Polynomial Chaos Expansion Coefficients Research Articles

Ship Robust Design Optimization Based on Polynomial Chaos Expansions

Considerable parameter perturbations occur owing to the influence of uncertain factors in actual ship transportation, resulting in a substantial decline in ship performance. These parameters should not be regarded as certain values but uncertain variables. Ship robust design optimization (RDO) is a method in which various uncertainties are fully considered in the early stages of ship design to ensure that the optimal case adapts to the perturbation of the uncertain parameters. In this study, instead of the commonly used Monte Carlo method, polynomial chaos expansions (PCEs) are adopted to quantify the uncertainty, and an improved probabilistic collocation method (PCM) based on the linear independence principle is proposed to select sample points for calculating polynomial coefficients of PCE, which not only reduces the number of collocation points compared with the traditional statistical sampling method but also avoids the problem that arises with the traditional PCM, which cannot maintain high calculation accuracy even with considerable collocation points. Finally, to ensure ship robustness, in comparison with deterministic optimization design, the proposed RDO framework is applied to minimum Energy Efficiency Design Index (EEDI) KRISO Container Ship hull form design.

Uncertainty Quantification of Stochastic Impact Dynamic Oscillator Using a Proper Orthogonal Decomposition-Polynomial Chaos Expansion Technique

Abstract A proper orthogonal decomposition (POD)-based polynomial chaos expansion (PCE) is utilized in this article for the uncertainty quantification (UQ) of an impact dynamic oscillator. The time-dependent nonsmooth behavior and the uncertainties are decoupled using the POD approach. The uncertain response domain is reduced using the POD approach, and the dominant POD modes are utilized for the UQ of the response quantity. Furthermore, the PCE model is utilized for the propagation of the input uncertainties. Two different cases of impact oscillator are considered, namely, single impact and multiple impact. The contact between two bodies is modeled by Hertz’s law. For both the cases, UQ is performed on the projectile displacement, projectile velocity, and contact force. A highly nonsmooth behavior is noticed for the contact force. For that reason, most number of POD modes are required to assess the UQ of contact force. All the results are compared with the Monte Carlo simulation (MCS) and time domain PCE results. Very good accuracies are observed for the PCE and the POD-PCE predicted results using much less number of model evaluations compared to MCS. As the PCE coefficients are dependent on time, the PCE model is computed at each time step. On the contrary, for the POD-PCE model, the PCE coefficients are computed for the number of POD modes only: it is much less than the PCE model.

Long duration response evaluation of linear structural system with random system properties using time dependent polynomial chaos

Polynomial Chaos (PC) is one of the popular methods for approximate evaluation of responses of stochastic structural mechanics problems. In the case of a structural system under dynamic loading, the PC method is combined with a time integration scheme. The PC method with Galerkin projection converts the stochastic PDE to a set of simultaneous deterministic PDE, which can be solved using any standard time integration scheme. However, the convergence of responses is known to fail for long duration time integration problem due to the development of unacceptable error in calculated response. The probability density functions of the responses are not of constant type as the time progresses. However, in the case of generalized PC (gPC), these are approximated using initially considered gPC, thus loses its efficiency. Time-dependent general polynomial chaos (TDgPC) can be considered to overcome these difficulties, where polynomials are generated on the fly to match with the pdf of the responses. The present study considered an investigation of structural mechanics responses with both Gaussian and non-Gaussian random system properties under dynamic loading using TDgPC. An adaptive updation scheme based on the RMS value of the coefficients of PC expansion is investigated for a proper updation of PC expansion. The curse of dimensionality of PC expansion is addressed by considering only the dominant components of the responses evaluated using KL expansion. Further, starting with lower-order PC, the order of PC expansion is increased only after the first updation, thus reducing computational complexities.

NIPC-based uncertainty analysis of infrared radiation from rocket exhaust plumes caused by nozzle exit conditions

In this study, an uncertainty analysis for the infrared radiation characteristics of rocket exhaust plumes at representative trajectory points is performed. Owing to the instability of the rocket motor’s working characteristics, numerical rocket plume infrared radiation predictions possess significant uncertainties. In this study, four epistemic uncertain variables (freestream velocity, nozzle exit pressure, temperature, and velocity) are considered for uncertainty and sensitivity analyses. Based on the infrared signature analysis tool, the response surface of statistical samples is established through the point collocation nonintrusive polynomial chaos expansion method. Polynomial chaos expansion coefficients are solved using the quadrature method to calculate the statistical characteristics and uncertainty of random input variables. The tensor-product quadrature sparse grid method is utilized to reduce the number of samples for multiple input variables. Based on these models, the uncertainty quantification of infrared radiation for Atlas-IIA rocket plumes is analyzed, including the flows, radiation images, spectra, and radiance. The results show that the uncertainty mainly results from afterburning at low altitude, and the nozzle exit velocity has a significant influence on the radiation intensity of the plume. With an increase in altitude, the uncertainty of infrared radiation owing to the afterburning effect decreases, and the influence of the freestream velocity increases. In addition, the proportion of radiation intensity in the 4.3-μm band is higher than that in the 2.7-μm band, and the corresponding uncertainty band is gradually widened. The nozzle exit temperature is the dominant factor that affects the radiation characteristics of the plume at high altitudes. These results of uncertainty and sensitivity analyses are helpful for improving numerical models of the plume infrared signature.

Global sensitivity analysis: A Bayesian learning based polynomial chaos approach

A novel sparse polynomial chaos expansion (PCE) is proposed in this paper for global sensitivity analysis (GSA). The proposed model combines variational Bayesian (VB) inference and automatic relevance determination (ARD) with the PCE model. The VB inference is utilized to compute the PCE coefficients. The PCE coefficients are obtained through a simple optimization procedure in the VB framework. On the other hand, the curse of dimensionality issue of PCE model is tackled using the ARD which reduces the number of polynomial bases significantly. The applicability of the proposed approach is illustrated by performing GSA on five numerical examples. The results show that the proposed approach outperforms a similar state-of-art surrogate model in obtaining an accurate sensitivity index using limited number of model evaluations. For all the examples, the PCE models are highly sparse, which require very few polynomial bases to assess an accurate sensitivity index.

Adaptive multi-fidelity probabilistic collocation-based Kalman filter for subsurface flow data assimilation: numerical modeling and real-world experiment

The ensemble Kalman filter (EnKF) has received substantial attention in hydrologic data assimilation due to its ease of implementation. In EnKF, a large enough ensemble size is often required to ensure accuracy, which may result in considerable computational overhead, especially for large-scale problems. Motivated by recent developments in multi-fidelity simulation, we develop a novel data assimilation method that provides an alternative to EnKF, namely adaptive multi-fidelity probabilistic collocation-based Kalman filter (AMF-PCKF). The appealing feature is to approximate the system response with polynomial chaos expansion (PCE) using the adaptive multi-fidelity probabilistic collocation method, which improves the computational efficiency without sacrificing accuracy. This constitutes the forecast step of AMF-PCKF, while the analysis step is established by sequentially updating the PCE coefficients. As demonstrated by a synthetic numerical case of heat transport in unsaturated flow and a real-world two-phase flow experiment, AMF-PCKF can provide more accurate estimations than EnKF under the same amount of computation, even when the number of unknown parameters is as high as 100.

PoCET: a Polynomial Chaos Expansion Toolbox for Matlab

We introduce PoCET: a free and open-scource Polynomial Chaos Expansion Toolbox for Matlab, featuring the automatic generation of polynomial chaos expansion (PCE) for linear and nonlinear dynamic systems with time-invariant stochastic parameters or initial conditions, as well as several simulation tools. It offers a built-in handling of Gaussian, uniform, and beta probability density functions, projection and collocation-based calculation of PCE coefficients, and the calculation of stochastic moments from a PCE. Efficient algorithms for the calculation of the involved integrals have been designed in order to increase its applicability. PoCET comes with a variety of introductory and instructive examples. Throughout the paper we show how to perform a polynomial chaos expansion on a simple ordinary differential equation using PoCET, as well as how it can be used to solve the more complex task of optimal experimental design.

Regression-based sparse polynomial chaos for uncertainty quantification of subsurface flow models

Surrogate-modeling techniques including Polynomial Chaos Expansion (PCE) is commonly used for statistical estimation (aka. Uncertainty Quantification) of quantities of interests obtained from expensive computational models. PCE is a data-driven regression-based technique that relies on spectral polynomials as basis-functions. In this technique, the outputs of few numerical simulations are used to estimate the PCE coefficients within a regression framework combined with regularization techniques where the regularization parameters are estimated using standard cross-validation as applied in supervised machine learning methods.In the present work, we introduce an efficient method for estimating the PCE coefficients combining Elastic Net regularization with a data-driven feature ranking approach. Our goal is to increase the probability of identifying the most significant PCE components by assigning each of the PCE coefficients a numerical value reflecting the magnitude of the coefficient and its stability with respect to perturbations in the input data. In our evaluations, the proposed approach has shown high convergence rate for high-dimensional problems, where standard feature ranking might be challenging due to the curse of dimensionality.The presented method is implemented within a standard machine learning library (scikit-learn [1]) allowing for easy experimentation with various solvers and regularization techniques (e.g. Tikhonov, LASSO, LARS, Elastic Net) and enabling automatic cross-validation techniques using a widely used and well tested implementation. We present a set of numerical tests on standard analytical functions, a two-phase subsurface flow model and a simulation dataset for CO2 sequestration in a saline aquifer. For all test cases, the proposed approach resulted in a significant increase in PCE convergence rates.

Dimension-adaptive algorithm-based PCE for models with many model parameters

Purpose To present the models with many model parameters by polynomial chaos expansion (PCE), and improve the accuracy, this paper aims to present dimension-adaptive algorithm-based PCE technique and verify the feasibility of the proposed method through taking solid rocket motor ignition under low temperature as an example. Design/methodology/approach The main approaches of this work are as follows: presenting a two-step dimension-adaptive algorithm; through computing the PCE coefficients using dimension-adaptive algorithm, improving the accuracy of PCE surrogate model obtained; and applying the proposed method to uncertainty quantification (UQ) of solid rocket motor ignition under low temperature to verify the feasibility of the proposed method. Findings The result indicates that by means of comparing with some conventional non-invasive method, the proposed method is able to raise the computational accuracy significantly on condition of meeting the efficiency requirement. Originality/value This paper proposes an approach in which the optimal non-uniform grid that can avoid the issue of overfitting or underfitting is obtained.

Quantification of predictive uncertainty with a metamodel: toward more efficient hydrologic simulations

Hydrologic flood prediction has been a quite complex and difficult task because of various sources of inherent uncertainty. Accurately quantifying these uncertainties plays a significant role in providing flood warnings and mitigating risk, but it is time-consuming. To offset the cost of quantifying the uncertainty, we adopted a highly efficient metamodel based on polynomial chaos expansion (PCE) theory and applied it to a lumped, deterministic rainfall–runoff model (Nedbor–Afstromnings model, NAM) combined with generalized likelihood uncertainty estimation (GLUE). The central conclusions are: (1) the subjective aspects of GLUE (e.g., the cutoff threshold values of likelihood function) are investigated for 8 flood events that occurred in the Thu bon river watershed in Vietnam, resulting that the values of 0.82 for Nash–Sutcliffe efficiency, 4.05% for peak error, and 4.35% for volume error are determined as the acceptance thresholds. Moreover, the number of ensemble behavioral sets required to maintain the sufficient range of uncertainty but to avoid any unnecessary computation was set to 500. (2) The number of experiment designs (N) and degree of polynomial (p) are key factors in estimating PCE coefficients, and values of N = 50 and p = 4 are preferred. (3) The results computed using a PCE model consisting of polynomial bases are as good as those given by the NAM, while the total times required for making an ensemble in the PCE model are approximately seventeen times faster. (4) Two parameters (“CQOF” and “CK12”) turned out to be most dominant based on a visual inspection of the posterior distribution and the mathematical computations of the Sobol’ and Morris sensitivity analysis. Identification of the posterior parameter distributions from the calibration process helps to find the behavioral sets even faster. The unified framework that presents the most efficient ways of predicting flow regime and quantifying the uncertainty without deteriorating accuracy will ultimately be helpful for providing warnings and mitigating flood risk in a timely manner.

Polynomial chaos expansions for dependent random variables

Polynomial chaos expansions (PCE) are well-suited to quantifying uncertainty in models parameterized by independent random variables. The assumption of independence leads to simple strategies for building multivariate orthonormal bases and for sampling strategies to evaluate PCE coefficients. In contrast, the application of PCE to models of dependent variables is much more challenging. Three approaches can be used to construct PCE of models of dependent variables. The first approach uses mapping methods where measure transformations, such as the Nataf and Rosenblatt transformation, can be used to map dependent random variables to independent ones; however we show that this can significantly degrade performance since the Jacobian of the map must be approximated. A second strategy is the class of dominating support methods. In these approaches a PCE is built using independent random variables whose distributional support dominates the support of the true dependent joint density; we provide evidence that this approach appears to produce approximations with suboptimal accuracy. A third approach, the novel method proposed here, uses Gram–Schmidt orthogonalization (GSO) to numerically compute orthonormal polynomials for the dependent random variables. This approach has been used successfully when solving differential equations using the intrusive stochastic Galerkin method, and in this paper we use GSO to build PCE using a non-intrusive stochastic collocation method. The stochastic collocation method treats the model as a black box and builds approximations of the input–output map from a set of samples. Building PCE from samples can introduce ill-conditioning which does not plague stochastic Galerkin methods. To mitigate this ill-conditioning we generate weighted Leja sequences, which are nested sample sets, to build accurate polynomial interpolants. We show that our proposed approach, GSO with weighted Leja sequences, produces PCE which are orders of magnitude more accurate than PCE constructed using mapping or dominating support methods.

Uncertainty and sensitivity analysis of SST turbulence model on hypersonic flow heat transfer

Due to the lacking of knowledge and incomplete information about the closure coefficients in Menter Shear Stress Transport (SST) turbulence model, numerical predictions of hypersonic flow heat transfer possess some uncertainties. The objective of this work is to investigate the uncertainty and identify the key parameters contributing most to the uncertainty in hypersonic aeroheating predictions. In the current study, the nine closure coefficients in SST turbulence model are treated as uncertainty variables represented with intervals, meanwhile stochastic expansion based on non-intrusive polynomial chaos (NIPC) expansion is utilized to represent and propagate the uncertainties, afterwards the Sobol indices evaluated by the polynomial chaos expansion coefficients are used to rank the relative contribution of each closure coefficient. Totally, 110 CFD evaluations are adopted to quantify the uncertainty and sensitivity in hypersonic flow heat transfer. Specifically, uncertainty and sensitivity analysis are performed for hypersonic flow over the double-ellipsoid model and the X-33 flight vehicle. The numerical results show that heat flux is more sensitive to uncertainty of closure coefficients compared to pressure. For the SST model, the coefficients which contribute most to uncertainty in hypersonic flow heat are σω1, κ and a1. While a1 is one of the dominant source of heat flux and pressure in the shock region near the wall.

Uncertainty Quantification of Geo-Magnetically Induced Currents in UHV Power Grid

Geo-magnetically induced currents (GICs) have attracted more attention since many Ultra-High Voltage (UHV) transmission lines have been built, or are going to be built in the world. However, when calculating GICs based on the classical model, some input parameters, such as the earth conductivity and dc resistances of the grid, are uncertain or very hard to be determined in advance. Taking this into account, the uncertainty quantification (UQ) model of the geo-electric fields and GICs is proposed in this paper. The UQ of the maximums of the geo-electric fields and GICs during storms is carried out based on the polynomial chaos (PC) method. The results of the UHV grid, 1000 kV Sanhua Grid, were presented and compared to the Monte Carlo method. The total Sobol indices are calculated by using the PC expansion coefficients. The sensitivities of geo-electric fields and GICs to the input variables are analyzed based on the total Sobol indices. Results show that the GICs and geo-electric fields can be effectively simulated by the proposed model, which may offer a better understanding of the sensitivities to input uncertain variables and further give a reasonable evaluation of the geomagnetic threat to the grid.

Performance of ensemble Kalman filter and Markov chain Monte Carlo under uncertainty in forecast model

Abstract Ensemble Kalman filter (EnKF) and Markov chain Monte Carlo (MCMC) are popular methods to obtain the posterior distribution of unknown parameters in the reservoir model. However, millions of simulation runs may be required in MCMC for accurate sampling of posterior as subsurface flow problems are highly nonlinear and non-Gaussian. Similarly, EnKF formulated on the basis of linear and Gaussian assumptions may also require a large number of realizations to correctly map the solution space of the unknown model parameters, ultimately resulting in the high computational cost. Data-driven meta/surrogate/proxy models provide an alternative solution to alleviate the issue of high computational cost. Since these models are not as accurate as numerical solutions of partial differential equations (PDE), their implementation may add an uncertainty in the forecast model. In literature, the effect of uncertainty in forecast model on data assimilation is not well studied, especially with field-scale reservoir models. In this work, we take the opportunity to evaluate and compare the performance of EnKF and MCMC using polynomial chaos expansion (PCE) based forecast model. Proposed forecast model relies on reducing parameter space using Karhunen–Loeve (KL) expansion which preserves the two-point statistics of the field. Random variables from KL expansion and orthogonal polynomials corresponding to the prior probability density function (pdf) form the set of input parameters in PCE. Further, non-intrusive probabilistic collocation method (PCM) is used to compute PCE coefficients. PCE forecast model is then used in EnKF and MCMC to calculate the likelihood of the samples in place of high fidelity full physics simulation runs. A case study is performed using a 3D field scale model of a reservoir located near Fort McMurray in northern Alberta, Canada. Performance of EnKF and MCMC are assessed under forecast model uncertainty using rigorous qualitative and quantitative analysis and posterior distribution characterization. Results clearly depict that, although EnKF provided reliable mean and variance estimates of model parameters, MCMC outperformed the former even under the uncertainty associated with PCE metamodel. Inaccurate initial assumptions of model parameters were successfully handled by MCMC, although, with a longer burn-in period. Furthermore, characterization of posterior demonstrated reduced uncertainty in the estimation of model parameters using MCMC as compared to EnKF. Practical implications of the proposed approach and performance assessment under forecast model uncertainty will be consequential in designing accurate and computationally efficient reservoir characterization workflows and hence, improved decision-making in reservoir management.

Sensitivity Analysis Based on Polynomial Chaos Expansions and Its Application in Ship Uncertainty‐Based Design Optimization

In order to truly reflect the ship performance under the influence of uncertainties, uncertainty‐based design optimization (UDO) for ships that fully considers various uncertainties in the early stage of design has gradually received more and more attention. Meanwhile, it also brings high dimensionality problems, which may result in inefficient and impractical optimization. Sensitivity analysis (SA) is a feasible way to alleviate this problem, which can qualitatively or quantitatively evaluate the influence of the model input uncertainty on the model output, so that uninfluential uncertain variables can be determined for the descending dimension to achieve dimension reduction. In this paper, polynomial chaos expansions (PCE) with less computational cost are chosen to directly obtain Sobol′ global sensitivity indices by its polynomial coefficients; that is, once the polynomial of the output variable is established, the analysis of the sensitivity index is only the postprocessing of polynomial coefficients. Besides, in order to further reduce the computational cost, for solving the polynomial coefficients of PCE, according to the properties of orthogonal polynomials, an improved probabilistic collocation method (IPCM) based on the linear independence principle is proposed to reduce sample points. Finally, the proposed method is applied to UDO of a bulk carrier preliminary design to ensure the robustness and reliability of the ship.

Mori-Zwanzig reduced models for uncertainty quantification

In many time-dependent problems of practical interest the parameters and/or initial conditions entering the equations describing the evolution of the various quantities exhibit uncertainty. One way to address the problem of how this uncertainty impacts the solution is to expand the solution using polynomial chaos expansions and obtain a system of differential equations for the evolution of the expansion coefficients. We present an application of the Mori-Zwanzig (MZ) formalism to the problem of constructing reduced models of such systems of differential equations. In particular, we construct reduced models for a subset of the polynomial chaos expansion coefficients that are needed for a full description of the uncertainty caused by uncertain parameters or initial conditions. Even though the MZ formalism is exact, its straightforward application to the problem of constructing reduced models for estimating uncertainty involves the computation of memory terms whose cost can become prohibitively expensive. For those cases, we present a Markovian reformulation of the MZ formalism which can lead to approximations that can alleviate some of the computational expense while retaining an accuracy advantage over reduced models that discard the memory altogether. Our results support the conclusion that successful reduced models need to include memory effects.

Improving parameter estimation with an efficient sequential probabilistic collocation-based optimal design method

Accurate estimation of soil hydraulic parameters is essential for modeling of water flow and solute transport in unsaturated soil. Ensemble-based methods have been proposed to provide the optimal monitoring (data-collection) strategy to improve hydraulic parameter estimation. To guarantee the performance, however, these methods usually require relatively large ensemble sizes, leading to high computational cost. To this end, we develop an efficient sequential probabilistic collocation-based optimal design (SPCOD) method, which integrates the adaptive ANOVA (analysis of variance)-based probabilistic collocation-based Kalman filter with the sequential optimal design. At the forecast step, the probabilistic collocation method is used to construct a reduced-order surrogate for system model with the adaptive ANOVA-based polynomial chaos expansion (PCE). At the design step, an information metric in the PCE form is derived to evaluate candidate monitoring strategies. The one with the maximum expected information metric value is chosen as the optimal. At the analysis step, the PCE coefficients are sequentially updated by a square-root Kalman filter with the real-time measurements obtained from the optimal monitoring strategy. Two synthetic numerical cases involving water flow and solute transport in the vadose zone are employed to illustrate the performance of the proposed method. The results show that: (1) compared with conventional fixed monitoring strategies, the optimal monitoring strategy designed by SPCOD provides more accurate parameter estimation and state prediction; (2) compared with our previously developed sequential ensemble-based optimal design method (Man et al., 2016b), the new approach provides a more robust and accurate optimal monitoring design with nearly the same computational cost.

Uncertainty analysis of composite laminated plate with data-driven polynomial chaos expansion method under insufficient input data of uncertain parameters

The uncertainty information related to uncertain structural, material and geometric parameters is included in the available input uncertainty data, and there are multiple uncertainty types when only insufficient input data is acquired from experimental or numerical analyses. In this study, an uncertainty analysis method for composite laminated plate is proposed using a data-driven polynomial chaos expansion (PCE) method under insufficient input data related to uncertain design parameters. An identification criterion for uncertain types of design parameters is constructed based on the Akaike information criterion and observation significance level method, and the uncertain design parameters are subsequently divided into strong statistical variables, sparse variables, and interval variables. A data-driven PCE model of composite laminated plate is constructed by simultaneously considering the three uncertainty types, in which the polynomial coefficients are calculated based on the statistical moment of both strong statistical variables and sparse variables, and the PCE coefficients are quadratic polynomial functions of interval variables. The uncertainty analysis problems of the natural frequency of 5-layer and 10-layer composite laminated plates are presented to verify the effectiveness of the proposed algorithm.

Sparsity-promoting elastic net method with rotations for high-dimensional nonlinear inverse problem

An elastic-net (EN) based polynomial chaos (PC) ensemble Kalman filter (PC-EnKF) with iterative PC-basis rotations is developed for high-dimensional nonlinear inverse modeling. To avoid the huge computational cost of estimating PC expansion coefficients and the Kalman gain matrix in PC-EnKF, this paper focuses mainly on solving the minimization problem of the elastic-net (EN) cost function with the fast iterative shrinkage-thresholding algorithm (FISTA). To further enhance the sparsity and accuracy, an iterative PC-basis rotation method is employed. When performing the rotation technique, two key issues need to be addressed to accommodate the computation of the inverse problem. One is the derivation of a new multi-dimensional random variable. This can be realized by exploring the construction of the gradient matrix used in a multi-parameter and vector-valued response model. The other issue is the selection of the number of iterative rotations during the process of each data assimilation, which can be addressed by resorting to a curve of sparsity versus the number of iterations. As for the regularization parameters, they can be tuned by calculating the information criteria (IC). Through the numerical examples, we demonstrate that EN-based PC-EnKF combined with the iterative PC-basis rotation method is well suited in the high-dimensional nonlinear inverse modeling, and has great potential in the high-dimensional nonlinear inverse modeling of real-world complex systems.

A gradient-based uncertainty optimization framework utilizing dimensional adaptive polynomial chaos expansion

To improve the efficiency of solving uncertainty design optimization problems, a gradient-based optimization framework is herein proposed, which combines the dimension adaptive polynomial chaos expansion (PCE) and sensitivity analysis. The dimensional adaptive PCE is used to quantify the quantities of interest (e.g., reliability, robustness metrics) and the sensitivity. The dimensional adaptive property is inherited from the dimension adaptive sparse grid, which is used to evaluate the PCE coefficients. Robustness metrics, referred to as statistical moments, and their gradients with respect to design variables are easily derived from the PCE, whereas the evaluation of the reliability and its gradient require integrations. To quantify the reliability, the framework uses the Heaviside step function to eliminate the failure domain and calculates the integration by Monte Carlo simulation with the function replaced by PCE. The PCE is further combined with Taylor’s expansion and the finite difference to compute the reliability sensitivity. Since the design vector may affect the sample set determined by dimension adaptive sparse grid, the update of the sample set is controlled by the norm variations of the design vector. The optimization framework is formed by combining reliability, robustness quantification and sensitivity analysis, and the optimization module. The accuracy and efficiency of the reliability quantification, as well as the reliability sensitivity, are verified through a mathematical example, a system of springs, and a cantilever beam. The effectiveness of the framework in solving optimization problems is validated by multiple limit states example, a truss optimization example, an airfoil optimization example, and an ONERA M6 wing optimization problem. The results demonstrate that the framework can obtain accurate solutions at the expense of a manageable computational cost.