Mathematical foundation of Information Field Dynamics (revised version)

Abstract

Information Field Dynamics (IFD) by Torsten En{\ss}lin provides a tool to construct simulation schemes for data vectors $d(T)$ from measurements $d(0)$ which describe certain features of a physical process (signal), without any concrete assumptions about the subgrid structure of the problem. In this work, the measure theoretical fundament and the necessary probabilistic framework for IFD are introduced first. The notation for the rest of the work is established and essential properties are developed. Afterwards, the general setting for IFD is described, consisting of the signal with its evolution equation and its linear connection to the data by a response operator. The developed environment is then used, to step by step imbed the physical language, used in Torsten En{\ss}lin's work, into a mathematical framework. Also a general approach for the approximation of an evolution equation is given, serving as a base to describe the construction of a simulation scheme with IFD. Simulation errors in the various steps are pointed out, to give an idea where inaccuracies come in and which steps therefore lead to the necessity of many degrees of freedom of the signal and of many time steps within the simulation. In the end, IFD is illustrated in an example scenario. A matrix relation for the update steps of IFD is derived. It allows to simulate a data vector $d(T)$ which averages a Klein-Gordon field with one dimension in space and periodic over $[0,2\pi)$. Also a non-iterative equation for the direct computation of $d(T)$ from $d(0)$ is constructed. This is reached by the fact that the original problem converts into an ordinary differential equation for the data, if the simulation time steps get infinitesimally small.