  In this paper, we propose a space-time least-squares isogeometric method to
solve parabolic evolution problems, well suited for high-degree smooth splines
in the space-time domain. We focus on the linear solver and its computational
efficiency: thanks to the proposed formulation and to the tensor-product
construction of space-time splines, we can design a preconditioner whose
application requires the solution of a Sylvester-like equation, which is
performed efficiently by the fast diagonalization method. The preconditioner is
robust w.r.t. spline degree and mesh size. The computational time required for
its application, for a serial execution, is almost proportional to the number
of degrees-of-freedom and independent of the polynomial degree. The proposed
approach is also well-suited for parallelization.
