Operator Equations , Multiscale Concepts and Complexity

Adaptively Compressed Polarizability Operator for Accelerating Large Scale <i>Ab Initio</i> Phonon Calculations | Multiscale Modeling & Simulation | Vol. 15, No. 1 | Society for Industrial and Applied Mathematics

Phonon calculations based on first principle electronic structure theory, such as the Kohn–Sham density functional theory, have wide applications in physics, chemistry, and material science. The computational cost of first principle phonon calculations typically scales steeply as O(N4 e ), where Ne is the number of electrons in the system. In this work, we develop a new method for reducing the ...

Partial differential equations and the finite element method

Partial Differential Equations and the Finite Element Method provides a much–needed, clear, and systematic introduction to modern theory of partial differential equations (PDEs) and finite element methods (FEM). Both nodal and hierachic concepts of the FEM are examined. Reflecting the growing complexity and multiscale nature of current engineering and scientific problems, the author emphasizes ...

Operator Based Multiscale Method for Compressible Flow

Accurate simulation of subsurface flow with detailed geologic description is of great academic and industrial interest. Fully fine-scale simulation is usually too expensive. The multiscale method is developed to capture fine-scale information without solving fine-scale equations. It is more efficient than fine-scale simulation methods and more accurate than traditional upscaling techniques. Pre...

Particle Systems and Kinetic Equations Modeling Interacting Agents in High Dimension

In this paper we explore how concepts of high-dimensional data compression via random projections onto lower-dimensional spaces can be applied for tractable simulation of certain dynamical systems modeling complex interactions. In such systems, one has to deal with a large number of agents (typically millions) in spaces of parameters describing each agent of high dimension (thousands or more). ...

Wavelet and Multiscale Methods for Operator Equations *