Markus Bachmayr

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Many real world problems are high-dimensional in that their solution is a function which depends on many variables or parameters. This presents a computational challenge since traditional numerical techniques are built on model classes for functions based solely on smoothness. It is known that the approximation of smoothness classes of functions suffers(More)
We consider the linear elliptic equation −div(a∇u) = f on some bounded domain D, where a has the affine form a = a(y) = ā + ∑ j≥1 yjψj for some parameter vector y = (yj)j≥1 ∈ U = [−1, 1]N. We study the summability properties of polynomial expansions of the solution map y 7→ u(y) ∈ V := H 0 (D). We consider both Taylor series and Legendre series. Previous(More)
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(More)
Bases of atomic-like functions provide a natural, physically motivated description of electronic states, and Gaussian-type orbitals are the most widely used basis functions in molecular simulations. This paper aims at developing a systematic analysis of numerical approximations based on linear combinations of some Gaussian-type orbitals. We give a priori(More)
Low-rank tensor methods for the approximate solution of second-order elliptic partial differential equations in high dimensions have recently attracted significant attention. A critical issue is to rigorously bound the error of such approximations, not with respect to a fixed finite dimensional discrete background problem, but with respect to the exact(More)
This paper is concerned with the development and analysis of an iterative solver for high-dimensional second-order elliptic problems based on subspace-based low-rank tensor formats. Both the subspaces giving rise to low-rank approximations and corresponding sparse approximations of lower-dimensional tensor components are determined adaptively. A principal(More)