1 Introduction to model reduction - University of （1介绍模型减少大学）.pdf

SIGGRAPH 2012 Course Notes FEM Simulation of 3D Deformable Solids: A practitioner’s guide to theory, discretization and model reduction. Part 2: Model Reduction (version: August 4, 2012) ˇ Jernej Barbic Course notes URL: 1 Introduction to model reduction Figure 1: Model reduction overview: a high-dimensional ordinary differential equation is approximated with a projection to a low-dimensional space. Model reduction (also called dimensional model reduction, or model order reduction (MOR)) is a tech- nique to simplify the simulation of dynamical systems described by differential equations. The idea is to project the original, high-dimensional, state-space onto a properly chosen low-dimensional subspace to arrive at a (much) smaller system having properties similar to the original system (see Figure 1). Complex systems can thus be approximated by simpler systems involving fewer equations and unknown variables, which can be solved much more quickly than the original problem. Such projection-based model reduction appears in literature under the names of Principal Orthogonal Directions (POD) Method, or Subspace Integra- tion Method, and it has a long history in the engineering and applied mathematics literature [29]. See [27] and [33] for good overviews of model reduction applied to linear and nonlinear problems, respectively. Model reduction has been used extensively in the fields of control theory, electrical circuit simulation, computational electromagnetics and microelectromechanical systems [28]. Most model reduction tech- niques in these fields, however, aim at linear systems, and linear time-invariant systems in particular, e.g., small perturbations of voltages in