The primary objective of this study is to develop new computational methods for solving a general random eigenvalue problem (REP) commonly encountered in modeling and simulation of high-dimensional, complex dynamic systems. Four major research directions, all anchored in polynomial dimensional decomposition (PDD), have been defined to meet the objective. They involve: (1) a rigorous comparison of accuracy, efficiency, and convergence properties of the polynomial chaos expansion (PCE) and PDD methods; (2) development of two novel multiplicative PDD methods for addressing multiplicative structures in REPs; (3) development of a new hybrid PDD method to account for the combined effects of the multiplicative and additive structures in REPs; and (4) development of adaptive and sparse algorithms in conjunction with the PDD methods. The major findings are as follows. First, a rigorous comparison of the PCE and PDD methods indicates that the infinite series from the two expansions are equivalent but their truncations endow contrasting dimensional structures, creating significant difference between the two approximations. When the cooperative effects of input variables on an eigenvalue attenuate rapidly or vanish altogether, the PDD approximation commits smaller error than does the PCE approximation for identical expansion orders. Numerical analysis reveal higher convergence rates and significantly higher efficiency of the PDD approximation than the PCE approximation. Second, two novel multiplicative PDD methods, factorized PDD and logarithmic PDD, were developed to exploit the hidden multiplicative structure of an REP, if it exists. Since a multiplicative PDD recycles the same component functions of the additive PDD, no additional cost is incurred. Numerical results show that indeed both the multiplicative PDD methods are capable of effectively utilizing the multiplicative structure of a random response. Third, a new hybrid PDD method was constructed for uncertainty quantification of high-dimensional complex systems. The method is based on a linear combination of an additive and a multiplicative PDD approximation. Numerical results indicate that the univariate hybrid PDD method, which is slightly more expensive than the univariate additive or multiplicative PDD approximations, yields more accurate stochastic solutions than the latter two methods. Last, two novel adaptive-sparse PDD methods were developed that entail global sensitivity analysis for defining the relevant pruning criteria. Compared with the past developments, the adaptive-sparse PDD methods do not require its truncation parameter(s) to be assigned a priori or arbitrarily. Numerical results reveal that an adaptive-sparse PDD method achieves a desired level of accuracy with considerably fewer coefficients compared with existing PDD approximations.

The primary objective of this study is to develop new computational methods for robust design optimization (RDO) and reliability-based design optimization (RBDO) of high-dimensional, complex engineering systems. Four major research directions, all anchored in polynomial dimensional decomposition (PDD), have been defined to meet the objective. They involve: (1) development of new sensitivity analysis methods for RDO and RBDO; (2) development of novel optimization methods for solving RDO problems; (3) development of novel optimization methods for solving RBDO problems; and (4) development of a novel scheme and formulation to solve stochastic design optimization problems with both distributional and structural design parameters. The major achievements are as follows. Firstly, three new computational methods were developed for calculating design sensitivities of statistical moments and reliability of high-dimensional complex systems subject to random inputs. The first method represents a novel integration of PDD of a multivariate stochastic response function and score functions, leading to analytical expressions of design sensitivities of the first two moments. The second and third methods, relevant to probability distribution or reliability analysis, exploit two distinct combinations built on PDD: the PDD-SPA method, entailing the saddlepoint approximation (SPA) and score functions; and the PDD-MCS method, utilizing the embedded Monte Carlo simulation (MCS) of the PDD approximation and score functions. For all three methods developed, both the statistical moments or failure probabilities and their design sensitivities are both determined concurrently from a single stochastic analysis or simulation. Secondly, four new methods were developed for RDO of complex engineering systems. The methods involve PDD of a high-dimensional stochastic response for statistical moment analysis, a novel integration of PDD and score functions for calculating the second-moment sensitivities with respect to the design variables, and standard gradient-based optimization algorithms. The methods, depending on how statistical moment and sensitivity analyses are dovetailed with an optimization algorithm, encompass direct, single-step, sequential, and multi-point single-step design processes. Thirdly, two new methods were developed for RBDO of complex engineering systems. The methods involve an adaptive-sparse polynomial dimensional decomposition (AS-PDD) of a high-dimensional stochastic response for reliability analysis, a novel integration of AS-PDD and score functions for calculating the sensitivities of the failure probability with respect to design variables, and standard gradient-based optimization algorithms, resulting in a multi-point, single-step design process. The two methods, depending on how the failure probability and its design sensitivities are evaluated, exploit two distinct combinations built on AS-PDD: the AS-PDD-SPA method, entailing SPA and score functions; and the AS-PDD-MCS method, utilizing the embedded MCS of the AS-PDD approximation and score functions. In addition, a new method, named as the augmented PDD method, was developed for RDO and RBDO subject to mixed design variables, comprising both distributional and structural design variables. The method comprises a new augmented PDD of a high-dimensional stochastic response for statistical moment and reliability analyses; an integration of the augmented PDD, score functions, and finite-difference approximation for calculating the sensitivities of the first two moments and the failure probability with respect to distributional and structural design variables; and standard gradient-based optimization algorithms, leading to a multi-point, single-step design process. The innovative formulations of statistical moment and reliability analysis, design sensitivity analysis, and optimization algorithms have achieved not only highly accurate but also computationally efficient design solutions. Therefore, these new methods are capable of performing industrial-scale design optimization with numerous design variables.