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A stochastic collocation approach for parabolic PDEs with random domain deformations
Julio E Castrillón-Candás1, Jie Xu1
1Boston University, Department of Mathematics and Statistics, 111 Cummington Mall, Boston, MA 02215.
Computers & Mathematics with Applications (Oxford, England : 1987)
|June 11, 2021
Summary
This study develops a numerical method for approximating statistical moments of quantities of interest in partial differential equations with random geometries. The approach confirms theoretical error estimates through numerical experiments.
Area of Science:
- Computational mathematics
- Numerical analysis
- Stochastic partial differential equations
Background:
- Analyzing partial differential equations with stochastic domain deformation is crucial for modeling complex systems.
- Accurate numerical approximation of statistical moments for quantities of interest (QoI) is a significant challenge.
Purpose of the Study:
- To develop and analyze a numerical method for approximating statistical moments of a QoI in a linear parabolic partial differential equation with stochastic domain deformation.
- To investigate the convergence rates of the proposed numerical method.
Main Methods:
- Remapping the stochastic parabolic problem to a fixed domain with random coefficients.
- Utilizing a collocation method combined with an isotropic Smolyak sparse grid for numerical approximation.
- Deriving theoretical sub-exponential convergence rates based on the number of collocation knots.
Main Results:
- The stochastic moments of the QoI were computed efficiently using the proposed method.
- Theoretical sub-exponential convergence rates were established.
- Numerical experiments validated the theoretical error estimates, confirming the method's accuracy.
Conclusions:
- The collocation method with Smolyak sparse grids provides an effective approach for computing statistical moments in stochastic partial differential equations.
- The study confirms the theoretical convergence rates and demonstrates the practical applicability of the numerical technique.
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