The aim of this book is to provide methods and algorithms for the optimization of input signals so as to estimate parameters in systems described by PDEs as accurate as possible under given constraints. The optimality conditions have their background in the optimal experiment design theory for regression functions and in simple but useful results on the dependence of eigenvalues of partial differential operators on their parameters. Examples are provided that reveal sometimes intriguing geometry of spatiotemporal input signals and responses to them.
An introduction to optimal experimental design for parameter estimation of regression functions is provided. The emphasis is on functions having a tensor product (Kronecker) structure that is compatible with eigenfunctions of many partial differential operators. New optimality conditions in the time domain and computational algorithms are derived for D-optimal input signals when parameters of ordinary differential equations are estimated. They are used as building blocks for constructing D-optimal spatiotemporal inputs for systems described by linear partial differential equations of the parabolic and hyperbolic types with constant parameters. Optimality conditions for spatially distributed signals are also obtained for equations of elliptic type in those cases where their eigenfunctions do not depend on unknown constant parameters. These conditions and the resulting algorithms are interesting in their own right and, moreover, they are second building blocks for optimality of spatiotemporal signals. A discussion of the generalizability and possible applications of the results obtained is presented. Contents
Frontmatter, Preface, Acknowledgements, Contents, List of Tables, List of Figures, Frequently Used Acronyms, Introduction, Optimal Experiment Design for Linear Regression Models—A Primer, Numerical Search for D-Optimal Designs—Basic Methods for a Linear Regression, The Product of Experiment Designs and Its Optimality, Optimal Excitations for Systems Described by Elliptic Equations, Optimal Input Signals for DPS—Time Domain Synthesis, Input Signal Design for Systems with Spatiotemporal Dynamics—Frequency Domain Approach, Final Comments, Bibliography, Index Related Topics