In this paper, we propose a method for the approximation of the solution ofhigh-dimensional weakly coercive problems formulated in tensor spaces using low-rankapproximation formats. The method can be seen as a perturbation of a minimal residualmethod with a measure of the residual corresponding to the error in a specified solutionnorm. The residual norm can be designed such that the resulting low-rank approximationsare optimal with respect to particular norms of interest, thus allowing to take intoaccount a particular objective in the definition of reduced order approximations ofhigh-dimensional problems. We introduce and analyze an iterative algorithm that is able toprovide an approximation of the optimal approximation of the solution in a given low-ranksubset, without any a priori information on this solution. We alsointroduce a weak greedy algorithm which uses this perturbed minimal residual method forthe computation of successive greedy corrections in small tensor subsets. We prove itsconvergence under some conditions on the parameters of the algorithm. The proposednumerical method is applied to the solution of a stochastic partial differential equationwhich is discretized using standard Galerkin methods in tensor product spaces.

Approximation theory in the context of probability density function turns out to go beyond the classical idea of orthogonal projection. Special tools have to be designed so as to respect the nonnegativity of the approximate function. We develop here and justify from the theoretical point of view an approximation procedure introduced by Levermore [Levermore, J. Stat. Phys.83 (1996) 1021–1065] and based on an entropy minimization principle under moment constraints. We prove in particular a global existence theorem for such an approximation and derive as a by-product a necessary and sufficient condition for the so-called problem of moment realizability. Applications of the above result are given in kinetic theory: first in the context of Levermore's approach and second to design generalized BGK models for Maxwellian molecules.