Uncertainty quantification via random domain decomposition and probabilistic collocation on sparse grids

• Published in: Journal of computational physics Vol. 229; no. 19; pp. 6995 - 7012
• Main Authors: Lin, G., Tartakovsky, A.M., Tartakovsky, D.M.
• Format: Journal Article
• Language: English
• Published: Kidlington Elsevier Inc 20.09.2010; Elsevier
• Subjects: ACCURACY; ANALYTICAL SOLUTION; Approximation; APPROXIMATIONS; Computation; Computational techniques; DATA COVARIANCES; Domain decomposition; Exact sciences and technology; MATHEMATICAL METHODS AND COMPUTING; Mathematical methods in physics; Mathematical models; MONTE CARLO METHOD; Physics; Polynomial chaos; Probabilistic methods; PROBABILITY; Probability theory; Random composite; RANDOMNESS; STATISTICS; Stochastic collocation method; Stochastic finite element; stochastic finite elements; Uncertainty; Uncertainty quantification; Uncertainty quantification; Stochastic collocation method; Stochastic finite element; Polynomial chaos; Random composite; Chaos; Uncertainty; Partial differential equations; Stochastic processes; Variability; Stochastic method; Spatial variation; Calculation methods; Deterministic system; Models; Calculation; Diffusion; Correlation length
• ISSN: 0021-9991; 1090-2716
• DOI: 10.1016/j.jcp.2010.05.036

Quantitative predictions of the behavior of many deterministic systems are uncertain due to ubiquitous heterogeneity and insufficient characterization by data. We present a computational approach to quantify predictive uncertainty in complex phenomena, which is modeled by (partial) differential equations with uncertain parameters exhibiting multi-scale variability. The approach is motivated by flow in random composites whose internal architecture (spatial arrangement of constitutive materials) and spatial variability of properties of each material are both uncertain. The proposed two-scale framework combines a random domain decomposition (RDD) and a probabilistic collocation method (PCM) on sparse grids to quantify these two sources of uncertainty, respectively. The use of sparse grid points significantly reduces the overall computational cost, especially for random processes with small correlation lengths. A series of one-, two-, and three-dimensional computational examples demonstrate that the combined RDD–PCM approach yields efficient, robust and non-intrusive approximations for the statistics of diffusion in random composites.