Application of local improvements to reduced-order models to sampling methods for nonlinear PDEs with noise

ABSTRACT

In this work, we extend upon the results of Raissi and Seshaiyer [A multi-fidelity stochastic collocation method for parabolic partial differential equations with random input data, Int. J. Uncertain. Quantif. 4(3) (2014), pp. 225–242]. In Raissi and Seshaiyer (2014), the authors propose to use deterministic model reduction techniques to enhance the performance of sampling methods like Monte-Carlo or stochastic collocation. However, in order to be able to apply the method proposed in Raissi and Seshaiyer (2014) to non-linear problems a crucial step needs to be taken. This step involves local improvements to reduced-order models. This paper is an illustration of the importance of this step. Local improvements to reduced-order models are achieved using sensitivity analysis of the proper orthogonal decomposition.

KEYWORDS:

• Sensitivity analysis
• multi-fidelity model reduction
• proper orthogonal decomposition
• sparse grid
• stochastic collocation method
• finite elements
• stochastic burgers equation
• Monte-Carlo

2010 MSC Subject Classifications:

• 65N30
• 65N35
• 65C20

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1. The expert reader will notice that the term $\sum _{k=1}^d{\xi }_k{h}_k\left(t\right)$ approximates $\dot{W}\left(t\right)$, where $W\left(t\right)$ is a Brownian motion.

2. Please refer to pages 10–11 of [21] for a more detailed exposure.