Eigenvalues of Toeplitz matrices associated with orthogonal polynomials
Journal of Approximation Theory
Stochastic finite elements: a spectral approach
Stochastic finite elements: a spectral approach
A stochastic projection method for fluid flow. I: basic formulation
Journal of Computational Physics
Modern Control Engineering
The Wiener--Askey Polynomial Chaos for Stochastic Differential Equations
SIAM Journal on Scientific Computing
Multi-resolution analysis of wiener-type uncertainty propagation schemes
Journal of Computational Physics
Numerical Challenges in the Use of Polynomial Chaos Representations for Stochastic Processes
SIAM Journal on Scientific Computing
Physical Systems with Random Uncertainties: Chaos Representations with Arbitrary Probability Measure
SIAM Journal on Scientific Computing
Inverse Problem Theory and Methods for Model Parameter Estimation
Inverse Problem Theory and Methods for Model Parameter Estimation
Multi-Element Generalized Polynomial Chaos for Arbitrary Probability Measures
SIAM Journal on Scientific Computing
Error analysis of generalized polynomial chaos for nonlinear random ordinary differential equations
Applied Numerical Mathematics
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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.