Interpolation methods are automatic techniques for reducing the size and other complexities of large complicated models. Modeling physical systems using differential equations and geometrical properties has been popular among scientists and engineers for the last three centuries. Using these models, computer simulations are done to resolve related problems. Some of these models pose high computational complexity problems. Interpolation techniques can be used to reduce such computational complexities. The book presents interpolatory techniques for both linear and nonlinear systems.
The book is organized into four parts spanning ten chapters. The first two parts cover the basics. Chapters 1 and 2 make up Part 1. Chapter 1 has some engineering problems as motivating examples. Chapter 2 introduces linear and nonlinear systems through equations. It gives an overview of transformations, for example, Laplace transformations and associated norms.
Part 2, comprising chapters 3, 4, and 5, presents a framework for interpolatory model reduction. The framework is data driven, that is, at a particular time, the response of the reduction is based on data parameters at that time. Tangential interpolation is used on the interpolation side. Techniques for projecting various data are used in the framework. The framework also has algorithms for interpolation in various types of systems. Further, the conditions for when to use interpolation for optimality are described. The iterative rational Krylov algorithm (IRKA) is given. In these chapters, the systems under consideration are linear.
In Part 3, chapter 6 is on parametric models and chapter 7 is on bilinear systems. For parametric systems, greedy parameter sampling is suggested to address the difficulty of representing parameters. The interpolation algorithm for parametric models uses frequency grid, parameter grid, reduction of orders, and partition of measurements. For bilinear systems, the entire framework described in Part 2 is adopted. The changes in equations and algorithms from linear to bilinear systems are described in detail.
Part 4 covers “advanced topics.” Chapter 8 presents some variations in the norms related to error approximations. It gives two numerical examples. Chapter 9 is on interpolatory reduction methods for differential algebraic systems. The chapter notes that the differential equations involving singularities are resolved through algebraic systems. It not only discusses the necessary equation changes for interpolatory reductions, but also discusses related computational aspects. The last chapter deals with inexact iterative solvers. It addresses how the interpolatory methods of the framework can be used for iterative solvers. The appendix covers some computational issues.
The authors clearly state that the book is not intended to be used as a textbook. Instead, it is a research monograph that will help research students begin their work. A background in basic mathematics, differential equation modeling, linear algebra, numerical analysis, and algorithms is needed to understand the material. Professionals working in optimization will find the book useful, especially when encountering models with large matrices. Performance analysis studies of the framework and techniques appear to be beyond the scope of the book.