Download or read book Unsteady Adaptive Stochastic Collocation Methods on Sparse Grids written by Bettina Schieche and published by . This book was released on 2012 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Download or read book An Adaptive Wavelet Stochastic Collocation Method for Irregular Solutions of Stochastic Partial Differential Equations written by and published by . This book was released on 2012 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: Accurate predictive simulations of complex real world applications require numerical approximations to first, oppose the curse of dimensionality and second, converge quickly in the presence of steep gradients, sharp transitions, bifurcations or finite discontinuities in high-dimensional parameter spaces. In this paper we present a novel multi-dimensional multi-resolution adaptive (MdMrA) sparse grid stochastic collocation method, that utilizes hierarchical multiscale piecewise Riesz basis functions constructed from interpolating wavelets. The basis for our non-intrusive method forms a stable multiscale splitting and thus, optimal adaptation is achieved. Error estimates and numerical examples will used to compare the efficiency of the method with several other techniques.

Download or read book The Analysis of a Sparse Grid Stochastic Collocation Method for Partial Differential Equations with High dimensional Random Input Data written by and published by . This book was released on 2007 with total page 56 pages. Available in PDF, EPUB and Kindle. Book excerpt: This work describes the convergence analysis of a Smolyak-type sparse grid stochastic collocation method for the approximation of statistical quantities related to the solution of partial differential equations with random coefficients and forcing terms (input data of the model). To compute solution statistics, the sparse grid stochastic collocation method uses approximate solutions, produced here by finite elements, corresponding to a deterministic set of points in the random input space. This naturally requires solving uncoupled deterministic problems and, as such, the derived strong error estimates for the fully discrete solution are used to compare the computational efficiency of the proposed method with the Monte Carlo method. Numerical examples illustrate the theoretical results and are used to compare this approach with several others, including the standard Monte Carlo.

Download or read book Sparse Grids and Applications Munich 2012 written by Jochen Garcke and published by Springer Science & Business Media. This book was released on 2014-04-11 with total page 345 pages. Available in PDF, EPUB and Kindle. Book excerpt: Sparse grids have gained increasing interest in recent years for the numerical treatment of high-dimensional problems. Whereas classical numerical discretization schemes fail in more than three or four dimensions, sparse grids make it possible to overcome the “curse” of dimensionality to some degree, extending the number of dimensions that can be dealt with. This volume of LNCSE collects the papers from the proceedings of the second workshop on sparse grids and applications, demonstrating once again the importance of this numerical discretization scheme. The selected articles present recent advances on the numerical analysis of sparse grids as well as efficient data structures, and the range of applications extends to uncertainty quantification settings and clustering, to name but a few examples.

Download or read book Uncertainty Quantification In Computational Science Theory And Application In Fluids And Structural Mechanics written by Sunetra Sarkar and published by World Scientific. This book was released on 2016-08-18 with total page 197 pages. Available in PDF, EPUB and Kindle. Book excerpt: During the last decade, research in Uncertainty Quantification (UC) has received a tremendous boost, in fluid engineering and coupled structural-fluids systems. New algorithms and adaptive variants have also emerged.This timely compendium overviews in detail the current state of the art of the field, including advances in structural engineering, along with the recent focus on fluids and coupled systems. Such a strong compilation of these vibrant research areas will certainly be an inspirational reference material for the scientific community.

Download or read book A Sparse Grid Stochastic Collocation Method for Partial Differential Equations with Random Input Data written by F. Nobile and published by . This book was released on 2007 with total page 36 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Download or read book Recent Trends in Computational Engineering CE2014 written by Miriam Mehl and published by Springer. This book was released on 2015-10-12 with total page 324 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book presents selected papers from the 3rd International Workshop on Computational Engineering held in Stuttgart from October 6 to 10, 2014, bringing together innovative contributions from related fields with computer science and mathematics as an important technical basis among others. The workshop discussed the state of the art and the further evolution of numerical techniques for simulation in engineering and science. We focus on current trends in numerical simulation in science and engineering, new requirements arising from rapidly increasing parallelism in computer architectures, and novel mathematical approaches. Accordingly, the chapters of the book particularly focus on parallel algorithms and performance optimization, coupled systems, and complex applications and optimization.

Download or read book Meshfree Methods for Partial Differential Equations VII written by Michael Griebel and published by Springer. This book was released on 2014-12-02 with total page 323 pages. Available in PDF, EPUB and Kindle. Book excerpt: Meshfree methods, particle methods, and generalized finite element methods have witnessed substantial development since the mid 1990s. The growing interest in these methods is due in part to the fact that they are extremely flexible numerical tools and can be interpreted in a number of ways. For instance, meshfree methods can be viewed as a natural extension of classical finite element and finite difference methods to scattered node configurations with no fixed connectivity. Furthermore, meshfree methods offer a number of advantageous features which are especially attractive when dealing with multiscale phenomena: a priori knowledge about particular local behavior of the solution can easily be introduced in the meshfree approximation space, and coarse-scale approximations can be seamlessly refined with fine-scale information. This volume collects selected papers presented at the Seventh International Workshop on Meshfree Methods, held in Bonn, Germany in September 2013. They address various aspects of this highly dynamic research field and cover topics from applied mathematics, physics and engineering.

Download or read book Models and Applications of Chaos Theory in Modern Sciences written by Elhadj Zeraoulia and published by CRC Press. This book was released on 2011-09-07 with total page 742 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book presents a select group of papers that provide a comprehensive view of the models and applications of chaos theory in medicine, biology, ecology, economy, electronics, mechanical, and the human sciences. Covering both the experimental and theoretical aspects of the subject, it examines a range of current topics of interest. It consid

Download or read book Stochastic Engineering Simulations Using Sparse Grid Collocation Method and Kriging Based Approaches written by D. Chandra Sekhar and published by . This book was released on 2017 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt:

Download or read book An Anisotropic Sparse Grid Stochastic Collocation Method for PDEs with Random Input Data written by and published by . This book was released on 2007 with total page 75 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Download or read book Spatially Adaptive Sparse Grids for High dimensional Problems written by Dirk Pflüger and published by . This book was released on 2010 with total page 181 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Download or read book Sparse Grid Stochastic Collocation for Partial Differential Equations with Random Data as Constraint for Optimal Control Problems written by Maximilian Reihn and published by . This book was released on 2023 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Download or read book Enhancing Adaptive Sparse Grid Approximations and Improving Refinement Strategies Using Adjoint based a Posteriori Error Estimates written by and published by . This book was released on 2015 with total page 18 pages. Available in PDF, EPUB and Kindle. Book excerpt: In this paper we present an algorithm for adaptive sparse grid approximations of quantities of interest computed from discretized partial differential equations. We use adjoint-based a posteriori error estimates of the interpolation error in the sparse grid to enhance the sparse grid approximation and to drive adaptivity. We show that utilizing these error estimates provides significantly more accurate functional values for random samples of the sparse grid approximation. We also demonstrate that alternative refinement strategies based upon a posteriori error estimates can lead to further increases in accuracy in the approximation over traditional hierarchical surplus based strategies. Throughout this paper we also provide and test a framework for balancing the physical discretization error with the stochastic interpolation error of the enhanced sparse grid approximation.

Download or read book Sparse Grid Methods for Higher Dimensional Approximation written by Christian Feuersänger and published by Sudwestdeutscher Verlag Fur Hochschulschriften AG. This book was released on 2010 with total page 184 pages. Available in PDF, EPUB and Kindle. Book excerpt: Approximation of functions depending on more than three variables is relevant for the simulation of dynamic models. Especially stochastic models lead to a wide range of applications like simulation of polymeric liquids, kinetic equations for forces acting on offshore wind parks, or even the pricing of financial products. Sparse Grids have been designed to significantly reduce the cost to approximate high-dimensional functions under certain smoothness conditions. This book presents tools to design and work with Sparse Grids, including construction properties, algorithms, data structures and error estimates. It also discusses inherent limitations and their benefits when it comes to the simulation of stochastic models, namely to the representation of probability density functions. Furthermore, the book reveals new space- and dimension-adaptive Sparse Grids which are suitable for functions that effectively depend on only a few of their input variables.

Download or read book Multi fidelity Stochastic Collocation Methods Using Model Reduction Techniques written by Maziar Raissi and published by . This book was released on 2013 with total page 70 pages. Available in PDF, EPUB and Kindle. Book excerpt: Over the last few years there have been dramatic advances in our understanding of mathematical and computational models of complex systems in the presence of uncertainty. This has led to a growth in the area of uncertainty quantification as well as the need to develop efficient, scalable, stable and convergent computational methods for solving differential equations with random inputs. Stochastic Galerkin methods based on polynomial chaos expansions have shown superiority to other non-sampling and many sampling techniques. For complicated governing equations numerical implementations of stochastic Galerkin methods can become non-trivial. Monte Carlo and other traditional sampling methods, are straightforward to implement. But they do not offer as fast convergence rates as stochastic Galerkin. Other numerical approaches are the stochastic collocation (SC) methods, which inherit both the ease of implementation of Monte Carlo and the robustness of stochastic Galerkin to a great deal. Stochastic collocation and its powerful extensions, e.g. sparse grid stochastic collocation, can simply fail to handle more levels of complication. The seemingly innocent Burgers equation driven by Brownian motion is such an example. We propose a novel enhancement to stochastic collocation methods using deterministic model reduction techniques that can handle this pathological example and hopefully other more complicated equations like Stochastic Navier Stokes. Our numerical results show the efficiency of the proposed technique. We also perform a mathematically rigorous study of linear parabolic partial differential equations with random forcing terms. Justified by the truncated Karhunen-Loeve expansions, the input data are assumed to be represented by a finite number of random variables. A rigorous convergence analysis of our method applied to parabolic partial differential equations with random forcing terms, supported by numerical results, shows that the proposed technique is not only reliable and robust but also very efficient.

Download or read book Frontiers in PDE Constrained Optimization written by Harbir Antil and published by Springer. This book was released on 2018-10-12 with total page 434 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume provides a broad and uniform introduction of PDE-constrained optimization as well as to document a number of interesting and challenging applications. Many science and engineering applications necessitate the solution of optimization problems constrained by physical laws that are described by systems of partial differential equations (PDEs)​. As a result, PDE-constrained optimization problems arise in a variety of disciplines including geophysics, earth and climate science, material science, chemical and mechanical engineering, medical imaging and physics. This volume is divided into two parts. The first part provides a comprehensive treatment of PDE-constrained optimization including discussions of problems constrained by PDEs with uncertain inputs and problems constrained by variational inequalities. Special emphasis is placed on algorithm development and numerical computation. In addition, a comprehensive treatment of inverse problems arising in the oil and gas industry is provided. The second part of this volume focuses on the application of PDE-constrained optimization, including problems in optimal control, optimal design, and inverse problems, among other topics.