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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… (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… (More)

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… (More)

- 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… (More)

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

Hierarchical tensors can be regarded as a generalisation, preserving many crucial features, of the singular value decomposition to higher-order tensors. For a given tensor product space, a recursive… (More)

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 yjψj where y = (yj) ∈ R are i.i.d.… (More)

- Markus Bachmayr, Albert Cohen, Dinh Anh Dung, Christoph Schwab
- SIAM J. Numerical Analysis
- 2017

It has recently been demonstrated that locality of spatial supports in the parametrization of coefficients in elliptic PDEs can lead to improved convergence rates of sparse polynomial expansions of… (More)

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

We construct a soft thresholding operation for rank reduction of hierarchical tensors and subsequently consider its use in iterative thresholding methods, in particular for the solution of… (More)

- Markus Bachmayr, Albert Cohen
- Math. Comput.
- 2016

Kolmogorov n-widths and low-rank approximations are studied for families of elliptic diffusion PDEs parametrized by the diffusion coefficients. The decay of the n-widths can be controlled by that of… (More)

In this work, we develop methods for the numerical approximation of higher-dimensional functions, given as solutions of linear elliptic operator equations or as eigenfunctions of such operators. The… (More)