No abstract

We consider a framework for the construction of iterative schemes for operator equations that combine low-rank approximation in tensor formats and adaptive approximation in a basis. Under fairly general assumptions, we obtain a rigorous convergence analysis, where all parameters required for the execution of the methods depend only on the underlying infinite-dimensional problem, but not on a concrete discretization. Under certain assumptions on the rates for the involved low-rank approximations and basis expansions, we can also give bounds on the computational complexity of the iteration as a function of the prescribed target error. Our theoretical findings are illustrated and supported by computational experiments. These demonstrate that problems in very high dimensions can be treated with controlled solution accuracy.

We consider the linear elliptic equation −div(a∇u) = f on some bounded domain D, where a has the form a = exp(b) with b a random function defined as b(y) = j≥1 y j ψ j where y = (y j ) ∈ R N are i.i.d. standard scalar Gaussian variables and (ψ j ) j≥1 is a given sequence of functions in L ∞ (D). We study the summability properties of Hermite-type expansions of the solution map y → u(y) ∈ V := H 1 0 (D), that is, expansions of the form u(y) = ν∈F u ν H ν (y), where H ν (y) = j≥1 H νj (y j ) are the tensorized Hermite polynomials indexed by the set F of finitely supported sequences of nonnegative integers. Previous results [19] have demonstrated that, for any 0 < p ≤ 1, the ℓ p summability of the sequence (j ψ j L ∞ ) j≥1 implies ℓ p summability of the sequence ( u ν V ) ν∈F . Such results ensure convergence rates n −s with s = 1 p − 1 2 of polynomial approximations obtained by best n-term truncation of Hermite series, where the error is measured in the mean-square sense, that is, in L 2 (R N , V, γ), where γ is the infinitedimensional Gaussian measure. In this paper we considerably improve these results by providing sufficient conditions for the ℓ p summability of ( u ν V ) ν∈F expressed in terms of the pointwise summability properties of the sequence (|ψ j |) j≥1 . This leads to a refined analysis which takes into account the amount of overlap between the supports of the ψ j . For instance, in the case of disjoint supports, our results imply that, for all 0 < p < 2 the ℓ p summability of ( u ν V ) ν∈F follows from the weaker assumption that ( ψ j L ∞ ) j≥1 is ℓ q summable for q := 2p 2−p > p. In the case of arbitrary supports, our results imply that the ℓ p summability of ( u ν V ) ν∈F follows from the ℓ p summability of (j β ψ j L ∞ ) j≥1 for some β > 1 2 , which still represents an improvement over the condition in [19]. We also explore intermediate cases of functions with local yet overlapping supports, such as wavelet bases. One interesting observation following from our analysis is that for certain relevant examples, the use of the Karhunen-Loève basis for the representation of b might be suboptimal compared to other representations, in terms of the resulting summability properties of ( u ν V ) ν∈F . While we focus on the diffusion equation, our analysis applies to other type of linear PDEs with similar lognormal dependence in the coefficients.