  We consider the numerical solution of parameterized linear systems where the
system matrix, the solution, and the right-hand side are parameterized by a set
of uncertain input parameters. We explore spectral methods in which the
solutions are approximated in a chosen finite-dimensional subspace. It has been
shown that the stochastic Galerkin projection technique fails to minimize any
measure of the solution error [20]. As a remedy for this, we propose a novel
stochastic least-squares Petrov--Galerkin (LSPG) method. The proposed method is
optimal in the sense that it produces the solution that minimizes a weighted
l2-norm of the residual over all solutions in a given finite-dimensional
subspace. Moreover, the method can be adapted to minimize the solution error in
different weighted l2-norms by simply applying a weighting function within the
least-squares formulation. In addition, a goal-oriented semi-norm induced by an
output quantity of interest can be minimized by defining a weighting function
as a linear functional of the solution. We establish optimality and error
bounds for the proposed method, and extensive numerical experiments show that
the weighted LSPG methods outperforms other spectral methods in minimizing
corresponding target weighted norms.
