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Stochastic collocation methods for nonlinear parabolic equations with random coefficients. (English) Zbl 1343.65005

Summary: We evaluate the performance of global stochastic collocation methods for solving nonlinear parabolic and elliptic problems (e.g., transient and steady nonlinear diffusion) with random coefficients. The robustness of these and other strategies based on a spectral decomposition of stochastic state variables depends on the regularity of the system’s response in outcome space. The latter is affected by statistical properties of the input random fields. These include variances of the input parameters, whose effect on the computational efficiency of this class of uncertainty quantification techniques has remained unexplored. Our analysis shows that if random coefficients have low variances and large correlation lengths, stochastic collocation strategies outperform Monte Carlo simulations (MCS). As variance increases, the regularity of the stochastic response decreases, which requires higher-order quadrature rules to accurately approximate the moments of interest and increases the overall computational cost above that of MCS.

MSC:

65C30 Numerical solutions to stochastic differential and integral equations
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
60H35 Computational methods for stochastic equations (aspects of stochastic analysis)
35R60 PDEs with randomness, stochastic partial differential equations
35K55 Nonlinear parabolic equations
35J60 Nonlinear elliptic equations
65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs
65N35 Spectral, collocation and related methods for boundary value problems involving PDEs
65C05 Monte Carlo methods
65Y20 Complexity and performance of numerical algorithms
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