Download or read book Forecast Uncertainty Quantification Using Monte Carlo Polynomial Chaos Expansion and Unscented Transformation Methods written by Junjun Hu and published by . This book was released on 2015 with total page 336 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Download or read book An Empirical Chaos Expansion Method for Uncertainty Quantification written by Gautam Andrew Wilkins and published by . This book was released on 2016 with total page 162 pages. Available in PDF, EPUB and Kindle. Book excerpt: Uncertainty quantification seeks to provide a quantitative means to understand complex systems that are impacted by uncertainty in their parameters. The polynomial chaos method is a computational approach to solve stochastic partial differential equations (SPDE) by projecting the solution onto a space of orthogonal polynomials of the stochastic variables and solving for the deterministic coefficients. Polynomial chaos can be more efficient than Monte Carlo methods when the number of stochastic variables is low, and the integration time is not too large. When performing long-term integration, however, achieving accurate solutions often requires the space of polynomial functions to become unacceptably large. This dissertation presents an alternative approach, where sets of empirical basis functions are constructed by examining the behavior of the solution for fixed values of the random variables. The empirical basis functions are evolved over time, which means that the total number can be kept small, even when performing long-term integration. We introduce this method of empirical chaos expansion, and apply it to a number of model equations, demonstrating that the computational time scales linearly with the final integration time. That is not the case for polynomial chaos in general, since achieving accuracy for long-term integration usually requires larger polynomial bases, causing a nonlinear scaling with the final integration time. We also present an analytical method that uses the dynamics of the SPDE to predict the evolution of the empirical basis functions and demonstrate how it can be applied to evolve the empirical basis functions without needing to resample realizations of the original SPDE.

Download or read book Framework for Analysis and Identification of Nonlinear Distributed Parameter Systems using Bayesian Uncertainty Quantification based on Generalized Polynomial Chaos written by Janya-anurak, Chettapong and published by KIT Scientific Publishing. This book was released on 2017-04-04 with total page 248 pages. Available in PDF, EPUB and Kindle. Book excerpt: In this work, the Uncertainty Quantification (UQ) approaches combined systematically to analyze and identify systems. The generalized Polynomial Chaos (gPC) expansion is applied to reduce the computational effort. The framework using gPC based on Bayesian UQ proposed in this work is capable of analyzing the system systematically and reducing the disagreement between the model predictions and the measurements of the real processes to fulfill user defined performance criteria.

Download or read book Uncertainty Quantification written by Luis Chase and published by Nova Science Publishers. This book was released on 2019 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: In recent times, polynomial chaos expansion has emerged as a dominant technique to determine the response uncertainties of a system by propagating the uncertainties of the inputs. In this regard, the opening chapter of Uncertainty Quantification: Advances in Research and Applications, an intrusive approach called Galerkin Projection as well as non-intrusive approaches (such as pseudo-spectral projection and linear regression) are discussed.Next, the authors introduce a new methodology to determine the uncertainties of input parameters using CIRCÉ software to overcome the reliance on expert judgment. The goal is to determinate and evaluate the uncertainty bounds for physical models related to reflood model of MARS-KS code Vessel module (coupled with COBRA-TF) using both CIRCÉ and the experimental data of FEBA.Lastly, uncertainties related to rheological model parameters of skeletal muscles are modeled and analyzed, and available data are acquired and fused for hyperelastic constitutive model parameters with Neo-Hookean and Mooney-Rivlin formulations.

Download or read book Efficient Algorithms for Uncertainty Quantification Using Polynomial Chaos Expansion and Its Applications to Composite Structures written by Mishal Thapa and published by . This book was released on 2019 with total page 618 pages. Available in PDF, EPUB and Kindle. Book excerpt: Uncertainty Quantification (UQ) deals with the study of variation in the response due to the presence of uncertainties in input parameters and governing models. Among the prevalent probabilistic techniques for UQ, non-intrusive Polynomial Chaos Expansion (PCE) has become more popular recently due to its mean square convergence property and ability to integrate deterministic codes as black-box. However, the number of basis terms in the expansion increases exponentially with the number of random inputs - 'curse of dimensionality,' and demands a huge number of function evaluations. Hence, this dissertation has attempted to extensively explore new robust algorithms for PCE while maintaining a proper balance between accuracy and computational efficiency. At first, a new non-intrusive method for PCE called Polynomial Chaos Decomposition with Differentiation (PCDD) is developed. The PCDD utilizes higher-order sensitivities of the responses and requires samples equal to the number of basis terms only. Secondly, the PCDD is utilized to develop a stochastic multi-scale modeling framework for composite structures since the response of composites is hugely influenced by the uncertainties existing at different scales such as micro-scale and macro-scale. Another framework for stochastic progressive failure analysis (PFA) of composites is also developed that allows performing global sensitivity analysis (GSA) to identify the relative importance of random inputs as a post-processing step. To further reduce the number of samples and make the stochastic problem more tractable, an adaptive L2-minimization algorithm that allows basis adaptivity along with sequential adaptive sampling is developed. Finally, an adaptive algorithm to obtain sparse PCE models with L1-minimization and sequential sampling is also proposed for high-dimensional problems. The L1-minimization is capable of solving the under-determined system when the number of samples is minuscule. It is also advantageous in terms of computational storage and memory because of its ability to provide a sparse solution. In general, the overarching goal of obtaining high-fidelity stochastic response models while maintaining a balance between accuracy and computational cost was successfully achieved by the novel algorithms developed in this dissertation. Furthermore, the invaluable information obtained with PCE for composite structures highlighted the benefits of its implementation in engineering problems.

Download or read book Uncertainty Quantification written by Christian Soize and published by Springer. This book was released on 2017-04-24 with total page 344 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book presents the fundamental notions and advanced mathematical tools in the stochastic modeling of uncertainties and their quantification for large-scale computational models in sciences and engineering. In particular, it focuses in parametric uncertainties, and non-parametric uncertainties with applications from the structural dynamics and vibroacoustics of complex mechanical systems, from micromechanics and multiscale mechanics of heterogeneous materials. Resulting from a course developed by the author, the book begins with a description of the fundamental mathematical tools of probability and statistics that are directly useful for uncertainty quantification. It proceeds with a well carried out description of some basic and advanced methods for constructing stochastic models of uncertainties, paying particular attention to the problem of calibrating and identifying a stochastic model of uncertainty when experimental data is available. This book is intended to be a graduate-level textbook for students as well as professionals interested in the theory, computation, and applications of risk and prediction in science and engineering fields.

Download or read book Uncertainty Quantification Via Polynomial Chaos Expansion Methods and Applications for Optimization of Power Systems written by Tillmann Mühlpfordt and published by . This book was released on 2020 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt:

Download or read book Uncertainty Quantification and Model Calibration written by Jan Peter Hessling and published by BoD – Books on Demand. This book was released on 2017-07-05 with total page 228 pages. Available in PDF, EPUB and Kindle. Book excerpt: Uncertainty quantification may appear daunting for practitioners due to its inherent complexity but can be intriguing and rewarding for anyone with mathematical ambitions and genuine concern for modeling quality. Uncertainty quantification is what remains to be done when too much credibility has been invested in deterministic analyses and unwarranted assumptions. Model calibration describes the inverse operation targeting optimal prediction and refers to inference of best uncertain model estimates from experimental calibration data. The limited applicability of most state-of-the-art approaches to many of the large and complex calculations made today makes uncertainty quantification and model calibration major topics open for debate, with rapidly growing interest from both science and technology, addressing subtle questions such as credible predictions of climate heating.

Download or read book Proceedings of the 5th International Symposium on Uncertainty Quantification and Stochastic Modelling written by José Eduardo Souza De Cursi and published by Springer Nature. This book was released on 2020-08-19 with total page 472 pages. Available in PDF, EPUB and Kindle. Book excerpt: This proceedings book discusses state-of-the-art research on uncertainty quantification in mechanical engineering, including statistical data concerning the entries and parameters of a system to produce statistical data on the outputs of the system. It is based on papers presented at Uncertainties 2020, a workshop organized on behalf of the Scientific Committee on Uncertainty in Mechanics (Mécanique et Incertain) of the AFM (French Society of Mechanical Sciences), the Scientific Committee on Stochastic Modeling and Uncertainty Quantification of the ABCM (Brazilian Society of Mechanical Sciences) and the SBMAC (Brazilian Society of Applied Mathematics).

Download or read book Data driven Polynomial Chaos Expansions for Uncertainty Quantification written by Zhanlin Liu and published by . This book was released on 2020 with total page 103 pages. Available in PDF, EPUB and Kindle. Book excerpt: Uncertainties exist in both physics-based and data-driven models of systems. Understanding how system inputs affect a system output's uncertainty is essential to improve system outputs such as quality and productivity. Variance-based sensitivity analysis, which is widely used for uncertainty quantification, characterizes how the output variance is propagated from inputs. To estimate the variance-based sensitivity indices of the output with respect to inputs, polynomial chaos expansions (PCEs) are widely used. However, a majority of existing PCEs impose parametric distributional assumptions on inputs. Furthermore, existing sensitivity indices for dependent inputs impose strong assumptions on the dependence structure of the inputs or lack interpretability. Although recent studies proposed fully data-driven PCEs without strong assumptions on inputs, these PCEs are generally inefficient because the minimally required number of observations increases exponentially in the number of the inputs. To address these challenges, three data-driven PCEs are proposed in this dissertation. We first propose the sparse network PCE (SN-PCE) model for a broad class of systems whose input-output relationships are expressed as directed acyclic graphs. The proposed SN-PCE model accurately estimates variance-based sensitivity indices with far fewer observations than state-of-the-art black-box methods. Next, we propose data-driven sensitivity indices by constructing ordered partitions of linearly independent polynomials of dependent inputs for PCEs. The proposed sensitivity indices provide intuitive interpretations of how the dependent inputs affect the variance of the output without a priori knowledge of the dependence structure of the inputs. Finally, we propose a data-driven algorithm to build sparse PCEs for models with dependent inputs. The proposed algorithm not only reduces the number of minimally required observations but also improves upon the numerical stability and estimation accuracy of a state-of-the-art sparse PCE.

Download or read book The Quadratic Point Estimate Method for Probabilistic Moment and Distribution Estimation for Uncertainty Quantification written by Minhyeok Ko and published by . This book was released on 2023 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: The study of probabilistic engineering mechanics has witnessed significant advancements, particularly in the domain of moment estimation and probability distribution evaluation. This work introduces and rigorously develops a novel method, the Quadratic Point Estimate Method (QPEM), which is verified to effectively evaluate moments of any input random variables up to the fifth order. The Monte Carlo (MC) method, a prevalent sampling-based technique for evaluating probabilistic integrals, has limitations, notably the slow convergence of estimation error. This work explores variance reduction techniques like Quasi-Monte Carlo and Latin Hypercube Sampling to enhance MC's efficiency. It also delves into Sparse Grid Quadrature, with a focus on the Smolyak scheme, addressing challenges in multidimensional integrals. QPEM's robustness and innovative approach in capturing moments stand out, offering a superior alternative to existing Point Estimate Methods (PEM) and the Unscented Transformation (UT). It presents an enhanced capability in numerical evaluation of probabilistic integrals, showing adaptability and precision across various applications in civil engineering, especially structural and geotechnical engineering. Integrated with the Rosenblatt transformation, the QPEM addresses applications involving complex multivariate input dependencies effectively. The work also examines the Pearson/Johnson Distribution Systems in relation to QPEM. These systems classify probability distributions using their moments, providing a framework to categorize various distributions. Integrating the QPEM with these systems facilitates both output distribution estimation and moments evaluation. Subsequently, the work explores the Polynomial Chaos Expansion (PCE) framework, a spectral representation of random processes. By leveraging the strengths of QPEM in estimating higher-order moments and the flexibility of PCE in representing random outputs, this combined approach aims to offer a more comprehensive solution for uncertainty quantification. In conclusion, this work offers a holistic view of the significance of the QPEM and its integration with renowned distribution systems. The versatility of the QPEM is further underscored by its successful application across problems in structural and geotechnical engineering. From statics to dynamics, elasticity to plasticity, and even for random fields, the methodologies presented in this work have proven their efficacy.

Download or read book Uncertainty Quantification written by Ralph C. Smith and published by SIAM. This book was released on 2013-12-02 with total page 400 pages. Available in PDF, EPUB and Kindle. Book excerpt: The field of uncertainty quantification is evolving rapidly because of increasing emphasis on models that require quantified uncertainties for large-scale applications, novel algorithm development, and new computational architectures that facilitate implementation of these algorithms. Uncertainty Quantification: Theory, Implementation, and Applications provides readers with the basic concepts, theory, and algorithms necessary to quantify input and response uncertainties for simulation models arising in a broad range of disciplines. The book begins with a detailed discussion of applications where uncertainty quantification is critical for both scientific understanding and policy. It then covers concepts from probability and statistics, parameter selection techniques, frequentist and Bayesian model calibration, propagation of uncertainties, quantification of model discrepancy, surrogate model construction, and local and global sensitivity analysis. The author maintains a complementary web page where readers can find data used in the exercises and other supplementary material.

Download or read book Uncertainty Quantification and Prediction for Non autonomous Linear and Nonlinear Systems written by Akash Phadnis and published by . This book was released on 2013 with total page 197 pages. Available in PDF, EPUB and Kindle. Book excerpt: The science of uncertainty quantification has gained a lot of attention over recent years. This is because models of real processes always contain some elements of uncertainty, and also because real systems can be better described using stochastic components. Stochastic models can therefore be utilized to provide a most informative prediction of possible future states of the system. In light of the multiple scales, nonlinearities and uncertainties in ocean dynamics, stochastic models can be most useful to describe ocean systems. Uncertainty quantification schemes developed in recent years include order reduction methods (e.g. proper orthogonal decomposition (POD)), error subspace statistical estimation (ESSE), polynomial chaos (PC) schemes and dynamically orthogonal (DO) field equations. In this thesis, we focus our attention on DO and various PC schemes for quantifying and predicting uncertainty in systems with external stochastic forcing. We develop and implement these schemes in a generic stochastic solver for a class of non-autonomous linear and nonlinear dynamical systems. This class of systems encapsulates most systems encountered in classic nonlinear dynamics and ocean modeling, including flows modeled by Navier-Stokes equations. We first study systems with uncertainty in input parameters (e.g. stochastic decay models and Kraichnan-Orszag system) and then with external stochastic forcing (autonomous and non-autonomous self-engineered nonlinear systems). For time-integration of system dynamics, stochastic numerical schemes of varied order are employed and compared. Using our generic stochastic solver, the Monte Carlo, DO and polynomial chaos schemes are inter-compared in terms of accuracy of solution and computational cost. To allow accurate time-integration of uncertainty due to external stochastic forcing, we also derive two novel PC schemes, namely, the reduced space KLgPC scheme and the modified TDgPC (MTDgPC) scheme. We utilize a set of numerical examples to show that the two new PC schemes and the DO scheme can integrate both additive and multiplicative stochastic forcing over significant time intervals. For the final example, we consider shallow water ocean surface waves and the modeling of these waves by deterministic dynamics and stochastic forcing components. Specifically, we time-integrate the Korteweg-de Vries (KdV) equation with external stochastic forcing, comparing the performance of the DO and Monte Carlo schemes. We find that the DO scheme is computationally efficient to integrate uncertainty in such systems with external stochastic forcing.

Download or read book Quantification of Uncertainty Improving Efficiency and Technology written by Marta D'Elia and published by Springer Nature. This book was released on 2020-07-30 with total page 290 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book explores four guiding themes – reduced order modelling, high dimensional problems, efficient algorithms, and applications – by reviewing recent algorithmic and mathematical advances and the development of new research directions for uncertainty quantification in the context of partial differential equations with random inputs. Highlighting the most promising approaches for (near-) future improvements in the way uncertainty quantification problems in the partial differential equation setting are solved, and gathering contributions by leading international experts, the book’s content will impact the scientific, engineering, financial, economic, environmental, social, and commercial sectors.

Download or read book Multilevel Monte Carlo Methods and Uncertainty Quantification written by Aretha Leonore Teckentrup and published by . This book was released on 2013 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: We consider the application of multilevel Monte Carlo methods to elliptic partial differential equations with random coefficients. Such equations arise, for example, in stochastic groundwater ow modelling. Models for random coefficients frequently used in these applications, such as log-normal random fields with exponential covariance, lack uniform coercivity and boundedness with respect to the random parameter and have only limited spatial regularity. To give a rigorous bound on the cost of the multilevel Monte Carlo estimator to reach a desired accuracy, one needs to quantify the bias of the estimator. The bias, in this case, is the spatial discretisation error in the numerical solution of the partial differential equation. This thesis is concerned with establishing bounds on this discretisation error in the practically relevant and technically demanding case of coefficients which are not uniformly coercive or bounded with respect to the random parameter. Under mild assumptions on the regularity of the coefficient, we establish new results on the regularity of the solution for a variety of model problems. The most general case is that of a coefficient which is piecewise Hölder continuous with respect to a random partitioning of the domain. The established regularity of the solution is then combined with tools from classical discretisation error analysis to provide a full convergence analysis of the bias of the multilevel estimator for finite element and finite volume spatial discretisations. Our analysis covers as quantities of interest several spatial norms of the solution, as well as point evaluations of the solution and its gradient and any continuously Fréchet differentiable functional. Lastly, we extend the idea of multilevel Monte Carlo estimators to the framework of Markov chain Monte Carlo simulations. We develop a new multilevel version of a Metropolis Hastings algorithm, and provide a full convergence analysis.

Download or read book Framework for Analysis and Identification of Nonlinear Distributed Parameter Systems Using Bayesian Uncertainty Quantification Based on Generalized Polynomial Chaos written by Chettapong Janya-anurak and published by . This book was released on 2020-10-09 with total page 238 pages. Available in PDF, EPUB and Kindle. Book excerpt: In this work, the Uncertainty Quantification (UQ) approaches combined systematically to analyze and identify systems. The generalized Polynomial Chaos (gPC) expansion is applied to reduce the computational effort. The framework using gPC based on Bayesian UQ proposed in this work is capable of analyzing the system systematically and reducing the disagreement between the model predictions and the measurements of the real processes to fulfill user defined performance criteria. This work was published by Saint Philip Street Press pursuant to a Creative Commons license permitting commercial use. All rights not granted by the work's license are retained by the author or authors.

Download or read book Computational Uncertainty Quantification for Inverse Problems written by Johnathan M. Bardsley and published by SIAM. This book was released on 2018-08-01 with total page 141 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is an introduction to both computational inverse problems and uncertainty quantification (UQ) for inverse problems. The book also presents more advanced material on Bayesian methods and UQ, including Markov chain Monte Carlo sampling methods for UQ in inverse problems. Each chapter contains MATLAB? code that implements the algorithms and generates the figures, as well as a large number of exercises accessible to both graduate students and researchers. Computational Uncertainty Quantification for Inverse Problems is intended for graduate students, researchers, and applied scientists. It is appropriate for courses on computational inverse problems, Bayesian methods for inverse problems, and UQ methods for inverse problems.