  Gelfand numbers represent a measure for the information complexity which is
given by the number of information needed to approximate functions in a subset
of a normed space with an error less than $\varepsilon$. More precisely,
Gelfand numbers coincide up to the factor 2 with the minimal error $ e^{\rm
wor}(n,\Lambda^{\rm all})$ which describes the error of the optimal
(non-linear) algorithm that is based on $n$ arbitrary linear functionals. This
explain the crucial role of Gelfand numbers in the study of approximation
problems. Let $S^t_{p_1,p_1}B((0,1)^d)$ be the Besov spaces with dominating
mixed smoothness on $(0,1)^d$. In this paper we consider the problem ${\rm
App}: S^t_{p_1,p_1}B((0,1)^d) \to L_{p_2}((0,1)^d)$ and investigate the
asymptotic behaviour of Gelfand numbers of this embedding. We shall give the
correct order of convergence of Gelfand numbers in almost all cases. In
addition we shall compare these results with the known behaviour of
approximation numbers which coincide with $ e^{\rm wor-lin}(n,\Lambda^{\rm
all})$ when we only allow linear algorithms.
