Adjoint-Based a Posteriori Error Estimation and Uncertainty Quantification for Transient Nonlinear Problems with Discontinuous Solutions PDF Download
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Languages : en
Pages : 73
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Author:
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Category :
Languages : en
Pages : 73
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Category :
Languages : en
Pages : 75
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Author: Rudiger Verfurth
Publisher: Wiley
ISBN: 9780471967958
Category : Mathematics
Languages : en
Pages : 134
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Wiley—Teubner Series Advances in Numerical Mathematics Editors Hans Georg Bock Mitchell Luskin Wolfgang Hackbusch Rolf Rannacher A Review of A Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques Rüdiger Verfürth Ruhr-Universität Bochum, Germany Self-adaptive discretization methods have gained an enormous importance for the numerical solution of partial differential equations which arise in physical and technical applications. The aim of these methods is to obtain a numerical solution within a prescribed tolerance using a minimal amount of work. The main tools utilised are a posteriori error estimators and indicators which are able to give global and local information on the error of the numerical solution, using only the computed numerical solution and known data of the problem. Presenting the most frequently used error estimators which have been developed by various scientists in the last two decades, this book demonstrates that they are all based on the same basic principles. These principles are then used to develop an abstract framework which is able to handle general nonlinear problems. The abstract results are applied to various classes of nonlinear elliptic partial differential equations from, for example, fluid and continuum mechanics, to yield reliable and easily computable error estimators. The book covers stationary problems but omits transient problems, where theory is often still too complex and not yet well developed.
Author: Curtis Curry Ober
Publisher:
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Category :
Languages : en
Pages : 228
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Sensitivity analysis is critically important to numerous analysis algorithms, including large scale optimization, uncertainty quantification, reduced order modeling, and error estimation. Our research focused on developing tools, algorithms and standard interfaces to facilitate the implementation of sensitivity type analysis into existing code and equally important, the work was focused on ways to increase the visibility of sensitivity analysis. We attempt to accomplish the first objective through the development of hybrid automatic differentiation tools, standard linear algebra interfaces for numerical algorithms, time domain decomposition algorithms and two level Newton methods. We attempt to accomplish the second goal by presenting the results of several case studies in which direct sensitivities and adjoint methods have been effectively applied, in addition to an investigation of h-p adaptivity using adjoint based a posteriori error estimation. A mathematical overview is provided of direct sensitivities and adjoint methods for both steady state and transient simulations. Two case studies are presented to demonstrate the utility of these methods. A direct sensitivity method is implemented to solve a source inversion problem for steady state internal flows subject to convection diffusion. Real time performance is achieved using novel decomposition into offline and online calculations. Adjoint methods are used to reconstruct initial conditions of a contamination event in an external flow. We demonstrate an adjoint based transient solution. In addition, we investigated time domain decomposition algorithms in an attempt to improve the efficiency of transient simulations. Because derivative calculations are at the root of sensitivity calculations, we have developed hybrid automatic differentiation methods and implemented this approach for shape optimization for gas dynamics using the Euler equations. The hybrid automatic differentiation method was applied to a first order approximation of the Euler equations and used as a preconditioner. In comparison to other methods, the AD preconditioner showed better convergence behavior. Our ultimate target is to perform shape optimization and hp adaptivity using adjoint formulations in the Premo compressible fluid flow simulator. A mathematical formulation for mixed-level simulation algorithms has been developed where different physics interact at potentially different spatial resolutions in a single domain. To minimize the implementation effort, explicit solution methods can be considered, however, implicit methods are preferred if computational efficiency is of high priority. We present the use of a partial elimination nonlinear solver technique to solve these mixed level problems and show how these formulation are closely coupled to intrusive optimization approaches and sensitivity analyses. Production codes are typically not designed for sensitivity analysis or large scale optimization. The implementation of our optimization libraries into multiple production simulation codes in which each code has their own linear algebra interface becomes an intractable problem. In an attempt to streamline this task, we have developed a standard interface between the numerical algorithm (such as optimization) and the underlying linear algebra. These interfaces (TSFCore and TSFCoreNonlin) have been adopted by the Trilinos framework and the goal is to promote the use of these interfaces especially with new developments. Finally, an adjoint based a posteriori error estimator has been developed for discontinuous Galerkin discretization of Poisson's equation. The goal is to investigate other ways to leverage the adjoint calculations and we show how the convergence of the forward problem can be improved by adapting the grid using adjoint-based error estimates. Error estimation is usually conducted with continuous adjoints but if discrete adjoints are available it may be possible to reuse the discrete version for error estimation. We investigate the advantages and disadvantages of continuous and discrete adjoints through a simple example.
Author: Weimin Han
Publisher: Springer Science & Business Media
ISBN: 038723537X
Category : Mathematics
Languages : en
Pages : 312
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This work provides a posteriori error analysis for mathematical idealizations in modeling boundary value problems, especially those arising in mechanical applications, and for numerical approximations of numerous nonlinear var- tional problems. An error estimate is called a posteriori if the computed solution is used in assessing its accuracy. A posteriori error estimation is central to m- suring, controlling and minimizing errors in modeling and numerical appr- imations. In this book, the main mathematical tool for the developments of a posteriori error estimates is the duality theory of convex analysis, documented in the well-known book by Ekeland and Temam ([49]). The duality theory has been found useful in mathematical programming, mechanics, numerical analysis, etc. The book is divided into six chapters. The first chapter reviews some basic notions and results from functional analysis, boundary value problems, elliptic variational inequalities, and finite element approximations. The most relevant part of the duality theory and convex analysis is briefly reviewed in Chapter 2.
Author: Johannes Neher
Publisher: Logos Verlag Berlin GmbH
ISBN: 3832530886
Category : Mathematics
Languages : en
Pages : 106
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There is a variety of finite element based methods applicable to the discretization of second order elliptic boundary value problems in mixed form. However, it is expensive to solve the resulting discrete linear system due to its size and its algebraic structure. Hybridization serves as a tool to circumvent these difficulties. Furthermore hybridization is an elegant concept to establish connections among various finite element methods. In this work connections between the methods and their hybridized counterparts are established after showing the link between three different formulations of the elliptic model problem. The main part of the work contains the development of a reliable a posteriori error estimator, which is applicable to all of the methods above. This estimator is the key ingredient of an adaptive numerical approximation of the original boundary value problem. Finally, a number of numerical tests is discussed in order to exhibit the performance of the adaptive hybridized methods.
Author: Corey Michael Bryant
Publisher:
ISBN:
Category :
Languages : en
Pages : 414
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The objective of this work is to develop a posteriori error estimates and adaptive strategies for the numerical solution to nonlinear systems of partial differential equations with uncertain data. Areas of application cover problems in fluid mechanics including a Bayesian model selection study of turbulence comparing different uncertainty models. Accounting for uncertainties in model parameters may significantly increase the computational time when simulating complex problems. The premise is that using error estimates and adaptively refining the solution process can reduce the cost of such simulations while preserving their accuracy within some tolerance. New insights for goal-oriented error estimation for deterministic nonlinear problems are first presented. Linearization of the adjoint problems and quantities of interest introduces higher-order terms in the error representation that are generally neglected. Their effects on goal-oriented adaptive strategies are investigated in detail here. Contributions on that subject include extensions of well-known theoretical results for linear problems to the nonlinear setting, computational studies in support of these results, and an extensive comparative study of goal-oriented adaptive schemes that do, and do not, include the higher-order terms. Approaches for goal-oriented error estimation for PDEs with uncertain coefficients have already been presented, but lack the capability of distinguishing between the different sources of error. A novel approach is proposed here, that decomposes the error estimate into contributions from the physical discretization and the uncertainty approximation. Theoretical bounds are proven and numerical examples are presented to verify that the approach identifies the predominant source of the error in a surrogate model. Adaptive strategies, that use this error decomposition and refine the approximation space accordingly, are designed and tested. All methodologies are demonstrated on benchmark flow problems: Stokes lid-driven cavity, 1D Burger's equation, 2D incompressible flows at low Reynolds numbers. The procedure is also applied to an uncertainty quantification study of RANS turbulence models in channel flows. Adaptive surrogate models are constructed to make parameter uncertainty propagation more efficient. Using surrogate models and adaptivity in a Bayesian model selection procedure, it is shown that significant computational savings can be gained over the full RANS model while maintaining similar accuracy in the predictions.
Author: Stefan Trenz
Publisher:
ISBN:
Category :
Languages : en
Pages :
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Author: Weimin Han
Publisher: Springer
ISBN: 9780387235363
Category : Mathematics
Languages : en
Pages : 302
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Book Description
This work provides a posteriori error analysis for mathematical idealizations in modeling boundary value problems, especially those arising in mechanical applications, and for numerical approximations of numerous nonlinear var- tional problems. An error estimate is called a posteriori if the computed solution is used in assessing its accuracy. A posteriori error estimation is central to m- suring, controlling and minimizing errors in modeling and numerical appr- imations. In this book, the main mathematical tool for the developments of a posteriori error estimates is the duality theory of convex analysis, documented in the well-known book by Ekeland and Temam ([49]). The duality theory has been found useful in mathematical programming, mechanics, numerical analysis, etc. The book is divided into six chapters. The first chapter reviews some basic notions and results from functional analysis, boundary value problems, elliptic variational inequalities, and finite element approximations. The most relevant part of the duality theory and convex analysis is briefly reviewed in Chapter 2.
Author:
Publisher:
ISBN:
Category :
Languages : en
Pages : 57
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