Enforcing positivity in intrusive PC-UQ methods for reactive ODE systems

Najm, Habib N.; Valorani, Mauro

doi:10.1016/j.jcp.2014.03.061

Title: Enforcing positivity in intrusive PC-UQ methods for reactive ODE systems

Abstract

We explore the relation between the development of a non-negligible probability of negative states and the instability of numerical integration of the intrusive Galerkin ordinary differential equation system describing uncertain chemical ignition. To prevent this instability without resorting to either multi-element local polynomial chaos (PC) methods or increasing the order of the PC representation in time, we propose a procedure aimed at modifying the amplitude of the PC modes to bring the probability of negative state values below a user-defined threshold. This modification can be effectively described as a filtering procedure of the spectral PC coefficients, which is applied on-the-fly during the numerical integration when the current value of the probability of negative states exceeds the prescribed threshold. We demonstrate the filtering procedure using a simple model of an ignition process in a batch reactor. This is carried out by comparing different observables and error measures as obtained by non-intrusive Monte Carlo and Gauss-quadrature integration and the filtered intrusive procedure. Lastly, the filtering procedure has been shown to effectively stabilize divergent intrusive solutions, and also to improve the accuracy of stable intrusive solutions which are close to the stability limits.

Authors:

Najm, Habib N. ^[1]; Valorani, Mauro ^[2]

Sandia National Lab. (SNL-CA), Livermore, CA (United States)
Sapienza Univ. of Rome (Italy)

Publication Date:: Sat Apr 12 00:00:00 EDT 2014

Research Org.:: Sandia National Lab. (SNL-CA), Livermore, CA (United States)

Sponsoring Org.:: USDOE Office of Science (SC), Basic Energy Sciences (BES)

OSTI Identifier:: 1106517

Report Number(s):: SAND2013-6540J
Journal ID: ISSN 0021-9991; PII: S0021999114002484

Grant/Contract Number:: AC04-94AL85000

Resource Type:: Journal Article: Accepted Manuscript

Journal Name:: Journal of Computational Physics

Additional Journal Information:: Journal Volume: 270; Related Information: Proposed for publication in Journal of Computational Physics.; Journal ID: ISSN 0021-9991

Publisher:: Elsevier

Country of Publication:: United States

Language:: English

Subject:: 97 MATHEMATICS AND COMPUTING; Uncertainty quanti cation; polynomial chaos; intrusive; ordinary di erential equations; stability

Citation Formats


                    Najm, Habib N., and Valorani, Mauro. Enforcing positivity in intrusive PC-UQ methods for reactive ODE systems.  United States: N. p., 2014. 
        Web.  doi:10.1016/j.jcp.2014.03.061.

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                    Najm, Habib N., & Valorani, Mauro. Enforcing positivity in intrusive PC-UQ methods for reactive ODE systems.  United States.  https://doi.org/10.1016/j.jcp.2014.03.061

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                    Najm, Habib N., and Valorani, Mauro. 2014.  
        "Enforcing positivity in intrusive PC-UQ methods for reactive ODE systems".  United States.  https://doi.org/10.1016/j.jcp.2014.03.061.  https://www.osti.gov/servlets/purl/1106517.

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@article{osti_1106517,

  title        = {Enforcing positivity in intrusive PC-UQ methods for reactive ODE systems},

  author       = {Najm, Habib N. and Valorani, Mauro},

  abstractNote = {We explore the relation between the development of a non-negligible probability of negative states and the instability of numerical integration of the intrusive Galerkin ordinary differential equation system describing uncertain chemical ignition. To prevent this instability without resorting to either multi-element local polynomial chaos (PC) methods or increasing the order of the PC representation in time, we propose a procedure aimed at modifying the amplitude of the PC modes to bring the probability of negative state values below a user-defined threshold. This modification can be effectively described as a filtering procedure of the spectral PC coefficients, which is applied on-the-fly during the numerical integration when the current value of the probability of negative states exceeds the prescribed threshold. We demonstrate the filtering procedure using a simple model of an ignition process in a batch reactor. This is carried out by comparing different observables and error measures as obtained by non-intrusive Monte Carlo and Gauss-quadrature integration and the filtered intrusive procedure. Lastly, the filtering procedure has been shown to effectively stabilize divergent intrusive solutions, and also to improve the accuracy of stable intrusive solutions which are close to the stability limits.},

  doi          = {10.1016/j.jcp.2014.03.061},

  url          = {https://www.osti.gov/biblio/1106517},
  journal      = {Journal of Computational Physics},

  issn         = {0021-9991},

  number       = ,

  volume       = 270,

  place        = {United States},

  year         = {Sat Apr 12 00:00:00 EDT 2014},

  month        = {Sat Apr 12 00:00:00 EDT 2014}

}

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Title: Enforcing positivity in intrusive PC-UQ methods for reactive ODE systems

Abstract

Citation Formats

Making best use of model evaluations to compute sensitivity indices journal, May 2002

Local and Global Uncertainty Analyses of a Methane Flame Model journal, November 2005

The role of sensitivity and uncertainty analysis in combustion modelling journal, January 2013

The Homogeneous Chaos journal, October 1938

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A Stochastic Projection Method for Fluid Flow journal, September 2002

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A new stochastic approach to transient heat conduction modeling with uncertainty journal, November 2003

Modeling uncertainty in flow simulations via generalized polynomial chaos journal, May 2003

Uncertainty quantification in reacting-flow simulations through non-intrusive spectral projection journal, February 2003

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Hybrid Stochastic Finite Elements and Generalized Monte Carlo Simulation journal, December 1998

Stochastic Finite-Element Analysis of Seismic Soil–Structure Interaction journal, January 2002

Stochastic Response Surface Methods (SRSMs) for Uncertainty Propagation: Application to Environmental and Biological Systems journal, June 1998

High-Order Collocation Methods for Differential Equations with Random Inputs journal, January 2005

Multiparameter Spectral Representation of Noise-Induced Competence in Bacillus Subtilis journal, November 2012

A Stochastic Multiscale Coupling Scheme to Account for Sampling Noise in Atomistic-to-Continuum Simulations journal, January 2012

A generalized spectral decomposition technique to solve a class of linear stochastic partial differential equations journal, September 2007

Generalized spectral decomposition method for solving stochastic finite element equations: Invariant subspace problem and dedicated algorithms journal, October 2008

A least-squares approximation of partial differential equations with high-dimensional random inputs journal, July 2009

A non-adapted sparse approximation of PDEs with stochastic inputs journal, April 2011

Finite elements for elliptic problems with stochastic coefficients journal, February 2005

Convergence rates for sparse chaos approximations of elliptic problems with stochastic coefficients journal, April 2007

Uncertainty propagation using Wiener–Haar expansions journal, June 2004

Multi-resolution analysis of Wiener-type uncertainty propagation schemes journal, July 2004

Multi‐Resolution‐Analysis Scheme for Uncertainty Quantification in Chemical Systems journal, January 2007

An adaptive multi-element generalized polynomial chaos method for stochastic differential equations journal, November 2005

Multi-Element Generalized Polynomial Chaos for Arbitrary Probability Measures journal, January 2006

Uncertainty Quantification given Discontinuous Model Response and a Limited Number of Model Runs journal, January 2012

Spectral and multiresolution Wiener expansions of oscillatory stochastic processes journal, July 2006

Uncertainty quantification of limit-cycle oscillations journal, September 2006

Wiener Chaos expansions and numerical solutions of randomly forced equations of fluid mechanics journal, August 2006

Modeling physical uncertainties in dynamic stall induced fluid–structure interaction of turbine blades using arbitrary polynomial chaos journal, June 2007

An alternative unsteady adaptive stochastic finite elements formulation based on interpolation at constant phase journal, December 2008

Effect of randomness on multi-frequency aeroelastic responses resolved by Unsteady Adaptive Stochastic Finite Elements journal, October 2009

Multiscale Stochastic Preconditioners in Non-intrusive Spectral Projection journal, April 2011

Time-dependent generalized polynomial chaos journal, November 2010

Stochastic Galerkin techniques for random ordinary differential equations journal, April 2012

Eigenvalues of the Jacobian of a Galerkin-Projected Uncertain ODE System journal, January 2011

On the convergence of generalized polynomial chaos expansions journal, October 2011

Uncertainty quantification in chemical systems journal, November 2009

Slow Manifold Structure in Explosive Kinetics. 1. Bifurcations of Points-at-Infinity in Prototypical Models journal, November 2006

On the Statistical Calibration of Physical Models: STATISTICAL CALIBRATION OF PHYSICAL MODELS journal, February 2015

Making best use of model evaluations to compute sensitivity indices
journal, May 2002

Local and Global Uncertainty Analyses of a Methane Flame Model
journal, November 2005

The role of sensitivity and uncertainty analysis in combustion modelling
journal, January 2013

The Homogeneous Chaos
journal, October 1938

A Stochastic Projection Method for Fluid Flow
journal, November 2001

The Wiener--Askey Polynomial Chaos for Stochastic Differential Equations
journal, January 2002

Numerical Challenges in the Use of Polynomial Chaos Representations for Stochastic Processes
journal, January 2004

Uncertainty Quantification and Polynomial Chaos Techniques in Computational Fluid Dynamics
journal, January 2009

Probabilistic characterization of transport in heterogeneous media
journal, June 1998

Adaptive polynomial chaos expansions applied to statistics of extremes in nonlinear random vibration
journal, April 1998

Incorporation of parametric uncertainty into complex kinetic mechanisms: Application to hydrogen oxidation in supercritical water
journal, January 1998

A Stochastic Projection Method for Fluid Flow
journal, September 2002

A multigrid solver for two-dimensional stochastic diffusion equations
journal, October 2003

A new stochastic approach to transient heat conduction modeling with uncertainty
journal, November 2003

Modeling uncertainty in flow simulations via generalized polynomial chaos
journal, May 2003

Uncertainty quantification in reacting-flow simulations through non-intrusive spectral projection
journal, February 2003

A Two-Scale Nonperturbative Approach to Uncertainty Analysis of Diffusion in Random Composites
journal, January 2004

Noisy Inflows Cause a Shedding-Mode Switching in Flow Past an Oscillating Cylinder
journal, April 2004

Spectral stochastic uncertainty quantification in chemical systems
journal, September 2004

Variational multiscale stabilized FEM formulations for transport equations: stochastic advection–diffusion and incompressible stochastic Navier–Stokes equations
journal, January 2005

Using stochastic analysis to capture unstable equilibrium in natural convection
journal, September 2005

A stochastic variational multiscale method for diffusion in heterogeneous random media
journal, November 2006

Long-term behavior of polynomial chaos in stochastic flow simulations
journal, August 2006

Stochastic analysis of transport in tubes with rough walls
journal, September 2006

Quantifying uncertainty in chemical systems modeling
journal, January 2005

Stochastic Approaches to Uncertainty Quantification in CFD Simulations
journal, March 2005

Galerkin methods for linear and nonlinear elliptic stochastic partial differential equations
journal, April 2005

A Stochastic Collocation Method for Elliptic Partial Differential Equations with Random Input Data
journal, January 2007

Sparse grid collocation schemes for stochastic natural convection problems
journal, July 2007

A Sparse Grid Stochastic Collocation Method for Partial Differential Equations with Random Input Data
journal, January 2008

An Anisotropic Sparse Grid Stochastic Collocation Method for Partial Differential Equations with Random Input Data
journal, January 2008

An adaptive hierarchical sparse grid collocation algorithm for the solution of stochastic differential equations
journal, May 2009

The multi-element probabilistic collocation method (ME-PCM): Error analysis and applications
journal, November 2008

Multi-element probabilistic collocation method in high dimensions
journal, March 2010

Hybrid Stochastic Finite Elements and Generalized Monte Carlo Simulation
journal, December 1998

Stochastic Finite-Element Analysis of Seismic Soil–Structure Interaction
journal, January 2002

Stochastic Response Surface Methods (SRSMs) for Uncertainty Propagation: Application to Environmental and Biological Systems
journal, June 1998

High-Order Collocation Methods for Differential Equations with Random Inputs
journal, January 2005

Multiparameter Spectral Representation of Noise-Induced Competence in Bacillus Subtilis
journal, November 2012

A Stochastic Multiscale Coupling Scheme to Account for Sampling Noise in Atomistic-to-Continuum Simulations
journal, January 2012

A generalized spectral decomposition technique to solve a class of linear stochastic partial differential equations
journal, September 2007

Generalized spectral decomposition method for solving stochastic finite element equations: Invariant subspace problem and dedicated algorithms
journal, October 2008

A least-squares approximation of partial differential equations with high-dimensional random inputs
journal, July 2009

A non-adapted sparse approximation of PDEs with stochastic inputs
journal, April 2011

Finite elements for elliptic problems with stochastic coefficients
journal, February 2005

Convergence rates for sparse chaos approximations of elliptic problems with stochastic coefficients
journal, April 2007

Uncertainty propagation using Wiener–Haar expansions
journal, June 2004

Multi-resolution analysis of Wiener-type uncertainty propagation schemes
journal, July 2004

Multi‐Resolution‐Analysis Scheme for Uncertainty Quantification in Chemical Systems
journal, January 2007

An adaptive multi-element generalized polynomial chaos method for stochastic differential equations
journal, November 2005

Multi-Element Generalized Polynomial Chaos for Arbitrary Probability Measures
journal, January 2006

Uncertainty Quantification given Discontinuous Model Response and a Limited Number of Model Runs
journal, January 2012

Spectral and multiresolution Wiener expansions of oscillatory stochastic processes
journal, July 2006

Uncertainty quantification of limit-cycle oscillations
journal, September 2006

Wiener Chaos expansions and numerical solutions of randomly forced equations of fluid mechanics
journal, August 2006

Modeling physical uncertainties in dynamic stall induced fluid–structure interaction of turbine blades using arbitrary polynomial chaos
journal, June 2007

An alternative unsteady adaptive stochastic finite elements formulation based on interpolation at constant phase
journal, December 2008

Effect of randomness on multi-frequency aeroelastic responses resolved by Unsteady Adaptive Stochastic Finite Elements
journal, October 2009

Multiscale Stochastic Preconditioners in Non-intrusive Spectral Projection
journal, April 2011

Time-dependent generalized polynomial chaos
journal, November 2010

Stochastic Galerkin techniques for random ordinary differential equations
journal, April 2012

Eigenvalues of the Jacobian of a Galerkin-Projected Uncertain ODE System
journal, January 2011

On the convergence of generalized polynomial chaos expansions
journal, October 2011

Uncertainty quantification in chemical systems
journal, November 2009

Slow Manifold Structure in Explosive Kinetics. 1. Bifurcations of Points-at-Infinity in Prototypical Models
journal, November 2006

On the Statistical Calibration of Physical Models: STATISTICAL CALIBRATION OF PHYSICAL MODELS
journal, February 2015