Contributions to System Reliability Assessment and Bayesian Inverse Problems

系統可靠性評估與貝葉斯逆問題中若干問題的研究

Student thesis: Doctoral Thesis

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Award date23 Jun 2020

Abstract

This dissertation presents three novel methodologies for problems in system reliability assessment and Bayesian inverse problems.

The first problem considered in this dissertation is exact methods for system reliability assessment. In consideration of high cost of lifetime testing, we often obtain a small sample of lifetime data, even only test data for components. Chapter 2 proposes an exact method for system reliability assessment (SRA) generalized from the Buehler confidence limit. We give a suggestion on the choice of statistic order based on an estimate of system reliability which only relies on the true value of components reliability even for multi-parameter model. In addition, we present an algorithm to solve the function with constraint set up in Buehler theorem. Numerical studies illustrate the performance of proposed method when applying to small sample problems.

The second problem considered in this dissertation is high order approximation for system reliability assessment. Chapter 3 presents a high order approximation approach for determining the confidence limits of system reliability from components lifetime data, called reliability-based Winterbottom-extended Cornish-Fisher (R-WCF) expansion method. The polynomial adjustment method is adopted to construct approximate confidence limit. We adopt R-WCF expansion to log-location-scale family for components lifetime model. Numerical studies are conducted to illustrate the effectiveness of the proposed approach, and results show that the R-WCF approach is more efficient than the delta method for highly reliable system assessment, especially with ultra-small sample size.

The third problem considered in this dissertation is calibration of a functional input to a time-consuming simulator. Chapter 4 proposes a methodology that solves calibration/inverse problems involving time consuming simulators with a functional input. In this research, a highly flexible nonparametric model given by a Gaussian process prior for the input together with a prior distribution for its correlation parameters is employed, which enables accurate posterior inference of highly nonlinear functional inputs. To alleviate computational burden in extremely high dimensional space, we propose to use a truncated KL expansion of the unconditional prior process for the functional input for dimension reduction. In addition, we propose a two-stage Gaussian process emulator-based Metropolis-within-Gibbs algorithm for computing the posterior density of the functional input, where a follow-up experiment design for improving the emulator is constructed from the posterior samples from the first stage. The follow up design significantly improves the accuracy of the Gaussian process emulator in high posterior regions, making posterior inferences minimally affected by emulator prediction uncertainty.

    Research areas

  • System reliability assessment, Buehler theorem, Gaussian processes, Bayesian inverse problems, Markov chain Monte Carlo, Karhunen-Loève expansion