This book reports recent mathematical developments in the Programme "Analysis, Modeling and Simulation of Multiscale Problems", which started as a German research initiative in 2006. Multiscale problems occur in many fields of science, such as microstructures in materials, sharp-interface models, many-particle systems and motions on different spatial and temporal scales in quantum mechanics or in molecular dynamics. The book presents current mathematical foundations of modeling, and proposes efficient numerical treatment.

Exponential Estimates in Averaging and Homogenisation.- Multiscale Problems in Solidification Processes.- Multiscale Modeling of Epitaxial Growth: From Discrete-Continuum to Continuum Equations.- On Asymptotic Limits of Cahn-Hilliard Systems with Elastic Misfit.- Simulations of Complex Microstructure Formations.- Multiple Scales in Phase Separating Systems with Elastic Misfit.- A General Theory for Elastic Phase Transitions in Crystals.- Relaxation and the Computation of Effective Energies and Microstructures in Solid Mechanics.- Derivation of Elastic Theories for Thin Sheets and the Constraint of Incompressibility.- Domain Walls and Vortices in Thin Ferromagnetic Films.- Wavelet-Based Multiscale Methods for Electronic Structure Calculations.- Multi-Scale Modeling of Quantum Semiconductor Devices.- Electronic States in Semiconductor Nanostructures and Upscaling to Semi-Classical Models.- Shape Optimization of Biomorphic Ceramics with Microstructures by Homogenization Modeling.- Discrete Free Energy Functionals for Elastic Materials with Phase Change.- Continuum Descriptions for the Dynamics in Discrete Lattices: Derivation and Justification.- Wave Trains, Solitons and Modulation Theory in FPU Chains.- On the Effect of Correlations, Fluctuations and Collisions in Ostwald Ripening.- Radiative Friction for Charges Interacting with the Radiation Field: Classical Many-Particle Systems.- Numerical Integrators for Highly Oscillatory Hamiltonian Systems: A Review.- Energy Level Crossings in Molecular Dynamics.- Motion of Electrons in Adiabatically Perturbed Periodic Structures.- Graph Algorithms for Dynamical Systems.- Conditional Averaging for Diffusive Fast-Slow Systems: A Sketch for Derivation.