  Understanding the dynamics of complex systems is a central task in many
different areas ranging form biology via epidemics to economics and
engineering. Unexpected behaviour of dynamic systems or even systems failure is
sometimes difficult to comprehend. Such unexpected dynamics can be caused by
systematic model errors, unknown inputs from the environment and systems
faults. Localizing the root cause of these errors or faults and reconstructing
their dynamics is only possible if the measured outputs of the system are
sufficiently informative. Here, we present a mathematical theory for the
measurements required to localize the position of error sources in large
dynamic networks. We assume, that faults or errors occur at a limited number of
positions in the network. This sparsity assumption facilitates the accurate
reconstruction of the dynamic timecourses of the errors by solving a convex
optimal control problem. For cases, where the sensor measurements are not
sufficiently informative to pinpoint the error position exactly, we provide
methods to restrict the error location to a smaller subset of network nodes. We
also suggest strategies to efficiently select additional measurements for
narrowing down the error location.
