Tractability of Linear Multivariate Problems in the Average Case Setting

Summary

We study the average case setting for linear multivariate problems defined over a separable Banach space of functions f of d variables. The Banach space is equipped with a Gaussian measure. We approximate linear multivariate problems by computing finitely many information evaluations. An information evaluation is defined as an evaluation of a continuous linear functional from a given class Λ. We consider two classes of information evaluations; the first class Λ ^all consists of all continuous linear functionals, and the second class Λ ^std consists of function evaluations.

We investigate the minimal number n(ε, d,Λ) of information evaluations needed to reduce the initial average case error by a factor ε. The initial average case error is defined as the minimal error that can be achieved without any information evaluations.

We study tractability of linear multivariate problems in the average case setting. Tractability means that n(ε, d,Λ) is bounded by a polynomial in both ε ⁻¹ and d, and strong tractability means that n(ε, d,Λ) is bounded by a polynomial only in ε ⁻¹.

For the class Λ ^all, we provide necessary and sufficient conditions for tractability and strong tractability in terms of the eigenvalues of the covariance operator of a Gaussian measure on the space of solution elements. These conditions are simplified under additional assumptions on the measure. In particular, we consider measures with finite-order weights and product weights. For finite-order weights, we prove that linear multivariate problems are always tractable.