Hierarchical preconditioning for the stochastic Galerkin method: Upper bounds to the strengthened CBS constants

One of the popular methods for numerical solution of partial differential equations with uncertain data is the stochastic Galerkin method, where function spaces used for discretized problems are tensor products of finite element spaces of spatial variables and of sets of polynomials of random variables. Related systems of linear equations are thus usually huge. Studying relations between subspaces of the solution spaces is important for obtaining efficient preconditioning. For a hierarchy of polynomials of random variables we introduce upper bounds to the strengthened Cauchy–Bunyakowsky–Schwarz (CBS) constants with respect to the scalar product defined by the operator of the weak form of the underlying equation. Small values of the CBS constant indicate that certain additive or multiplicative two-by-two block preconditioners reduce enough the condition number of the system. Moreover, we show that a recursive multiplicative two-by-two block preconditioning can be used, resulting in the algebraic multilevel iterative (AMLI) method. We present the conditions under which the AMLI preconditioning is of an optimal order. Numerical experiments confirm the introduced theoretical estimates.