Polynomial Response Surfaces for Probabilistic Risk Assessment

Many engineering systems represent challenging classes of complex dynamic systems. Lacking information about their system properties leads to model uncertainties up to a level where quantification of uncertainties may become the dominant question in modeling, simulation and application tasks. Uncertainty quantification is the prerequisite for probabilistic risk assessment and related tasks. Current numerical simulationmodels are often too expensive for advanced application tasks that involve accurate uncertainty quantification, risk assessment and robust design. This Chapter will present recent approaches for these challenges based on polynomial response surface techniques, which reduce massively the initial complex model at surprising accuracy. The reduction is achieved via projections on orthonormal polynomial bases, which form a so-called response surface. This way, the model response to changes in uncertain parameters and design or control variables is represented by multi-variate polynomials for each output quantity of interest. This technique is known as polynomial chaos expansion (PCE) in the field of stochastic PDE solutions. The reduced model represented by the response surface is vastly faster than the original complex one, and thus provides a promising starting point for follow-up tasks: global sensitivity analysis, uncertainty quantification and probabilistic risk assessment and as well as robust design and control under uncertainty. Often, the fact that the response surface has known polynomial properties can further simplify these tasks. We will emphasize a more engineering-like language as compared to otherwise intense mathematical derivations found in the literature on PCE. Also we will make use of most recent developments in the theory of stochastic PDE solutions for engineering applications. The current Chapter provides tools based on PCE for global sensitivity analysis, uncertainty quantification and risk analysis as well as design under uncertainty (robust design).