33 citations · 62 across the 26 of their papers we have counts for
35 papers
Efficient localised model reduction for multiscale PDEs via Grassmannian interpolation
Christian Alber, Markus Bachmayr, Robert Scheichl +1
Multiscale, parameter-dependent partial differential equations (PDEs) pose severe computational challenges due to strong coefficient heterogeneity and high-dimensional parameter sp…
Iterative thresholding low-rank time integration for high-dimensional problems
Markus Bachmayr, Tianyu Jin, Polina Sachsenmaier +1
This work analyzes a method for time integration of high-dimensional linear Schrödinger-type problems based on hierarchical tensor approximations. In particular, this method provid…
Low-rank eigenvalue solvers for block-sparse matrix product states
Markus Bachmayr, Sebastian Krämer, Max Pfeffer
We consider an iterative eigensolver for Schrödinger equations that constructs low-rank approximations of eigenfunctions with accuracy-adapted ranks, with particular focus on fermi…
Preconditioning and Numerical Stability in Neural Network Training for Parametric PDEs
Markus Bachmayr, Wolfgang Dahmen, Chenguang Duan +1
In the context of training neural network-based approximations of solutions of parameter-dependent PDEs, we investigate the effect of preconditioning via well-conditioned frame rep…
Iterative thresholding low-rank time integration
Markus Bachmayr, Matthieu Dolbeault, Polina Sachsenmaier
We develop time integration methods in low-rank representation that can adaptively adjust approximation ranks to achieve a prescribed accuracy, while ensuring that these ranks rema…
Sparse and low-rank approximations of parametric elliptic PDEs: the best of both worlds
Markus Bachmayr, Huqing Yang
A new approximation format for solutions of partial differential equations depending on infinitely many parameters is introduced. By combining low-rank tensor approximation in a se…
Adaptive stochastic Galerkin finite element methods: Optimality and non-affine coefficients
Markus Bachmayr, Henrik Eisenmann, Igor Voulis
Near-optimal computational complexity of an adaptive stochastic Galerkin method with independently refined spatial meshes for elliptic partial differential equations is shown. The…
Fluid flow channeling and mass transport with discontinuous porosity distribution
Simon Boisserée, Evangelos Moulas, Markus Bachmayr
The flow of fluids within porous rocks is an important process with numerous applications in Earth sciences. Modeling the compaction-driven fluid flow requires the solution of coup…
An adaptive space-time method for nonlinear poroviscoelastic flows with discontinuous porosities
Markus Bachmayr, Simon Boisserée
This paper is concerned with a space-time adaptive numerical method for instationary porous media flows with nonlinear interaction between porosity and pressure, with focus on prob…
Variationally Correct Neural Residual Regression for Parametric PDEs: On the Viability of Controlled Accuracy
Markus Bachmayr, Wolfgang Dahmen, Mathias Oster
This paper is about learning the parameter-to-solution map for systems of partial differential equations (PDEs) that depend on a potentially large number of parameters covering all…
Low-rank tensor product Richardson iteration for radiative transfer in plane-parallel geometry
Markus Bachmayr, Riccardo Bardin, Matthias Schlottbom
The radiative transfer equation (RTE) has been established as a fundamental tool for the description of energy transport, absorption and scattering in many relevant societal applic…
A convergent adaptive finite element stochastic Galerkin method based on multilevel expansions of random fields
Markus Bachmayr, Martin Eigel, Henrik Eisenmann +1
The subject of this work is an adaptive stochastic Galerkin finite element method for parametric or random elliptic partial differential equations, which generates sparse product p…
Dynamical low-rank tensor approximations to high-dimensional parabolic problems: existence and convergence of spatial discretizations
Markus Bachmayr, Henrik Eisenmann, André Uschmajew
We consider dynamical low-rank approximations to parabolic problems on higher-order tensor manifolds in Hilbert spaces. In addition to existence of solutions and their stability wi…
A space-time adaptive low-rank method for high-dimensional parabolic partial differential equations
Markus Bachmayr, Manfred Faldum
An adaptive method for parabolic partial differential equations that combines sparse wavelet expansions in time with adaptive low-rank approximations in the spatial variables is co…
Polynomial Approximation of Symmetric Functions
Markus Bachmayr, Geneviève Dusson, Christoph Ortner +1
We study the polynomial approximation of symmetric multivariate functions and of multi-set functions. Specifically, we consider , where ,…
An adaptive stochastic Galerkin method based on multilevel expansions of random fields: Convergence and optimality
Markus Bachmayr, Igor Voulis
The subject of this work is a new stochastic Galerkin method for second-order elliptic partial differential equations with random diffusion coefficients. It combines operator compr…
Particle Number Conservation and Block Structures in Matrix Product States
Markus Bachmayr, Michael Götte, Max Pfeffer
The eigenvectors of the particle number operator in second quantization are characterized by the block sparsity of their matrix product state representations. This is shown to gene…
Multilevel Representations of Isotropic Gaussian Random Fields on the Sphere
Markus Bachmayr, Ana Djurdjevac
Series expansions of isotropic Gaussian random fields on with independent Gaussian coefficients and localized basis functions are constructed. Such representations w…
Existence of dynamical low-rank approximations to parabolic problems
Markus Bachmayr, Henrik Eisenmann, Emil Kieri +1
The existence and uniqueness of weak solutions to dynamical low-rank evolution problems for parabolic partial differential equations in two spatial dimensions is shown, covering al…
Atomic Cluster Expansion: Completeness, Efficiency and Stability
Genevieve Dusson, Markus Bachmayr, Gabor Csanyi +4
The Atomic Cluster Expansion (Drautz, Phys. Rev. B 99, 2019) provides a framework to systematically derive polynomial basis functions for approximating isometry and permutation inv…
Adaptive Low-Rank Approximations for Operator Equations: Accuracy Control and Computational Complexity
Markus Bachmayr, Wolfgang Dahmen
The challenge of mastering computational tasks of enormous size tends to frequently override questioning the quality of the numerical outcome in terms of accuracy. By this we do no…
Unified Analysis of Periodization-Based Sampling Methods for Matérn Covariances
Markus Bachmayr, Ivan G. Graham, Van Kien Nguyen +1
The periodization of a stationary Gaussian random field on a sufficiently large torus comprising the spatial domain of interest is the basis of various efficient computational meth…
Identifiability of Diffusion Coefficients for Source Terms of Non-Uniform Sign
Markus Bachmayr, Van Kien Nguyen
The problem of recovering a diffusion coefficient in a second-order elliptic partial differential equation from a corresponding solution for a given right-hand side is…
Sequential sampling for optimal weighted least squares approximations in hierarchical spaces
Benjamin Arras, Markus Bachmayr, Albert Cohen
We consider the problem of approximating an unknown function from its evaluations at given sampling points , where is a…