group 195 Report and its use for validation and teaching of Monte Carlo simulations in medical imaging.

60 APPLIED LIFE SCIENCES↗
DOE OSTI2017

Partial differential equations preconditioner resilient to soft and hard faults

We present a domain-decomposition-based preconditioner for the solution of partial differential equations (PDEs) that is resilient to both soft and hard faults. The algorithm reformulates the PDE as a sampling problem, followed by a solution update through data manipulation that is resilient to both soft and hard faults. This reformulation allows us to recast the problem as a set of independent tasks, and exploit data locality to reduce global communication. We discuss two different parallel implementations: (a) a single program multiple data (SPMD) version based on a one-to-one mapping between subdomain and MPI processes responsible for both state and computation; and (b) an asynchronous server–client implementation where all state information is held by the servers and clients are designed solely as computational units. We present a scalability comparison of both implementations under nominal conditions, showing efficiency within ~80% for up to 12,000 cores. We present a resilience analysis under different fault scenarios based on the server–client implementation. This framework provides resiliency to hard faults such that if a client crashes, it stops asking for work, and the servers simply distribute the work among all of the other clients alive. Erroneous subdomain solves (e.g. due to soft faults) appear as corrupted data, which is either rejected if that causes a task to fail, or is seamlessly filtered out during the regression stage through a suitable noise model. Three different types of faults are modeled: hard faults modeling nodes (or clients) crashing; soft faults occurring during the communication of the tasks between server and clients; and soft faults occurring during task execution. We demonstrate the resiliency of the approach for a 2D elliptic PDE, and explore the effect of the faults at various failure rates.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2014

Report on FY11 Extensions to MeshKit and RGG

One of the barriers to performing high-fidelity computational simulation of reactor core phenomena is the production of good-quality geometry and mesh models required by these simulations. Although a variety of geometry and meshing tools are available, they suffer from shortcomings in usability, robustness, or generality which makes them difficult to apply to reactor applications. The SHARP frameworks project is addressing these deficiencies by developing a library of mesh generation algorithms, and tools based on that library. The library is known as MeshKit, and one of the tools being developed is named RGG, for Reactor Geometry (and mesh) Generator. Over the past year, our meshing-related work was split between general design improvements in MeshKit, enhancing RGG to run in parallel, and the application of RGG to several specific reactor designs. These activities are described in this report.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2017

Final Report: Subcontract B623868 Algebraic Multigrid solvers for coupled PDE systems

The Pennsylvania State University (“Subcontractor”) continued to work on the design of algebraic multigrid solvers for coupled systems of partial differential equations (PDEs) arising in numerical modeling of various applications, with a main focus on solving the Dirac equation arising in Quantum Chromodynamics (QCD). The goal of the proposed work was to develop combined geometric and algebraic multilevel solvers that are robust and lend themselves to efficient implementation on massively parallel heterogeneous computers for these QCD systems. The research in these areas built on previous works, focusing on the following three topics: (1) the development of parallel full-multigrid (PFMG) and non-Galerkin coarsening techniques in this frame work for solving the Wilson Dirac system; (2) the use of these same Wilson MG solvers for preconditioning the Overlap and Domain Wall formulations of the Dirac equation; and (3) the design and analysis of algebraic coarsening algorithms for coupled PDE systems including Stokes equation, Maxwell equation and linear elasticity.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2016

Covering Resilience: A Recent Development for Binomial Checkpointing

In terms of computing time, adjoint methods offer a very attractive alternative to compute gradient information, required, e.g., for optimization purposes. However, together with this very favorable temporal complexity result comes a memory requirement that is in essence proportional with the operation count of the underlying function, e.g., if algorithmic differentiation is used to provide the adjoints. For this reason, checkpointing approaches in many variants have become popular. This paper analyzes an extension of the so-called binomial approach to cover also possible failures of the computing systems. Such a measure of precaution is of special interest for massive parallel simulations and adjoint calculations where the mean time between failure of the large scale computing system is smaller than the time needed to complete the calculation of the adjoint information. We describe the extensions of standard checkpointing approaches required for such resilience, provide a corresponding implementation and discuss first numerical results.

Walther, Andrea↗
DOE OSTI2022

m-CUBES An efficient and portable implementation of multi-dimensional integration for gpus

The task of multi-dimensional numerical integration is frequently encountered in physics and other scientific fields, e.g., in modeling the effects of systematic uncertainties in physical systems and in Bayesian parameter estimation. Multi-dimensional integration is often time-prohibitive on CPUs. Efficient implementation on many-core architectures is challenging as the workload across the integration space cannot be predicted a priori. We propose m-Cubes, a novel implementation of the well-known Vegas algorithm for execution on GPUs. Vegas transforms integration variables followed by calculation of a Monte Carlo integral estimate using adaptive partitioning of the resulting space. m-Cubes improves performance on GPUs by maintaining relatively uniform workload across the processors. As a result, our optimized Cuda implementation for Nvidia GPUs outperforms parallelization approaches proposed in past literature. We further demonstrate the efficiency of m-Cubes by evaluating a six-dimensional integral from a cosmology application, achieving significant speedup and greater precision than the CUBA library's CPU implementation of VEGAS. We also evaluate m-Cubes on a standard integrand test suite. m-Cubes outperforms the serial implementations of the Cuba and GSL libraries by orders of magnitude speedup while maintaining comparable accuracy. Our approach yields a speedup of at least 10 when compared against publicly available Monte Carlo based GPU implementations. In summary, m-Cubes can solve integrals that are prohibitively expensive using standard libraries and custom implementations. A modern C++ interface header-only implementation makes m-Cubes portable, allowing its utilization in complicated pipelines with easy to define stateful integrals. Compatibility with non-Nvidia GPUs is achieved with our initial implementation of m-Cubes using the Kokkos framework.

Sakiotis, Ioannis↗
DOE OSTI2023

PETSc/TAO Users Manual (Rev. 3.19)

This manual describes the use of the Portable, Extensible Toolkit for Scientific Computation (PETSc) and the Toolkit for Advanced Optimization (TAO) for the numerical solution of partial differential equations and related problems on high-performance computers. PETSc/TAO is a suite of data structures and routines that provide the building blocks for the implementation of large-scale application codes on parallel (and serial) computers. PETSc uses the MPI standard for all distributed memory communication. PETSc/TAO includes a large suite of parallel linear solvers, nonlinear solvers, time integrators, and opti mization that may be used in application codes written in Fortran, C, C++, and Python (via petsc4py; see Getting Started). PETSc provides many of the mechanisms needed within parallel application codes, such as parallel matrix and vector assembly routines. The library is organized hierarchically, enabling users to employ the level of abstraction that is most appropriate for a particular problem. By using techniques of object-oriented programming, PETSc provides enormous flexibility for users. PETSc is a sophisticated set of software tools; as such, for some users it initially has a much steeper learning curve than packages such as MATLAB or a simple subroutine library. In particular, for individuals without some computer science background, experience programming in C, C++, python, or Fortran and experience using a debugger such as gdb or lldb, it may require a significant amount of time to take full advantage of the features that enable efficient software use. However, the power of the PETSc design and the algorithms it incorporates may make the efficient implementation of many application codes simpler than “rolling them” yourself. For many tasks a package such as MATLAB is often the best tool; PETSc is not intended for the classes of problems for which effective MATLAB code can be written. There are several packages, built on PETSc, that may satisfy your needs without requiring directly using PETSc. We recommend reviewing these packages functionality before starting to code directly with PETSc. PETSc can be used to provide a “MPI parallel linear solver” in an otherwise sequential, or OpenMP parallel code. This approach cannot provide extremely large improvements in the application time by utilizing large numbers of MPI processes but can still improve the performance. Certainly all parts of a previously sequential code need not be parallelized but the matrix generation portion must be parallelized to expect true scalability to large numbers of MPI processes. See PCMPI for details on how to utilize the PETSc MPI linear solver server. Since PETSc is under continued development, small changes in usage and calling sequences of routines will occur. PETSc has been supported for twenty-five years; see mailing list information on our website for information on contacting support.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2015

Performance analysis of distributed symmetric sparse matrix vector multiplication algorithm for multi-core architectures

In this article, sparse matrix vector multiply (SpMVM) is an important kernel that frequently arises in high performance computing applications. Due to its low arithmetic intensity, several approaches have been proposed in literature to improve its scalability and efficiency in large scale computations. In this paper, our target systems are high end multi-core architectures and we use messaging passing interface + open multiprocessing hybrid programming model for parallelism. We analyze the performance of recently proposed implementation of the distributed symmetric SpMVM, originally developed for large sparse symmetric matrices arising in ab initio nuclear structure calculations. We also study important features of this implementation and compare with previously reported implementations that do not exploit underlying symmetry. Our SpMVM implementations leverage the hybrid paradigm to efficiently overlap expensive communications with computations. Our main comparison criterion is the "CPU core hours" metric, which is the main measure of resource usage on supercomputers. We analyze the effects of topology-aware mapping heuristic using simplified network load model. Furthermore, we have tested the different SpMVM implementations on two large clusters with 3D Torus and Dragonfly topology. Our results show that the distributed SpMVM implementation that exploits matrix symmetry and hides communication yields the best value for the "CPU core hours" metric and significantly reduces data movement overheads.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2017

Analysis of Monte Carlo accelerated iterative methods for sparse linear systems: Analysis of Monte Carlo accelerated iterative methods for sparse linear systems

Here, we consider hybrid deterministic-stochastic iterative algorithms for the solution of large, sparse linear systems. Starting from a convergent splitting of the coefficient matrix, we analyze various types of Monte Carlo acceleration schemes applied to the original preconditioned Richardson (stationary) iteration. We expect that these methods will have considerable potential for resiliency to faults when implemented on massively parallel machines. We also establish sufficient conditions for the convergence of the hybrid schemes, and we investigate different types of preconditioners including sparse approximate inverses. Numerical experiments on linear systems arising from the discretization of partial differential equations are presented.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2021

Sparse Binary Matrix-Vector Multiplication on Neuromorphic Computers

Neuromorphic computers offer the opportunity for low-power, efficient computation. Though they have been primarily applied to neural network tasks, there is also the opportunity to leverage the inherent characteristics of neuromorphic computers (low power, massive parallelism, collocated processing and memory) to perform non-neural network tasks. Here, we demonstrate how an approach for performing sparse binary matrix-vector multiplication on neuromorphic computers. We describe the approach, which relies on the connection between binary matrix-vector multiplication and breadth first search, and we introduce the algorithm for performing this calculation in a neuromorphic way. We validate the approach in simulation. Finally, we provide a discussion of the runtime of this algorithm and discuss where neuromorphic computers in the future may have a computational advantage when performing this computation.

Schuman, Catherine↗
DOE OSTI2023

A Surrogate-Based Asynchronous Decomposition Technique for Realistic Security-Constrained Optimal Power Flow Problems

Here we present a decomposition approach for obtaining good feasible solutions for the security-constrained, alternating-current, optimal power flow (SC-AC-OPF) problem at an industrial scale and under real-world time and computational limits. The approach was designed while preparing and participating in ARPA-E’s Grid Optimization Competition (GOC) Challenge 1. The challenge focused on a near-real-time version of the SC-AC-OPF problem, where a base operating point is optimized, taking into account possible single-element contingencies, after which the system adapts its operating point following the response of automatic frequency droop controllers and voltage regulators. Our solution approach for this problem relies on state-of-the-art nonlinear programming algorithms, and it employs nonconvex relaxations for complementarity constraints, a specialized two-stage decomposition technique with sparse approximations of recourse terms and contingency ranking and prescreening. The paper describes and justifies our approach and outlines the features of its implementation, including functions and derivatives evaluation, warm-starting strategies, and asynchronous parallelism. We discuss the results of the independent benchmark of our approach by ARPA-E’s GOC team in Challenge 1, where it was found to consistently produce high-quality solutions across a wide range of network sizes and difficulty, and conclude by outlining future extensions of the approach.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2021

TEAM Project Review, Year 2

This report summarizes our research activities within the TEAM project between December 2020 and December 2021, funded by the ASCR Advanced Research in Quantum Computing program. During the reporting period the LLNL-MSU team has made progress on several fronts. An overarching goal of the team is to provide a comprehensive suite of software tools that can be used for the Characterize-Optimize-Compute loop needed to implement and execute algorithms on quantum devices. We are concurrently developing lightweight solvers that can be used on desktop computers to find optimal control pulses and to characterize small quantum systems (consisting of a few transmons and cavities). However, desktop computers are insufficient for simulating and characterizing larger quantum systems. We have therefore also developed parallel, distributed memory, simulators and optimization solvers, both for open and closed quantum systems. These parallel solvers have, for example, been used to study quantum optimal control for pure-state preparation, utilizing 1000’s of cores on a modern high-performance computing (HPC) platform.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2023

Targeting tissues via dynamic human systems modeling in generative design

Drug discovery is a complex, costly process with high failure rates. A successful drug should bind to a target, be deliverable to an intended site of activity, and promote a desired pharmacological effect without causing toxicity. Typically, these factors are evaluated in series over the course of a pipeline where the number of candidates is sequentially whittled down from a very large initial pool. One promise of AI-driven discovery is the opportunity to evaluate multiple facets of drug performance in parallel. However, despite ML-driven advancements, current models for pharmacological property prediction are exclusively trained to predict molecular properties, ignoring important, dynamic biodistribution and bioactivity effects. Here, we present our progress towards incorporating quantitative systems physiology models into an AI-enabled drug design and molecular generation pipeline. Within a genetic algorithm, we include human-relevant physiologically based pharmacokinetic (PBPK) models. These PBPK models leverage properties that are predicted by a fine-tuned molecular language model. Together, these models will aid in capturing the mapping between molecules and therapeutic outcomes that is necessary to accelerate the drug discovery process.

Fox, Zach↗
DOE OSTI2026

Two-Tower Quantum Matrix Chain Multiplication: Trading Qubits for Depth

Matrix chain multiplication -- computing $\mathcal{W} = M^{(0)}\cdots M^{(K-1)}$ where $M^{(k)} \in \mathbb{R}^{P_k \times P_{k+1}}$-- arises in scientific computing, machine learning, and graph analysis. Despite the importance of this problem, for chains of distinct matrices, the classical number of operations grows linearly with the chain length $K$ and polynomially in the matrix dimensions. We present \emph{Two-Tower Matrix Multiplication}, a quantum subroutine that encodes the product $\mathcal{W}$ of the $K$ matrices into a quantum state in circuit depth $\mathcal{O}(\max_{k} \mathrm{polylog} (P_k P_{k+1}))$, which is independent of~$K$ within the QRAM-based state-preparation model, whereas the qubit count is $\mathcal{O}\bigl(\sum_{k} \log P_k \bigr)$; the total gate count remains linear in $K$, so the gain is in the circuit depth. The construction interleaves state-preparation operators across two layers; within each layer, all operators act on disjoint registers and execute in parallel. This subroutine can be specialized for the chain-vector case, which computes the product of $K-1$ matrices applied to a vector. We prove the correctness of the subroutine for all $K$ and provide two implementations using the Qiskit and QCLAB frameworks. The subroutine is applicable to any downstream quantum algorithm that operates on a matrix encoded in the statevector, including norm estimation, graph-matrix powers, linear system solving, and quantum machine learning kernels.

Antonioli, Giacomo [Pisa U.] (ORCID:00090000668703↗
DOE OSTI2017

Multigroup Monte Carlo on GPUs: Comparison of history- and event-based algorithms

This article presents an investigation of the performance of different multigroup Monte Carlo transport algorithms on GPUs with a discussion of both history-based and event-based approaches. Several algorithmic improvements are introduced for both approaches. By modifying the history-based algorithm that is traditionally favored in CPU-based MC codes to occasionally filter out dead particles to reduce thread divergence, performance exceeds that of either the pure history-based or event-based approaches. The impacts of several algorithmic choices are discussed, including performance studies on Kepler and Pascal generation NVIDIA GPUs for fixed source and eigenvalue calculations. Single-device performance equivalent to 20–40 CPU cores on the K40 GPU and 60–80 CPU cores on the P100 GPU is achieved. Last, in addition, nearly perfect multi-device parallel weak scaling is demonstrated on more than 16,000 nodes of the Titan supercomputer.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2016

Institute for Sustained Performance, Energy, and Resilience (SuPER)

The University of Tennessee (UTK) and University of Texas at El Paso (UTEP) partnership supported the three main thrusts of the SUPER project---performance, energy, and resilience. The UTK-UTEP effort thus helped advance the main goal of SUPER, which was to ensure that DOE's computational scientists can successfully exploit the emerging generation of high performance computing (HPC) systems. This goal is being met by providing application scientists with strategies and tools to productively maximize performance, conserve energy, and attain resilience. The primary vehicle through which UTK provided performance measurement support to SUPER and the larger HPC community is the Performance Application Programming Interface (PAPI). PAPI is an ongoing project that provides a consistent interface and methodology for collecting hardware performance information from various hardware and software components, including most major CPUs, GPUs and accelerators, interconnects, I/O systems, and power interfaces, as well as virtual cloud environments. The PAPI software is widely used for performance modeling of scientific and engineering applications---for example, the HOMME (High Order Methods Modeling Environment) climate code, and the GAMESS and NWChem computational chemistry codes---on DOE supercomputers. PAPI is widely deployed as middleware for use by higher-level profiling, tracing, and sampling tools (e.g., CrayPat, HPCToolkit, Scalasca, Score-P, TAU, Vampir, PerfExpert), making it the de facto standard for hardware counter analysis. PAPI has established itself as fundamental software infrastructure in every application domain (spanning academia, government, and industry), where improving performance can be mission critical. Ultimately, as more application scientists migrate their applications to HPC platforms, they will benefit from the extended capabilities this grant brought to PAPI to analyze and optimize performance in these environments, whether they use PAPI directly, or via third-party performance tools. Capabilities added to PAPI through this grant include support for new architectures such as the lastest GPU and Xeon Phi accelerators, and advanced power measurement and management features. Another important topic for the UTK team was providing support for a rich ecosystem of different fault management strategies in the context of parallel computing. Our long term efforts have been oriented toward proposing flexible strategies and providing building boxes that application developers can use to build the most efficient fault management technique for their application. These efforts span across the entire software spectrum, from theoretical models of existing strategies to easily assess their performance, to algorithmic modifications to take advantage of specific mathematical properties for data redundancy and to extensions to widely used programming paradigms to empower the application developers to deal with all types of faults. We have also continued our tight collaborations with users to help them adopt these technologies to ensure their application always deliver meaningful scientific data. Large supercomputer systems are becoming more and more power and energy constrained, and future systems and applications running on them will need to be optimized to run under power caps and/or minimize energy consumption. The UTEP team contributed to the SUPER energy thrust by developing power modeling methodologies and investigating power management strategies. Scalability modeling results showed that some applications can scale better with respect to an increasing power budget than with respect to only the number of processors. Power management, in particular shifting power to processors on the critical path of an application execution, can reduce perturbation due to system noise and other sources of runtime variability, which are growing problems on large-scale power-constrained computer systems.

42 ENGINEERING↗