A Sparse Data-Driven Polynomial Chaos Expansion Method for Uncertainty Propagation

Fuzzy uncertainty propagation in composites using Gram–Schmidt polynomial chaos expansion

Surrogate modeling of high-dimensional problems via data-driven polynomial chaos expansions and sparse partial least square

Data-driven polynomial chaos expansions: A weighted least-square approximation

Computational fire models are a valuable tool for engineers, investigators, and researchers, but the governing equations and solution approaches typically are too complicated, and the models rely on too many variables to allow for direct uncertainty analysis and error propagation. Without these necessary steps in hypothesis testing and model validation, the true accuracy of predictions cannot be completely quantified. Additionally, there is no standard accepted methodology to determine the kinetics of thermal degradation of a material, which can further complicate uncertainty quantification, validation, and interlaboratory studies. Recently, the polynomial chaos expansion method has emerged as a means to conduct uncertainty quantification on complicated models with relatively low computational cost. A method is presented in this work to determine the kinetics of pyrolysis for polycarbonate and to quantify the uncertainty in the mass loss rate prediction. Using this methodology, predictions of mass loss rate measured in themogravimetric experiments showed relative standard deviations of the peak value ranging from 16% to 23% for heating rates ranging from 30 to 3 K/min. The experimental data were within the bounds of uncertainty of the mass loss rate predictions. This study demonstrated a successful parameterization methodology and validated the use of generalized polynomial chaos to inexpensively quantify uncertainty in pyrolysis model predictions.

Fatigue crack propagation analysis in structures with random parameters based on polynomial chaos expansion method

Considering uncertainty is inarguably a crucial aspect of dynamic analysis, design, and control of a mechanical system. When it comes to multibody problems, the effect of uncertainty in the system’s parameters and excitations becomes even more significant due to the accumulation of inaccuracies. For this reason, this paper presents a detailed research on the use of the Polynomial Chaos Expansion (PCE) method for the control of nondeterministic multibody systems. PCE is essentially a way to compactly represent random variables. In this scheme, each stochastic response output and random input is projected onto the space of appropriate independent orthogonal polynomial basis functions. In the field of robotics, a required task is to force robotic arms to follow designated paths. Controlling such systems usually leads to difficulties since the dynamic equations of multibody problems are highly nonlinear. Computed Torque Control Law (CTCL) is able to overcome these difficulties by using feedback linearization to evaluate the required torque/force at any time to make the system follow a trajectory. In this paper, a mathematical framework is introduced to apply the Computed Torque Control Law to a multibody system with uncertainty. Surprisingly, it is shown that using this control scheme, uncertainty in geometry does not affect the closed-loop equations of controlled systems. Both the intrusive PCE method and the Monte Carlo approach are used to control a fully actuated two-link planar elbow arm where each link is required to follow a specified path. Lastly, a comparison of the time efficiency and accuracy between the traditionally used Monte Carlo method and the intrusive PCE is presented. The results indicate that the intrusive PCE approach can provide better accuracy with much less computation time than the Monte Carlo method.

A wide variety of uncertainty propagation methods exist in literature; however, there is a lack of good understanding of their relative merits. In this paper, a comparative study on the performances of several representative uncertainty propagation methods, including a few newly developed methods that have received growing attention, is performed. The full factorial numerical integration, the univariate dimension reduction method, and the polynomial chaos expansion method are implemented and applied to several test problems. They are tested under different settings of the performance nonlinearity, distribution types of input random variables, and the magnitude of input uncertainty. The performances of those methods are compared in moment estimation, tail probability calculation, and the probability density function construction, corresponding to a wide variety of scenarios of design under uncertainty, such as robust design, and reliability-based design optimization. The insights gained are expected to direct designers for choosing the most applicable uncertainty propagation technique in design under uncertainty.

SummaryDecision making under uncertainty can be quite challenging, especially when complex numerical simulations are considered in the work flow and the decision has to be made relatively fast (e.g., in hours). This is the case when one needs to rank a given field portfolio within a limited budget and with acquisition constraints. If the ranking measure associated with each field is properly and rapidly evaluated, new prospect opportunities, which may lead to a favorable strategic position, can be readily identified.In this paper, we propose an efficient methodology for computing a “production-potential” measure that can be used to rank greenfield portfolios in the presence of geological uncertainty, quantifying both uncertainty and risk propagation. Next, we briefly describe the basics of the method proposed. First, uncertainty in sedimentary variability and flow behavior has to be characterized by a number of representative geological realizations. Sampling techniques are used to significantly reduce the number of realizations while preserving accuracy in the description and uncertainty propagation. Thereafter, multiple and varied field-development plans, based on primary/secondary-recovery mechanisms, are automatically generated while accounting for key parameters related to the number, drilling locations, and drilling sequence of wells. In these plans the reservoir is clustered by areas with similar production/injection potential, and the well locations and drilling schedules are obtained accordingly. The well controls are determined through estimations of the field-recovery factor. By means of experimental-design techniques a relatively small number of field-development plans are selected to capture the most significant production profiles. Each of these development plans is simulated for the realizations sampled previously, and the production-potential measure [e.g., average net present value (NPV) over all sampled realizations] is computed for all the plans. The highest of these measures (i.e., the best development plan) can be used for ranking the greenfield in the portfolio. Response-surface procedures are considered to perform additional analysis computations within iterative optimization procedures. It is important to note that other statistics related to the exploitation potential (e.g., standard deviation of the NPV) can also be used to complement the ranking, thereby mitigating the decision makers’ risk tolerance. The methodology has been tested on the Brugge Field benchmark, which presents 104 realizations of multiple geological parameters. The benchmark has been modified to simulate a greenfield scenario. The ranking measure is the (discounted) NPV averaged over the 104 realizations. The proposed work flow yields a ranking measure of USD 5.43 billion, and the computational cost is approximately 1,900 simulations (performed in a parallel-computing environment). This NPV is somewhat higher than those found for the Brugge benchmark (with similar modified settings) by other researchers. To validate the results, we performed more-exhaustive checking by use of approximately 17,000 simulations, and the ranking measure found was USD 5.51 billion.The new work flow presented allows one to efficiently and in a sufficiently accurate manner support decision making in greenfield-portfolio evaluation. Fast reservoir-performance-evaluation engines open new prospect opportunities that, with traditional decision-making techniques, may be frequently lost.

Collision Probability with Gaussian Mixture Orbit Uncertainty

This study quantifies the hazard associated with extreme sea levels due to eastern Adriatic meteotsunamis—long waves generated by traveling atmospheric disturbances—and assesses the sensitivity of the ocean response to the disturbances responsible for those events. In this spirit, a surrogate model of meteotsunami maximum elevation based on generalized polynomial chaos expansion (gPCE) methods, is implemented. The approach relies on the definition of a synthetic pressure disturbance—depending on six different stochastic parameters known to be important for meteotsunami generation, which is used as forcing to produce series of meteotsunami simulations defined with sparse grid methods (up to 10 689 used in this study). The surrogate model and the sensitivity study are then obtained with a pseudo-spectral approximation (PSA) method based on the chosen meteotsunami simulations. This study mainly presents the developed methodology and discusses the feasibility of implementing such gPCE-based surrogate models to assess the hazard and to study the sensitivity of meteorologically driven extreme sea levels.

This work introduces a practical approach for accelerating probabilistic, high-fidelity prognostics using the stochastic reduced order model (SROM) method and its availability in the open-source Python package, SROMPy. SROMs are used as an efficient Monte Carlo simulation (MCS) method, providing low-dimensional representations of random model inputs enabling rapid and non-intrusive uncertainty propagation. This study represents the first application of the SROM approach in the field of prognostics and health management and serves as a tutorial demonstration of the SROMPy software package. The relative ease of applying SROMs with SROMPy for uncertainty propagation is demonstrated on an example of probabilistic, non-planar crack growth simulation. Results show that the SROM approach agrees well with results from MCS while providing orders of magnitude computational speedup. The complete source code and input data required to reproduce the results in this paper are available online to facilitate further evaluation and adoption of the SROM method by researchers in the field.

Non-intrusive polynomial chaos expansion (NPCE) method is a fast algorithm with the best performances for an uncertain acoustic filed currently, in which the selection of collocation points is an important factor for the computational accuracy, and some special processing methods such as piecewise probabilistic collocation method, should be adopted when the outputs of acoustic field vary severely with uncertain ocean environmental parameters. A new fast algorithm for uncertain acoustic filed in shallow-water is proposed based on Kriging model. The theoretical description of the new algorithm is given, and numerical simulations are conducted to verify the performances of the proposed algorithm. The physical interpretations are given in detail. The results demonstrate that the proposed algorithm is more accurate than the NPCE method under the same conditions, and any special processing method does not need to be adopted when the outputs of acoustic field vary severely with the uncertain ocean environmental parameters. The weakness of NPCE method can be overcome by the proposed algorithm, which is that the computational cost increases with the stochastic polynomial expanding to a higher order for enhancing the computational accuracy. The selection of sample point of the proposed algorithm is simpler and easier than that of NPCE method, and the calculation errors can be given directly. Thus, the proposed algorithm is more universal than NPCE method.

Uncertainty quantification has proven to be an indispensable study for enhancing reliability and robustness of engineering systems in the early design phase. Single and multi-fidelity surrogate modelling methods have been used to replace the expensive high fidelity analyses which must be repeated many times for uncertainty quantification. However, since the number of analyses required to build an accurate surrogate model increases exponentially with the number of random input variables, most surrogate modelling methods suffer from the curse of dimensionality. As an alternative approach, the Low-Rank Approximation method can be applied to high-dimensional uncertainty quantification studies with a low computational cost, where the number of coefficients for building the surrogate model increases only linearly with the number of random input variables. In this study, the Low-Rank Approximation method is implemented for multi-fidelity applications with additive and multiplicative correction approaches to make the high-dimensional uncertainty quantification analysis more efficient and accurate. The developed uncertainty quantification methodology is tested on supersonic aircraft design problems and its predictions are compared with the results of single- and multi-fidelity Polynomial Chaos Expansion and Monte Carlo methods. For the same computational cost, the Low-Rank Approximation method outperformed both in surrogate modeling and uncertainty quantification cases for all the benchmarks and real-world engineering problems addressed in the present study.

To obtain reliable designs, aleatory and epistemic uncertainties are considered recently in the structural analysis and design optimization. The reliability based de-sign optimization (RBDO) method [1] is used when the amount of input data is sufficient enough to create accurate statistical distribution. On the other hand, when the sufficient input data are not available due to limitations in time, human, and facility resources, the optimum design may not be reliable if RBDO method is used. To deal with the situation that input uncertainties have insufficient information, a possibility (or fuzzy set) method can be used for structural analysis and possibility based design optimization (PBDO) [2]. However, in many industry design problems, we may have to deal with design problems that involve with the mixed input statistical random and fuzzy variables simultaneously. For these problems, RBDO may yield unreliable optimum designs because of insufficient data. On the other hand, treating the random variables as fuzzy variables and invoking PBDO to solve the mixed design variable problem may yield too conservative designs with higher optimum costs. This paper proposes a new mixed variable design optimization (MVDO) problem based on the performance measure approach (PMA) [1]. To evaluate the possibilistic constraint in MVDO, a sub-optimization problem for inverse analysis is carried out using a hyper-cylinder domain. To solve this sub-problem efficiently and effectively, a new numerical algorithm, maximum failure search (MFS) method, is proposed in this paper by combining the enhanced hybrid mean value (HMV+) method [3] for the inverse reliability analysis in RBDO and the maximal possibility search (MPS) method [2] for the inverse possibility analysis in PBDO. Some mathematical examples are used to demonstrate the efficiency and effectiveness of the proposed numerical MFS method. Some physical design examples are used to compare the proposed MVDO results with RBDO and PBDO results.

Uncertainty propagation in risk and resilience analysis of hierarchical systems