Local-Global Model Reduction Techniques Using Balanced Truncation

Abstract

Many applications such as porous media and material science possess multiscale nature of media properties and high number of variables. As a result, forward problems on today’s computer architectures require high cost of computations. The dissertation is devoted to creating novel model reduction techniques in order to carry out these prohibitively expensive computations.

Model reduction techniques can be divided into two categories, local reduced-order modeling techniques and global reduced-order modeling techniques. Local reduced-order modeling techniques include many multiscale and homogenization type methods. Multiscale techniques provide significant computational savings since the same multiscale basis functions are used for all forward simulations. While homogenization methods allow reducing cost of computations by solving cell problems with varying order of accuracy. As for global model reduction technique, we consider the earlier approach called Balanced Truncation (BT), where the system is written in terms of a mapping from input to output. In the dissertation local model reduction techniques and the global model reduction technique are combined.

Local-global model reduction techniques are designed for different problem settings. First, we examine the flow in porous media with separable scales. We employ hierarchical approaches for solving local problems. Then the obtained coarse-grid models are coupled with BT approach.

Next problem formulation describes the flow in porous media without scale separation. Two cases of media properties are considered: general heterogeneous media with a parameter and a time-varying heterogeneous media, where the media properties depend on time. For these type of problems we use appropriate form of BT and combine it with the offline - online GMsFEM procedure.

Finally, we consider a coupled flow and transport problem, where the transport equation is convection dominated. Both flow and transport equations are discretized on a coarse grid using GMsFEM. We bring together the mixed coarse-grid discretization of convection-diffusion equation and BT approaches to obtain an accurate local-global model reduction.