Download or read book Uncertainty Quantification for Hyperbolic and Kinetic Equations written by Shi Jin and published by Springer. This book was released on 2018-03-20 with total page 282 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book explores recent advances in uncertainty quantification for hyperbolic, kinetic, and related problems. The contributions address a range of different aspects, including: polynomial chaos expansions, perturbation methods, multi-level Monte Carlo methods, importance sampling, and moment methods. The interest in these topics is rapidly growing, as their applications have now expanded to many areas in engineering, physics, biology and the social sciences. Accordingly, the book provides the scientific community with a topical overview of the latest research efforts.
Download or read book Uncertainty Quantification and Sensitivity Analysis for Multiscale Kinetic Equations with Random Inputs written by Ruiwen Shu and published by . This book was released on 2018 with total page 151 pages. Available in PDF, EPUB and Kindle. Book excerpt: This thesis gives an overview of the current results on uncertainty quantification and sensitivity analysis for multiscale kinetic equations with random inputs, with an emphasis on the author's contribution to this field. In the first part of this thesis we consider a kinetic-fluid model for disperse two-phase flows with uncertainty in the fine particle regime. We propose a stochastic asymptotic-preserving (s-AP) scheme in the generalized polynomial chaos stochastic Galerkin (gPC-sG) framework, which allows the efficient computation of the problem in both kinetic and hydrodynamic regimes. The s-AP property is proved by deriving the equilibrium of the gPC version of the Fokker-Planck operator. The coefficient matrices that arise in a Helmholtz equation and a Poisson equation, essential ingredients of the algorithms, are proved to be positive definite under reasonable and mild assumptions. The computation of the gPC version of a translation operator that arises in the inversion of the Fokker-Planck operator is accelerated by a spectrally accurate splitting method. Numerical examples illustrate the s-AP property and the efficiency of the gPC-sG method in various asymptotic regimes. In the second part of this thesis we consider the same kinetic-fluid model with random initial inputs in the light particle regime. Using energy estimates, we prove the uniform regularity in the random space of the model for random initial data near the global equilibrium in some suitable Sobolev spaces, with the randomness in the initial particle distribution and fluid velocity. By hypocoercivity arguments, we prove that the energy decays exponentially in time, which means that the long time behavior of the solution is insensitive to such randomness in the initial data. Then we consider the gPC-sG method for the same model. For initial data near the global equilibrium and smooth enough in the physical and random spaces, we prove that the gPC-sG method has spectral accuracy, uniformly in time and the Knudsen number, and the error decays exponentially in time. In the third part of this thesis we propose a stochastic Galerkin method using sparse wavelet bases for the Boltzmann equation with multi-dimensional random inputs. The method uses locally supported piecewise polynomials as an orthonormal basis of the random space. By a sparse approach, only a moderate number of basis functions is required to achieve good accuracy in multi-dimensional random spaces. We discover a sparse structure of a set of basis-related coefficients, which allows us to accelerate the computation of the collision operator. Regularity of the solution of the Boltzmann equation in the random space and an accuracy result of the stochastic Galerkin method are proved in multi-dimensional cases. The efficiency of the method is illustrated by numerical examples with uncertainties from the initial data, boundary data and collision kernel. In the fourth part of this thesis we explore the possibility of using Generalized polynomial chaos (gPC) for uncertainty quantification in hyperbolic problems. GPC has been extensively used in uncertainty quantification problems to handle random variables. For gPC to be valid, one requires high regularity on the random space that hyperbolic type problems usually cannot provide, and thus it is believed to behave poorly in those systems. We provide a counter-argument, and show that despite the solution profile itself develops singularities in the random space, which prevents the use of gPC, the physical quantities such as shock emergence time, shock location, and shock width are all smooth functions of random variables in the initial data: with proper shifting, the solution's polynomial interpolation approximates with high accuracy. The studies were inspired by the stability results from hyperbolic systems. We use the Burgers' equation as an example for thorough analysis, and the analysis could be extended to general conservation laws with convex fluxes.
Download or read book Uncertainty Quantification for Multiscale Kinetic Equations and Quantum Dynamics written by Liu Liu and published by . This book was released on 2017 with total page 330 pages. Available in PDF, EPUB and Kindle. Book excerpt: In the first part of the thesis, we develop a generalized polynomial chaos approach based stochastic Galerkin (gPC-SG) method for the linear semi-conductor Boltzmann equation with random inputs and diffusive scalings. The random inputs are due to uncertainties in the collision kernel or initial data. We study the regularity (uniform in the Knudsen number) of the solution in the random space, and prove the spectral accuracy of the gPC-SG method. We then use the asymptotic-preserving framework for the deterministic counterpart to come up with the stochastic asymptotic-preserving (sAP) gPC-SG method for the problem under study which is efficient in the diffusive regime. Numerical experiments are conducted to validate the accuracy and asymptotic properties of the method. In the second part, we study the linear transport equation under diffusive scaling and with random inputs. The method is based on the gPC-SG framework. Several theoretical aspects will be addressed. A uniform numerical stability with respect to the Knudsen number and a uniform error estimate is given. For temporal and spatial discretizations, we apply the implicit-explicit (IMEX) scheme under the micro-macro decomposition framework and the discontinuous Galerkin (DG) method. A rigorous proof of the sAP property is given. Extensive numerical experiments that validate the accuracy and sAP of the method are shown. In the last part, we study a class of highly oscillatory transport equations that arise in semiclassical modeling of non-adiabatic quantum dynamics. These models contain uncertainties, particularly in coefficients that correspond to the potentials of the molecular system. We first focus on a highly oscillatory scalar model with random uncertainty. Our method is built upon the nonlinear geometrical optics (NGO) based method for numerical approximations of deterministic equations, which can obtain accurate pointwise solution even without numerically resolving spatially and temporally the oscillations. With the random uncertainty, we show that such a method has oscillatory higher order derivatives in the random space, thus requires a frequency dependent discretization in the random space. We modify this method by introducing a new "time" variable based on the phase, which is shown to be non-oscillatory in the random space, based on which we develop a gPC-SG method that can capture oscillations with the frequency-independent time step, mesh size as well as the degree of polynomial chaos. A similar approach is then extended to a semiclassical surface hopping model system with a similar numerical conclusion. Various numerical examples attest that these methods indeed capture accurately the solution statistics pointwisely even though none of the numerical parameters resolve the high frequencies of the solution.
Download or read book Advances in Numerical Methods for Hyperbolic Balance Laws and Related Problems written by Giacomo Albi and published by Springer Nature. This book was released on 2023-06-02 with total page 241 pages. Available in PDF, EPUB and Kindle. Book excerpt: A broad range of phenomena in science and technology can be described by non-linear partial differential equations characterized by systems of conservation laws with source terms. Well known examples are hyperbolic systems with source terms, kinetic equations, and convection-reaction-diffusion equations. This book collects research advances in numerical methods for hyperbolic balance laws and kinetic equations together with related modelling aspects. All the contributions are based on the talks of the speakers of the Young Researchers’ Conference “Numerical Aspects of Hyperbolic Balance Laws and Related Problems”, hosted at the University of Verona, Italy, in December 2021.
Download or read book Realizability preserving Discretization Strategies for Hyperbolic and Kinetic Equations with Uncertainty written by Jonas Kusch 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 Mathematical Descriptions of Traffic Flow Micro Macro and Kinetic Models written by Gabriella Puppo and published by Springer Nature. This book was released on 2021-03-31 with total page 102 pages. Available in PDF, EPUB and Kindle. Book excerpt: The book originates from the mini-symposium "Mathematical descriptions of traffic flow: micro, macro and kinetic models" organised by the editors within the ICIAM 2019 Congress held in Valencia, Spain, in July 2019. The book is composed of five chapters, which address new research lines in the mathematical modelling of vehicular traffic, at the cutting edge of contemporary research, including traffic automation by means of autonomous vehicles. The contributions span the three most representative scales of mathematical modelling: the microscopic scale of particles, the mesoscopic scale of statistical kinetic description and the macroscopic scale of partial differential equations.The work is addressed to researchers in the field.
Download or read book Uncertainty Quantification in Variational Inequalities written by Joachim Gwinner and published by CRC Press. This book was released on 2021-12-24 with total page 405 pages. Available in PDF, EPUB and Kindle. Book excerpt: Uncertainty Quantification (UQ) is an emerging and extremely active research discipline which aims to quantitatively treat any uncertainty in applied models. The primary objective of Uncertainty Quantification in Variational Inequalities: Theory, Numerics, and Applications is to present a comprehensive treatment of UQ in variational inequalities and some of its generalizations emerging from various network, economic, and engineering models. Some of the developed techniques also apply to machine learning, neural networks, and related fields. Features First book on UQ in variational inequalities emerging from various network, economic, and engineering models Completely self-contained and lucid in style Aimed for a diverse audience including applied mathematicians, engineers, economists, and professionals from academia Includes the most recent developments on the subject which so far have only been available in the research literature
Download or read book Trails in Kinetic Theory written by Giacomo Albi and published by Springer Nature. This book was released on 2021-07-15 with total page 251 pages. Available in PDF, EPUB and Kindle. Book excerpt: In recent decades, kinetic theory - originally developed as a field of mathematical physics - has emerged as one of the most prominent fields of modern mathematics. In recent years, there has been an explosion of applications of kinetic theory to other areas of research, such as biology and social sciences. This book collects lecture notes and recent advances in the field of kinetic theory of lecturers and speakers of the School “Trails in Kinetic Theory: Foundational Aspects and Numerical Methods”, hosted at Hausdorff Institute for Mathematics (HIM) of Bonn, Germany, 2019, during the Junior Trimester Program “Kinetic Theory”. Focusing on fundamental questions in both theoretical and numerical aspects, it also presents a broad view of related problems in socioeconomic sciences, pedestrian dynamics and traffic flow management.
Download or read book Recent Advances in Numerical Methods for Hyperbolic PDE Systems written by María Luz Muñoz-Ruiz and published by Springer Nature. This book was released on 2021-05-25 with total page 269 pages. Available in PDF, EPUB and Kindle. Book excerpt: The present volume contains selected papers issued from the sixth edition of the International Conference "Numerical methods for hyperbolic problems" that took place in 2019 in Málaga (Spain). NumHyp conferences, which began in 2009, focus on recent developments and new directions in the field of numerical methods for hyperbolic partial differential equations (PDEs) and their applications. The 11 chapters of the book cover several state-of-the-art numerical techniques and applications, including the design of numerical methods with good properties (well-balanced, asymptotic-preserving, high-order accurate, domain invariant preserving, uncertainty quantification, etc.), applications to models issued from different fields (Euler equations of gas dynamics, Navier-Stokes equations, multilayer shallow-water systems, ideal magnetohydrodynamics or fluid models to simulate multiphase flow, sediment transport, turbulent deflagrations, etc.), and the development of new nonlinear dispersive shallow-water models. The volume is addressed to PhD students and researchers in Applied Mathematics, Fluid Mechanics, or Engineering whose investigation focuses on or uses numerical methods for hyperbolic systems. It may also be a useful tool for practitioners who look for state-of-the-art methods for flow simulation.
Download or read book First Congress of Greek Mathematicians written by Ioannis Emmanouil and published by Walter de Gruyter GmbH & Co KG. This book was released on 2020-03-23 with total page 270 pages. Available in PDF, EPUB and Kindle. Book excerpt: The series is aimed specifically at publishing peer reviewed reviews and contributions presented at workshops and conferences. Each volume is associated with a particular conference, symposium or workshop. These events cover various topics within pure and applied mathematics and provide up-to-date coverage of new developments, methods and applications.
Download or read book Theory Numerics and Applications of Hyperbolic Problems II written by Christian Klingenberg and published by Springer. This book was released on 2018-06-27 with total page 698 pages. Available in PDF, EPUB and Kindle. Book excerpt: The second of two volumes, this edited proceedings book features research presented at the XVI International Conference on Hyperbolic Problems held in Aachen, Germany in summer 2016. It focuses on the theoretical, applied, and computational aspects of hyperbolic partial differential equations (systems of hyperbolic conservation laws, wave equations, etc.) and of related mathematical models (PDEs of mixed type, kinetic equations, nonlocal or/and discrete models) found in the field of applied sciences.
Download or read book Predicting Pandemics in a Globally Connected World Volume 1 written by Nicola Bellomo and published by Springer Nature. This book was released on 2022-09-22 with total page 314 pages. Available in PDF, EPUB and Kindle. Book excerpt: This contributed volume investigates several mathematical techniques for the modeling and simulation of viral pandemics, with a special focus on COVID-19. Modeling a pandemic requires an interdisciplinary approach with other fields such as epidemiology, virology, immunology, and biology in general. Spatial dynamics and interactions are also important features to be considered, and a multiscale framework is needed at the level of individuals and the level of virus particles and the immune system. Chapters in this volume address these items, as well as offer perspectives for the future.
Download or read book Handbook of Numerical Methods for Hyperbolic Problems written by Remi Abgrall and published by Elsevier. This book was released on 2017-01-16 with total page 612 pages. Available in PDF, EPUB and Kindle. Book excerpt: Handbook on Numerical Methods for Hyperbolic Problems: Applied and Modern Issues details the large amount of literature in the design, analysis, and application of various numerical algorithms for solving hyperbolic equations that has been produced in the last several decades. This volume provides concise summaries from experts in different types of algorithms, so that readers can find a variety of algorithms under different situations and become familiar with their relative advantages and limitations. - Provides detailed, cutting-edge background explanations of existing algorithms and their analysis - Presents a method of different algorithms for specific applications and the relative advantages and limitations of different algorithms for engineers or those involved in applications - Written by leading subject experts in each field, the volumes provide breadth and depth of content coverage
Download or read book The Cauchy Problem in Kinetic Theory written by Robert T. Glassey and published by SIAM. This book was released on 1996-01-01 with total page 246 pages. Available in PDF, EPUB and Kindle. Book excerpt: Studies the basic equations of kinetic theory in all of space, and contains up-to-date, state-of-the-art treatments of initial-value problems for the major kinetic equations. This is the only existing book to treat Boltzmann-type problems and Vlasov-type problems together. Although describing very different phenomena, these equations share the same streaming term.
Download or read book Quantification of Uncertainty Improving Efficiency and Technology written by Marta D'Elia and published by Springer. This book was released on 2021-08-01 with total page 282 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 Uncertainty Quantification with Shallow Water Equations written by Dishi Liu and published by . This book was released on 2010 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt:
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.
