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During recent years a great deal of interest has been devoted to large scale computing applications. The book contains survey articles as well as chapters on specific research applications, development and analysis of numerical algorithms, and performance evaluation of algorithms on advanced architectures.
Mathematical models are used to simulate, and sometimes control, the behavior of physical and artificial processes such as the weather and very large-scale integration (VLSI) circuits. The increasing need for accuracy has led to the development of highly complex models. However, in the presence of limited computational accuracy and storage capabilities model reduction (system approximation) is often necessary. Approximation of Large-Scale Dynamical Systems provides a comprehensive picture of model reduction, combining system theory with numerical linear algebra and computational considerations. It addresses the issue of model reduction and the resulting trade-offs between accuracy and complexity. Special attention is given to numerical aspects, simulation questions, and practical applications.
This book is thesecond volume of a three volume series recording the "Radon Special Semester 2011 on Multiscale Simulation & Analysis in Energy and the Environment" that took placein Linz, Austria, October 3-7, 2011. This volume addresses the common ground in the mathematical and computational procedures required for large-scale inverse problems and data assimilation in forefront applications. The solution of inverse problems is fundamental to a wide variety of applications such as weather forecasting, medical tomography, and oil exploration. Regularisation techniques are needed to ensure solutions of sufficient quality to be useful, and soundly theoretically based. This book addresses the common techniques required for all the applications, and is thus truly interdisciplinary. Thiscollection of surveyarticlesfocusses onthe large inverse problems commonly arising in simulation and forecasting in the earth sciences. For example, operational weather forecasting models have between 107 and 108 degrees of freedom. Even so, these degrees of freedom represent grossly space-time averaged properties of the atmosphere. Accurate forecasts require accurate initial conditions. With recent developments in satellite data, there are between 106 and 107 observations each day. However, while these also represent space-time averaged properties, the averaging implicit in the measurements is quite different from that used in the models. In atmosphere and ocean applications, there is a physically-based model available which can be used to regularise the problem. We assume that there is a set of observations with known error characteristics available over a period of time. The basic deterministic technique is to fit a model trajectory to the observations over a period of time to within the observation error. Since the model is not perfect the model trajectory has to be corrected, which defines the data assimilation problem. The stochastic view can be expressed by using an ensemble of model trajectories, and calculating corrections to both the mean value and the spread which allow the observations to be fitted by each ensemble member. In other areas of earth science, only the structure of the model formulation itself is known and the aim is to use the past observation history to determine the unknown model parameters. The book records the achievements of Workshop2 "Large-Scale Inverse Problems and Applications in the Earth Sciences". Itinvolves experts in the theory of inverse problems together with experts working on both theoretical and practical aspects of the techniques by which large inverse problems arise in the earth sciences.
These days, computer-based simulation is considered the quintessential approach to exploring new ideas in the different disciplines of science, engineering and technology (SET). To perform simulations, a physical system needs to be modeled using mathematics; these models are often represented by linear time-invariant (LTI) continuous-time (CT) systems. Oftentimes these systems are subject to additional algebraic constraints, leading to first- or second-order differential-algebraic equations (DAEs), otherwise known as descriptor systems. Such large-scale systems generally lead to massive memory requirements and enormous computational complexity, thus restricting frequent simulations, which are required by many applications. To resolve these complexities, the higher-dimensional system may be approximated by a substantially lower-dimensional one through model order reduction (MOR) techniques. Computational Methods for Approximation of Large-Scale Dynamical Systems discusses computational techniques for the MOR of large-scale sparse LTI CT systems. Although the book puts emphasis on the MOR of descriptor systems, it begins by showing and comparing the various MOR techniques for standard systems. The book also discusses the low-rank alternating direction implicit (LR-ADI) iteration and the issues related to solving the Lyapunov equation of large-scale sparse LTI systems to compute the low-rank Gramian factors, which are important components for implementing the Gramian-based MOR. Although this book is primarly aimed at post-graduate students and researchers of the various SET disciplines, the basic contents of this book can be supplemental to the advanced bachelor's-level students as well. It can also serve as an invaluable reference to researchers working in academics and industries alike. Features: Provides an up-to-date, step-by-step guide for its readers. Each chapter develops theories and provides necessary algorithms, worked examples, numerical experiments and related exercises. With the combination of this book and its supplementary materials, the reader gains a sound understanding of the topic. The MATLAB® codes for some selected algorithms are provided in the book. The solutions to the exercise problems, experiment data sets and a digital copy of the software are provided on the book's website; The numerical experiments use real-world data sets obtained from industries and research institutes.
This edited volume provides insights into and tools for the modeling, analysis, optimization, and control of large-scale networks in the life sciences and in engineering. Large-scale systems are often the result of networked interactions between a large number of subsystems, and their analysis and control are becoming increasingly important. The chapters of this book present the basic concepts and theoretical foundations of network theory and discuss its applications in different scientific areas such as biochemical reactions, chemical production processes, systems biology, electrical circuits, and mobile agents. The aim is to identify common concepts, to understand the underlying mathematical ideas, and to inspire discussions across the borders of the various disciplines. The book originates from the interdisciplinary summer school “Large Scale Networks in Engineering and Life Sciences” hosted by the International Max Planck Research School Magdeburg, September 26-30, 2011, and will therefore be of interest to mathematicians, engineers, physicists, biologists, chemists, and anyone involved in the network sciences. In particular, due to their introductory nature the chapters can serve individually or as a whole as the basis of graduate courses and seminars, future summer schools, or as reference material for practitioners in the network sciences.
The LES-method is rapidly developing in many practical applications in engineering The mathematical background is presented here for the first time in book form by one of the leaders in the field
Despite the large research effort in both public and commercial companies, no textbook has yet been written on this subject. This book aims to provide an overview to the topic of Carbon Capture and Storage (CSS), while at the same time focusing on the dominant processes and the mathematical and numerical methods that need to be employed in order to analyze the relevant systems. The book clearly states the carbon problem and the role of CCS and carbon storage. Thereafter, it provides an introduction to single phase and multi-phase flow in porous media, including some of the most common mathematical analysis and an overview of numerical methods for the equations. A considerable part of the book discusses the appropriate scales of modeling, and how to formulate consistent governing equations at these scales. The book also illustrates real world data sets and how the ideas in the book can be exploited through combinations of analytical and numerical approaches.