Accurate construction of high dimensional model representation with applications to uncertainty quantification

Surrogate modelling has been employed to substitute the computational costs that high-fidelity models HFMs offers due to many-query nature of uncertainty quantification UQ and global sensitivity analysis GSA.

High dimensional model representation, a surrogate model approach was developed having two expansions commonly used in literature; Cut-high dimensional model representation and ANOVA-high dimensional model representation (or random sampling- high dimensional model representation, RS-HDMR). In order to improve accuracy of high dimensional model representation, approximation errors caused by truncation and numerical process associated with low-order components functions must be reduced.

Research by Professors Giray Ökten and Yousuff Hussaini, with their former Ph.D. student Dr. Yaning Liu from Department of Mathematics at Florida State University developed various methods that improve accuracy of a particular class of surrogate models and high dimensional model representation HDMR thereby observing their performances in uncertainty quantification and variance-based global sensitivity analysis. The study is published in Reliability Engineering and System Safety.

The research team proposed non-uniform optimal grids such as Legendre-Gauss, Chebysev-Gauss and Clenshaw-Curtis as sample points in cases of cut-high dimensional model representation. In order to alleviate slow convergence of Monte Carlo MC sampling in case of random sampling- high dimensional model representation, two variance reduction techniques; correlation control variate method and ratio control variate method were proposed.

However, correlation control variate method can be divergent if sample size is small, hence an optimization process which guarantees maximum variance reduction each step was introduced. The  randomized quasi-Monte Carlo RQMC samples that leads to convergence rates proportional to the reciprocal of sample size was utilized after which the equivalence of cut- high dimensional model representation and random sampling-high dimensional model representation was demonstrated and proven.

For ANOVA-high dimensional model representation cases, a third-order random sampling- high dimensional model representation for polynomial function using Monte Carlo and randomized quasi-Monte Carlo method was constructed. Liu et al. (2016) also considered coupling the procedure to select the optimal polynomial orders and variance techniques such as ratio control variate methods RCVM and correlation control variate method CCVM to improve accuracy of random sampling- randomized quasi-Monte Carlo.

The effect of using thresholds with optimal polynomial order selection showed that no clear distinctions could be observed when thresholds are applied for either Monte Carlo and randomized quasi-Monte Carlo with selection of optimal polynomial orders.

When comparing performance of correlation control variate method and optimal correlation control variate method, sample size started from 2⁷ and optimal correlation control method variate method showed slight improvement over correlation control variate method. Monte Carlo sampling with sampling sizes of 2¹⁰ and 2¹¹ showed that optimal correlation control variate method had clear improvement in reducing errors of correlation control variate method by 10% and 22% respectively.

At different estimations of Monte Carlo sampling, the optimal polynomial orders reduced crude Monte Carlo error by a factor of 10 for sample sizes 2⁶, 2⁷ and 2¹¹ and at least 5 by other sizes. For sample sizes smaller than 2¹², Monte Carlo coupled with optimal correlation control variate was less accurate than Monte Carlo and ratio control variate methods but showed more accuracy otherwise. Using optimal polynomial order for sampling of Monte Carlo and optimal correlation control variate method further reduced the error and provided best performance among all estimators for sample size of 2¹³.

Randomized quasi-Monte Carlo estimators uniformly outperformed Monte Carlo estimators. For sample size smaller than 2¹⁰, randomized quasi-Monte Carlo and ratio control variate methods sampling was more accurate while ratio control variate methods and optimal correlation control variate method performs the best for sample size starting from 2¹⁰.

The improved cut-high dimensional model representation was applied to a chemical kinetic model of H₂/air combustion for quantifying the uncertainty of the ignition delay time which showed that cut-high dimensional model representation surrogate model is highly accurate even with small number of sample points.

The findings of this study can play an important role in studying fuel and combustion system reliability and safety.