A hybrid collocation-perturbation approach for PDEs with random domains

Area of Science:

• Computational mathematics
• Numerical analysis
• Scientific computing

Background:

• Quantifying uncertainty in quantities of interest (QoI) for PDEs with stochastic geometry is crucial for many applications.
• High-dimensional stochastic problems pose significant computational challenges.

Purpose of the Study:

• To develop an efficient method for uncertainty quantification in linear elliptic PDEs with stochastic geometry.
• To reduce the dimensionality of stochastic problems for improved computational feasibility.

Main Methods:

• A hybrid approach combining sparse grid stochastic collocation for large domain variations and stochastic collocation-perturbation for small variations.
• Splitting the stochastic domain into large and small variation components for targeted approximation.
• Deriving convergence rates for the variance of the QoI.

Main Results:

• The proposed method significantly reduces the dimensionality of the stochastic problem.
• Convergence rates for the QoI variance were derived and validated through numerical experiments.
• Computational cost scales at most quadratically with the number of small variation dimensions, and linearly for independent variations.

Conclusions:

• The hybrid method offers an efficient and scalable solution for uncertainty quantification in high-dimensional stochastic PDEs.
• This approach makes complex stochastic geometry problems more amenable to analysis and computation.