In this chapter the data assimilation problem is introduced as a control theory problem for partial differential equations, with initial conditions, model error, and empirical model parameters as optional control variables. An alternative interpretation of data assimilation as a processing of information in a dynamic-stochastic system is also introduced. Both approaches will be addressed in more detail throughout this book. The historical development of data assimilation has been documented, starting from the early nineteenth-century works by Legendre, Gauss, and Laplace, to optimal interpolation and Kalman filtering, to modern data assimilation based on variational and ensemble methods, and finally to future methods such as particle filters. This suggests that data assimilation is not a very new concept, given that it has been of scientific and practical interest for a long time. Part of the chapter focuses on introducing the common terminologies and notation used in data assimilation, with special emphasis on observation equation, observation errors, and observation operators. Finally, a basic linear estimation problem based on least squares is presented.

To get predictions from theoretical models of complex mechanical systems, the numerical tools are essential, as very few results can be obtained using analytical methods, especially when large deformations are involved. Variational methods are the preferred (or probably the most powerful) tool to formulate the numerical codes to be used, also in the study of metamaterials. A presentation focused on some aspects of numerical techniques, relevant to the considered class of problems, is presented.