Outstanding Issues in Model Order Reduction

Abstract

With roots dating back many years and applications in a wide variety of areas, model order reduction has emerged in the last few decades as a crucial step in the simulation, control, and optimization of complex physical processes. Reducing the order or dimension of models of such systems, is paramount to enabling their simulation and verification. While much progress has been achieved in the last few years regarding the robustness, efficiency and applicability of these techniques, certain problems of relevance still pose difficulties or renewed challenges that are not satisfactorily solved with the existing approaches. Furthermore, new applications for which dimension reduction is crucial, are becoming increasingly important, raising new issues in our quest for increased performance. This talk is aimed at presenting an overview of some of those issues and at sparking renewed discussion about how to tackle them. I. Basics of MOR Model reduction algorithms are now standard techniques in many areas, including the integrated circuits community. Such techniques are commonly used for approximation and analysis of models arising from interconnect and electromagnetic structure analysis. The need for higher accuracy while accounting for all relevant physical effects, implies that the mathematical formulations used to describe and model such systems often result in very large scale systems. Reducing the order or dimension of such models is paramount to enabling the simulation and verification of these systems. In recent years the need for reduction techniques for large-scale systems has triggered a revival of research activities in model order reduction. Initial interest in model reduction techniques stemmed from efforts to accelerate analysis of circuit interconnect. More recently, however, model reduction has come to be viewed as a method for generating compact models from all sorts of physical system modeling tools. Such techniques are now routinely used to generate lumped-circuit approximations of distributed electronic circuit elements, such as the interconnect or package of VLSI chips, or in simulations of microelectromechanical systems (MEMS), which have both electrical and mechanical components. While enormous progress has been achieved both from a theoretical as well as practical standpoint, still greater challenges lie ahead as new and exciting applications are being researched for which order reduction is again a crucial step. Model order reductions seeks to replace a large-scale models of a physical system by a system of lower dimension which exhibits similar behavior, typically measured in terms of its frequency or time response characteristics. Existing methods for model reduction can be broadly characterized into two types: those that are based on projection methods, and those based on balancing techniques. Among the first, Krylov subspace projection methods such as PVL [1] and PRIMA [2] have been the most widely studied over the past decade. They are very appealing due mostly to their simplicity and their strong performance in terms of efficiency and accuracy. However, Krylov projection methods are known to have a few drawbacks in practical application. First, there is no general agreement on how to control error in these methods. Error estimators do exist for some methods but they are seldom used in practice as they require additional computations, which can be expensive and awkward to implement. Second, moment based methods such as PRIMA, are known in some cases to produce models that are "too high" in order with the obvious consequences in terms of analysis or simulation cost [3], [4]. Third, any guarantees on retaining relevant properties of the underlying system, such as passivity of the models, are dependent on the structure of the system representation, a non-issue for RLC-type models, but a troublesome problem for other types of models. An alternate class of model reduction schemes are the truncated balanced realization (TBR) family [5]. These are purported to produce "nearly optimal" models and have easy to compute a-posteriori error bounds. However, they are expensive to apply, as they require the computation of the system Gramians by solving a large Lyapunov or Sylvester-type equation. This is an expensive procedure, which limits their applicability to small to medium sized systems. Balanced realization-based techniques are also awkward to implement correctly and Both types of methods have carved their niche in specific segments or applications. Furthermore hybrid techniques that combine some of the features of each type of method have also been presented. Examples of these include the solution of the large Lyapunov equations via a Krylov subspace method [6], [7]. Other approaches include a two-step method where an initial reduced model is first obtained via projection and then further compressed using a TBR method [3]. Recently a new technique has been proposed that attempts to establish a connection between