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In this paper we discuss the construction, analysis, and implementation of iterative schemes for the solution of inverse problems based on total variation regularization. Via different approximations of the nonlinearity we derive three different schemes resembling three well-known methods for nonlinear inverse problems in Hilbert spaces, namely iterated… (More)

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)

- Markus Bachmayr, Reinhold Schneider, André Uschmajew
- Foundations of Computational Mathematics
- 2016

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)

- Markus Bachmayr, Wolfgang Dahmen
- Foundations of Computational Mathematics
- 2015

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)

- Markus Bachmayr, Huajie Chen, Reinhold Schneider
- Numerische Mathematik
- 2014

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)

- Markus Bachmayr, Reinhold Schneider
- Foundations of Computational Mathematics
- 2017

- Markus Bachmayr, Wolfgang Dahmen
- SIAM J. Numerical Analysis
- 2016

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)