Article2011

Eigenvalues of the Jacobian of a Galerkin-Projected Uncertain ODE System

Benjamin E. Sonday · Robert Dan Berry · Habib N. Najm · Bert Debusschere

37 citations0/0 files storedSIAM Journal on Scientific Computing

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Probabilistic and Robust Engineering Design Model Reduction and Neural Networks Numerical methods for differential equations

Abstract

Projection onto polynomial chaos (PC) basis functions is often used to reformulate a system of ordinary differential equations (ODEs) with uncertain parameters and initial conditions as a deterministic ODE system that describes the evolution of the PC modes. The deterministic Jacobian of this projected system is different and typically much larger than the random Jacobian of the original ODE system. This paper shows that the location of the eigenvalues of the projected Jacobian is largely determined by the eigenvalues of the original Jacobian, regardless of PC order or choice of orthogonal polynomials. Specifically, the eigenvalues of the projected Jacobian always lie in the convex hull of the numerical range of the Jacobian of the original system.

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References

2002The Wiener--Askey Polynomial Chaos for Stochastic Differential Equations 2004Physical Systems with Random Uncertainties: Chaos Representations with Arbitrary Probability Measure 2006Multi-Element Generalized Polynomial Chaos for Arbitrary Probability Measures 2004Numerical Challenges in the Use of Polynomial Chaos Representations for Stochastic Processes 2001Solution of stochastic partial differential equations using Galerkin finite element techniques 2001A Stochastic Projection Method for Fluid Flow 2004Multi-resolution analysis of Wiener-type uncertainty propagation schemes 1997An efficient method for parametric uncertainty analysis of numerical geophysical models

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