Gaussian Process Regression For Uncertainty Quantification

There exists many kinds of uncertainties in the real-world systems or computer simulation models.The reasons causing these uncertainties include the unknown exact values of the model parameters,the lack of knowledge of the underlying physics,the numerical errors,the vari-ability of experimental measurements and the lack of available data collected from computer model simulations and/or experimental measurements.Any decision or control policy based on the prediction of the uncertainty system would significantly improve if the uncertainty was quantified.Traditional uncertainty quantification methods need a great number of data collected by accessing the experimental systems or computer models.When the systems or models are extremely expensive computation,the total time cost for tranditinal uncertainty quantification methods becomes unaffordable.Constructing efficient surrogate models for the expensive ex-perimental systems or computer models is a straightforward method to solve the time consuming problem.Gaussian process regression model is an effective and flexible method that in partic-ular can provide the uncertainty of the prediction.These advantages make Gaussian Process regression model become a good choice among so many kinds of surrogates.This paper aims at exploring the applications of gaussian process in various uncertainty information quantifica-tion problems by using the gaussian process surrogates instead of expensive systems or models.Specifically we study the application of gaussian process in rare probability estimation problem,posterior density estimation problem and experimental design problem.Real-life engineering systems are unavoidably subject to various uncertainties and these uncertainties may cause undesired events,in particular,system failures or malfunctions,to occur.Accurate identification of failure modes and evaluation of failure probability of a given system is an essential task in many fields of engineering.Here we propose to use an experimental design method to construct a high precision surrogate of the limit state for failure probability estimation of expensive system.This method abandons constructing high precision surrogate over all parameter space,and instead constructs a high-precision surrogate over the parameter space of limit state.This method reduces the demand for data and the times of accessing the real expensive system(because the parameter space of limit state is only a part of the whole parameter space).In numerical implement,we reduce the tested integral which is expressed in the formal experimental design frame to the single integral expression by use of the excellent analytic properties of normal distribution.The computational speed of new experimental design frame is efficiently increased and we obtain a high-precision surrogate of limit state.Multi experimental points design which is beneficial to parallel system is allowed in this experimental frame.Bayesian inference is a method which infers the the unknow parameters by combining mathematical model and data.Here we estimate the posterior distribution of the parameter by B ayesian inference in the situation where the likelihood function is expensive.We use the prod-uct of exponential gaussian process regression model and the proposed distribution approximate the product of likelihood function and the prior of parameter.This method effectively decreases the difficulty of surrogate construction of the likelihood function.We design an active exper-imental design strategy which employs the posterior of parameter in the last iteration as the proposed distribution in this iteration and then obtain a more precise Gaussian Process surro-gate.Finally,we verify the availability of gaussian process regression surrogate for the posterior density estimation problem by three numerical examples.Often researchers are more interest in the experimental parameters than the design param-eters in the experimental systems.But the different design parameters would more or less affect the quality of experimental data and the precision of statistical models.This thesis studies the optimal design problem whose aim is to find the best design parameters.We use multi-task Gaussian process to approximate the inverse process of expensive experiments and determine the optimal design by maximizing the expectated utility.The D-optimal design criterion and A-optimal design criterion are respectively applied as the utility function to quantify the model uncertainties.Our method reduces the uncertainties of data and gives an accurate posterior dis-tribution estimation of the experimental parameters.The method is validated by comparing the posterior marginal distribution of experimental parameters by using different design parameters.