Achieving algorithmic resilience for temporal integration through spectral deferred corrections

Grout, Ray; Kolla, Hemanth; Minion, Michael; Bell, John

doi:10.2140/camcos.2017.12.25

Title: Achieving algorithmic resilience for temporal integration through spectral deferred corrections

Abstract

Spectral deferred corrections (SDC) is an iterative approach for constructing higher-order-accurate numerical approximations of ordinary differential equations. SDC starts with an initial approximation of the solution defined at a set of Gaussian or spectral collocation nodes over a time interval and uses an iterative application of lower-order time discretizations applied to a correction equation to improve the solution at these nodes. Each deferred correction sweep increases the formal order of accuracy of the method up to the limit inherent in the accuracy defined by the collocation points. In this paper, we demonstrate that SDC is well suited to recovering from soft (transient) hardware faults in the data. A strategy where extra correction iterations are used to recover from soft errors and provide algorithmic resilience is proposed. Specifically, in this approach the iteration is continued until the residual (a measure of the error in the approximation) is small relative to the residual of the first correction iteration and changes slowly between successive iterations. Here, we demonstrate the effectiveness of this strategy for both canonical test problems and a comprehensive situation involving a mature scientific application code that solves the reacting Navier-Stokes equations for combustion research.

Authors:

Grout, Ray ^[1]; Kolla, Hemanth ^[2]; Minion, Michael ^[3]; Bell, John ^[3]

National Renewable Energy Lab. (NREL), Golden, CO (United States). Computational Science Center
Sandia National Lab. (SNL-CA), Livermore, CA (United States). Scalable Modeling and Analysis Dept.
Lawrence Berkeley National Lab. (LBNL), Berkeley, CA (United States). Computational Research Division

Publication Date:: Mon May 08 00:00:00 EDT 2017

Research Org.:: Lawrence Berkeley National Laboratory (LBNL), Berkeley, CA (United States)

Sponsoring Org.:: USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR)

OSTI Identifier:: 1436145

Grant/Contract Number:: AC02-05CH11231; AC36-08GO28308

Resource Type:: Accepted Manuscript

Journal Name:: Communications in Applied Mathematics and Computational Science

Additional Journal Information:: Journal Volume: 12; Journal Issue: 1; Journal ID: ISSN 1559-3940

Publisher:: Mathematical Sciences Publishers

Country of Publication:: United States

Language:: English

Subject:: 97 MATHEMATICS AND COMPUTING; SDC; resilience; time integration; deferred correction; exascale computing; combustion

Citation Formats


                    Grout, Ray, Kolla, Hemanth, Minion, Michael, and Bell, John. Achieving algorithmic resilience for temporal integration through spectral deferred corrections.  United States: N. p., 2017. 
Web.  doi:10.2140/camcos.2017.12.25.

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                    Grout, Ray, Kolla, Hemanth, Minion, Michael, & Bell, John. Achieving algorithmic resilience for temporal integration through spectral deferred corrections.  United States.  https://doi.org/10.2140/camcos.2017.12.25

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                    Grout, Ray, Kolla, Hemanth, Minion, Michael, and Bell, John. Mon .  
"Achieving algorithmic resilience for temporal integration through spectral deferred corrections".  United States.  https://doi.org/10.2140/camcos.2017.12.25.  https://www.osti.gov/servlets/purl/1436145.

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@article{osti_1436145,

  title        = {Achieving algorithmic resilience for temporal integration through spectral deferred corrections},

  author       = {Grout, Ray and Kolla, Hemanth and Minion, Michael and Bell, John},

  abstractNote = {Spectral deferred corrections (SDC) is an iterative approach for constructing higher-order-accurate numerical approximations of ordinary differential equations. SDC starts with an initial approximation of the solution defined at a set of Gaussian or spectral collocation nodes over a time interval and uses an iterative application of lower-order time discretizations applied to a correction equation to improve the solution at these nodes. Each deferred correction sweep increases the formal order of accuracy of the method up to the limit inherent in the accuracy defined by the collocation points. In this paper, we demonstrate that SDC is well suited to recovering from soft (transient) hardware faults in the data. A strategy where extra correction iterations are used to recover from soft errors and provide algorithmic resilience is proposed. Specifically, in this approach the iteration is continued until the residual (a measure of the error in the approximation) is small relative to the residual of the first correction iteration and changes slowly between successive iterations. Here, we demonstrate the effectiveness of this strategy for both canonical test problems and a comprehensive situation involving a mature scientific application code that solves the reacting Navier-Stokes equations for combustion research.},

  doi          = {10.2140/camcos.2017.12.25},

  journal      = {Communications in Applied Mathematics and Computational Science},

  number       = 1,

  volume       = 12,

  place        = {United States},

  year         = {Mon May 08 00:00:00 EDT 2017},

  month        = {Mon May 08 00:00:00 EDT 2017}

}

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Works referencing / citing this record:

Data recovery in computational fluid dynamics through deep image priors
preprint, January 2019

de Frahan, Marc T. Henry; Grout, Ray W.
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A scalable weakly-synchronous algorithm for solving partial differential equations
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Title: Achieving algorithmic resilience for temporal integration through spectral deferred corrections

Abstract

Citation Formats

Turbulent flame–wall interaction: a direct numerical simulation study journal, August 2010

Silent error detection in numerical time-stepping schemes journal, April 2014

Structure of a spatially developing turbulent lean methane–air Bunsen flame journal, January 2007

Terascale direct numerical simulations of turbulent combustion using S3D journal, January 2009

Conservative multi-implicit spectral deferred correction methods for reacting gas dynamics journal, March 2004

Cosmic rays don't strike twice: understanding the nature of DRAM errors and the implications for system design conference, January 2012

Comments on high-order integrators embedded within integral deferred correction methods journal, January 2009

Semi-implicit spectral deferred correction methods for ordinary differential equations journal, January 2003

The effect of threshold voltages on the soft error rate [memory and logic circuits] conference, January 2004

A direct numerical simulation study of turbulence and flame structure in transverse jets analysed in jet-trajectory based coordinates journal, July 2012

Design challenges of technology scaling journal, January 1999

Evaluation of models for flame stretch due to curvature in the thin reaction zones regime journal, January 2005

A study of DRAM failures in the field conference, November 2012

Numerical Analysis of Fixed Point Algorithms in the Presence of Hardware Faults journal, January 2015

Quantifying the Accuracy of High-Level Fault Injection Techniques for Hardware Faults conference, June 2014

Spectral Deferred Correction Methods for Ordinary Differential Equations journal, June 2000

Direct numerical simulation of flame stabilization downstream of a transverse fuel jet in cross-flow journal, January 2011

Implications of the Choice of Quadrature Nodes for Picard Integral Deferred Corrections Methods for Ordinary Differential Equations journal, June 2005

Scalar mixing in direct numerical simulations of temporally evolving plane jet flames with skeletal CO/H2 kinetics journal, January 2007

A deferred correction coupling strategy for low Mach number flow with complex chemistry journal, December 2012

Asynchronous finite-difference schemes for partial differential equations journal, October 2014

The effects of non-uniform temperature distribution on the ignition of a lean homogeneous hydrogen–air mixture journal, January 2005

Three-dimensional direct numerical simulation of a turbulent lifted hydrogen jet flame in heated coflow: flame stabilization and structure journal, December 2009

Accelerating S3D: A GPGPU Case Study book, January 2010

High-order multi-implicit spectral deferred correction methods for problems of reactive flow journal, August 2003

Toward an efficient parallel in time method for partial differential equations journal, January 2012

Solving Ordinary Differential Equations II book, September 1996

An algorithmic approach to error localization and partial recomputation for low-overhead fault tolerance conference, June 2013

Low-storage, explicit Runge–Kutta schemes for the compressible Navier–Stokes equations journal, November 2000

Cooperative Application/OS DRAM Fault Recovery book, January 2012

An updated comprehensive kinetic model of hydrogen combustion journal, January 2004

DRAM errors in the wild: a large-scale field study journal, February 2011

Direct numerical simulation of autoignition in non-homogeneous hydrogen-air mixtures journal, August 2003

Evaluating the Error Resilience of Parallel Programs conference, June 2014

Performance of Under-resolved Two-Dimensional Incompressible Flow Simulations journal, November 1995

Impact of deep submicron technology on dependability of VLSI circuits conference, January 2002

Design challenges of technology scaling journal, January 1999

Toward an efficient parallel in time method for partial differential equations journal, January 2012

Data recovery in computational fluid dynamics through deep image priors preprint, January 2019

A scalable weakly-synchronous algorithm for solving partial differential equations preprint, January 2019

Turbulent flame–wall interaction: a direct numerical simulation study
journal, August 2010

Silent error detection in numerical time-stepping schemes
journal, April 2014

Structure of a spatially developing turbulent lean methane–air Bunsen flame
journal, January 2007

Terascale direct numerical simulations of turbulent combustion using S3D
journal, January 2009

Conservative multi-implicit spectral deferred correction methods for reacting gas dynamics
journal, March 2004

Cosmic rays don't strike twice: understanding the nature of DRAM errors and the implications for system design
conference, January 2012

Comments on high-order integrators embedded within integral deferred correction methods
journal, January 2009

Semi-implicit spectral deferred correction methods for ordinary differential equations
journal, January 2003

The effect of threshold voltages on the soft error rate [memory and logic circuits]
conference, January 2004

A direct numerical simulation study of turbulence and flame structure in transverse jets analysed in jet-trajectory based coordinates
journal, July 2012

Design challenges of technology scaling
journal, January 1999

Evaluation of models for flame stretch due to curvature in the thin reaction zones regime
journal, January 2005

A study of DRAM failures in the field
conference, November 2012

Numerical Analysis of Fixed Point Algorithms in the Presence of Hardware Faults
journal, January 2015

Quantifying the Accuracy of High-Level Fault Injection Techniques for Hardware Faults
conference, June 2014

Spectral Deferred Correction Methods for Ordinary Differential Equations
journal, June 2000

Direct numerical simulation of flame stabilization downstream of a transverse fuel jet in cross-flow
journal, January 2011

Implications of the Choice of Quadrature Nodes for Picard Integral Deferred Corrections Methods for Ordinary Differential Equations
journal, June 2005

Scalar mixing in direct numerical simulations of temporally evolving plane jet flames with skeletal CO/H2 kinetics
journal, January 2007

A deferred correction coupling strategy for low Mach number flow with complex chemistry
journal, December 2012

Asynchronous finite-difference schemes for partial differential equations
journal, October 2014

The effects of non-uniform temperature distribution on the ignition of a lean homogeneous hydrogen–air mixture
journal, January 2005

Three-dimensional direct numerical simulation of a turbulent lifted hydrogen jet flame in heated coflow: flame stabilization and structure
journal, December 2009

Accelerating S3D: A GPGPU Case Study
book, January 2010

High-order multi-implicit spectral deferred correction methods for problems of reactive flow
journal, August 2003

Toward an efficient parallel in time method for partial differential equations
journal, January 2012

Solving Ordinary Differential Equations II
book, September 1996

An algorithmic approach to error localization and partial recomputation for low-overhead fault tolerance
conference, June 2013

Low-storage, explicit Runge–Kutta schemes for the compressible Navier–Stokes equations
journal, November 2000

Cooperative Application/OS DRAM Fault Recovery
book, January 2012

An updated comprehensive kinetic model of hydrogen combustion
journal, January 2004

DRAM errors in the wild: a large-scale field study
journal, February 2011

Direct numerical simulation of autoignition in non-homogeneous hydrogen-air mixtures
journal, August 2003

Evaluating the Error Resilience of Parallel Programs
conference, June 2014

Performance of Under-resolved Two-Dimensional Incompressible Flow Simulations
journal, November 1995

Impact of deep submicron technology on dependability of VLSI circuits
conference, January 2002

Design challenges of technology scaling
journal, January 1999

Toward an efficient parallel in time method for partial differential equations
journal, January 2012

Data recovery in computational fluid dynamics through deep image priors
preprint, January 2019

A scalable weakly-synchronous algorithm for solving partial differential equations
preprint, January 2019