[Paper Review] A multilevel adaptive sparse grid stochastic collocation approach to the non-smooth forward propagation of uncertainty in discretized problems

Robert L. Gates, Maximilian Bittens|arXiv (Cornell University)|Sep 4, 2015

Probabilistic and Robust Engineering Design4 citations

TL;DR

This paper proposes the Multilevel Adaptive Sparse Grid Collocation (MLASGC) method to efficiently solve non-smooth stochastic partial differential equations in computational mechanics. By combining adaptive sparse grid stochastic collocation with a hierarchy of spatial discretizations, MLASGC achieves a convergence rate of ε ∼ t⁻⁰·⁹⁵, significantly outperforming single-level ALSGC (ε ∼ t⁻⁰·⁶⁵) and MLMC (ε ≲ t⁻⁰·⁵).

ABSTRACT

This work proposes a scheme for significantly reducing the computational complexity of discretized problems involving the non-smooth forward propagation of uncertainty by combining the adaptive hierarchical sparse grid stochastic collocation method (ALSGC) with a hierarchy of successively finer spatial discretizations (e.g. finite elements) of the underlying deterministic problem. To achieve this, we build strongly upon ideas from the Multilevel Monte Carlo method (MLMC), which represents a well-established technique for the reduction of computational complexity in problems affected by both deterministic and stochastic error contributions. The resulting approach is termed the Multilevel Adaptive Sparse Grid Collocation (MLASGC) method. Preliminary results for a low-dimensional, non-smooth parametric ODE problem are promising: the proposed MLASGC method exhibits an error/cost-relation of $\\varepsilon \\sim t^{-0.95}$ and therefore significantly outperforms the single-level ALSGC ($\\varepsilon \\sim t^{-0.65}$) and MLMC methods ($\\varepsilon \\lesssim t^{-0.5}$).

Motivation & Objective

• Address the high computational cost of forward uncertainty propagation in non-smooth, parametric problems arising in computational mechanics.
• Overcome the limitations of classical stochastic collocation and Monte Carlo methods when solutions lack smoothness in the stochastic parameter domain.
• Reduce computational complexity by combining adaptive sparse grid collocation with multilevel spatial discretizations.
• Leverage multilevel Monte Carlo principles to balance deterministic and stochastic errors across hierarchical spatial grids.
• Develop a non-intrusive, parallelizable framework compatible with existing deterministic solvers for engineering applications.

Proposed method

• Adaptively refine a sparse grid in stochastic space using hierarchical basis functions and local error estimates to handle non-smooth solution behavior.
• Construct a hierarchy of successively finer finite element discretizations of the underlying deterministic problem to represent multiple levels of fidelity.
• Apply the adaptive hierarchical sparse grid collocation (ALSGC) method at each spatial discretization level to compute stochastic interpolants.
• Distribute the overall refinement tolerance across levels in a linear fashion to reduce cost on finer grids while maintaining accuracy.
• Use a multilevel correction strategy inspired by Multilevel Monte Carlo (MLMC), where differences between levels are computed to reduce overall variance.
• Balance discretization and interpolation errors by setting the refinement tolerance proportional to the global truncation error of the time integrator (Δt_R).

Experimental results

Research questions

• RQ1 Can the computational complexity of non-smooth stochastic forward problems be significantly reduced using a multilevel approach with adaptive sparse grids?
• RQ2 How does the error decay rate of the proposed MLASGC method compare to single-level ALSGC and MLMC in non-smooth problems?
• RQ3 Does a linear distribution of refinement tolerance across multilevel spatial discretizations yield better performance than uniform distribution?
• RQ4 Can the non-intrusive nature of the method be leveraged for efficient parallelization and integration into existing finite element frameworks?
• RQ5 Is the MLASGC method applicable to problems with moderate stochastic dimensionality and non-smooth solution behavior in the random space?

Key findings

• The MLASGC method achieves an error/cost convergence rate of ε ∼ t⁻⁰·⁹⁵, demonstrating superior efficiency compared to single-level ALSGC (ε ∼ t⁻⁰·⁶⁵).
• The MLASGC method outperforms the standard Multilevel Monte Carlo (MLMC) method, which exhibits a slower convergence rate of ε ≲ t⁻⁰·⁵.
• A linear distribution of refinement tolerance across multilevel interpolants leads to better performance than uniform distribution, reducing the number of collocation points required on finer levels.
• The method effectively handles non-smooth problems by combining adaptive refinement in stochastic space with multilevel spatial discretization, mitigating the loss of convergence due to lack of smoothness.
• The approach is non-intrusive and compatible with existing deterministic solvers, enabling straightforward implementation and parallelization.
• Despite promising results, the method lacks rigorous mathematical analysis of its convergence properties, leaving room for future theoretical development.