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The book integrates theoretical analysis, numerical simulation and modeling approaches for the treatment of singular phenomena. The projects covered focus on actual applied problems, and develop qualitatively new and mathematically challenging methods for various problems from the natural sciences. Ranging from stochastic and geometric analysis over nonlinear analysis and modelling to numerical analysis and scientific computation, the book is divided into the three sections: A) Scaling limits of diffusion processes and singular spaces, B) Multiple scales in mathematical models of materials science and biology and C) Numerics for multiscale models and singular phenomena. Each section addresses the key aspects of multiple scales and model hierarchies, singularities and degeneracies, and scaling laws and self-similarity.

Mathematics --- Mathematics --- Computer science --- Computer. Automation --- computers --- informatica --- externe fixatie (geneeskunde --- wiskunde --- wiskunde --- informaticaonderzoek --- computerkunde

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In the recent decade, there has been a growing interest in the numerical treatment of high-dimensional problems. It is well known that classical numerical discretization schemes fail in more than three or four dimensions due to the curse of dimensionality. The technique of sparse grids helps overcome this problem to some extent under suitable regularity assumptions. This discretization approach is obtained from a multi-scale basis by a tensor product construction and subsequent truncation of the resulting multiresolution series expansion. This volume of LNCSE is a collection of the papers from the proceedings of the workshop on sparse grids and its applications held in Bonn in May 2011. The selected articles present recent advances in the mathematical understanding and analysis of sparse grid discretization. Aspects arising from applications are given particular attention.    .

Business mathematics. --- Calculus, Integral. --- Mathematics. --- Numerical analysis. --- Computational grids (Computer systems) --- Mathematics --- Engineering & Applied Sciences --- Physical Sciences & Mathematics --- Mathematics - General --- Computer Science --- Numerical grid generation (Numerical analysis) --- Coordinate generation, Numerical (Numerical analysis) --- Generation of numerical grids (Numerical analysis) --- Grid generation, Numerical (Numerical analysis) --- Mesh generation, Numerical (Numerical analysis) --- Numerical coordinate generation (Numerical analysis) --- Numerical mesh generation (Numerical analysis) --- Sparse grids. --- Computer science --- Computer mathematics. --- Computational Mathematics and Numerical Analysis. --- Computational Science and Engineering. --- Mathematics of Computing. --- Boundary value problems --- Differential equations, Partial --- Nets (Mathematics) --- Numerical analysis --- Numerical solutions --- Computer science. --- Informatics --- Science --- Computer mathematics --- Discrete mathematics --- Electronic data processing --- Computer science—Mathematics.

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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.

Mathematics. --- Computational Mathematics and Numerical Analysis. --- Appl.Mathematics/Computational Methods of Engineering. --- Numeric Computing. --- Electronic data processing. --- Computer science --- Engineering mathematics. --- Mathématiques --- Informatique --- Mathématiques de l'ingénieur --- Differential equations, Partial -- Numerical solutions -- Congresses. --- Meshfree methods (Numerical analysis) -- Congresses. --- Numerical grid generation (Numerical analysis) -- Congresses. --- Mathematics --- Physical Sciences & Mathematics --- Mathematics - General --- Differential equations, Partial --- Meshfree methods (Numerical analysis) --- Gridless methods (Numerical analysis) --- Meshfree discretization techniques (Numerical analysis) --- Meshless methods (Numerical analysis) --- Numerical analysis --- Numerical solutions --- Conferences - Meetings --- Differential equations, Partial. --- Partial differential equations --- Numerical analysis. --- Computer mathematics. --- Applied mathematics. --- Engineering --- Engineering analysis --- Mathematical analysis --- Computer mathematics --- Discrete mathematics --- Electronic data processing --- Math --- Science --- Mathematical and Computational Engineering. --- ADP (Data processing) --- Automatic data processing --- Data processing --- EDP (Data processing) --- IDP (Data processing) --- Integrated data processing --- Computers --- Office practice --- Automation

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There have been substantial developments in meshfree methods, particle methods, and generalized finite element methods since the mid 1990s. The growing interest in these methods is in part due to the fact that they offer 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 have a number of advantageous features that are especially attractive when dealing with multiscale phenomena: A-priori knowledge about the solution’s particular local behavior can easily be introduced into the meshfree approximation space, and coarse scale approximations can be seamlessly refined by adding fine scale information. However, the implementation of meshfree methods and their parallelization also requires special attention, for instance with respect to numerical integration.

Mathematics. --- Numerical analysis. --- Computer simulation. --- Computer-aided engineering. --- Computer mathematics. --- Mathematical models. --- Computational Science and Engineering. --- Computer-Aided Engineering (CAD, CAE) and Design. --- Numeric Computing. --- Simulation and Modeling. --- Mathematical Modeling and Industrial Mathematics. --- Differential equations, Partial --- Meshfree methods (Numerical analysis) --- Numerical solutions --- Gridless methods (Numerical analysis) --- Meshfree discretization techniques (Numerical analysis) --- Meshless methods (Numerical analysis) --- Numerical analysis --- Computer science. --- Computer aided design. --- Electronic data processing. --- Computer modeling --- Computer models --- Modeling, Computer --- Models, Computer --- Simulation, Computer --- Electromechanical analogies --- Mathematical models --- Simulation methods --- Model-integrated computing --- CAD (Computer-aided design) --- Computer-assisted design --- Computer-aided engineering --- Design --- Informatics --- Science --- ADP (Data processing) --- Automatic data processing --- Data processing --- EDP (Data processing) --- IDP (Data processing) --- Integrated data processing --- Computers --- Office practice --- Automation --- Models, Mathematical --- Mathematical analysis --- CAE --- Engineering --- Computer mathematics --- Electronic data processing --- Mathematics