Successful modeling and analysis of dynamical systems within the natural sciences depend, as a first step, on a process of acquiring data from observations, tailored to the attributes in which one is interested. This process necessarily results with a representation of the system. It is therefore natural to ask how to acquire/represent a given system optimally from the point of view of simplicity, precision, efficiency, reconstructability, storage constraints or other parameters depending on the exact circumstances. We will present several problems arising in connection with these considerations.

Let us illustrate one of the problems. A renowned approach to the
problem of optimal acquiring and reconstruction of a sparse (=few non
zero entries) data vector $x\in \mathbb{R}^{N}$ is the technique of
*compressed sensing* developed by Candès, Donoho, Romberg and
Tao. Assuming the vector $x$ is *$s$-sparse*, it is shown that
applying a random matrix $A\in\mathbb{R}^{m\times N}$ with $m\approx
s\ln(\frac{N}{s})\ll N$, one is able to reconstruct $x$ from $y=A x$
efficiently with high probability. We are concerned with extending the
compressed sensing framework to the realistic scenario where the data
is given by an analog signal represented (as the result of sampling)
by a continuous-alphabet discrete-time stochastic process $\ldots,
x_{-1},x_0,x_1,\ldots$.
The main problem is finding *universal* algorithms in various
classes of stochastic processes. A universal algorithm is an algorithm
which performs \textit{optimally} despite not having prior knowledge
of the exact statistics of the process being compressed.