Publications


Preprints:

  • M. Bachmayr, H. Eisenmann and A. Uschmajew: Dynamical low-rank tensor approximations to high-dimensional parabolic problems: existence and convergence of spatial discretizations, arXiv:2308:16720.
  • M. Bachmayr, M. Eigel, H. Eisenmann and I. Voulis: A convergent adaptive finite element stochastic Galerkin method based on multilevel expansions of random fields, arXiv:2403.13770.
  • M. Bachmayr, R. Bardin and M. Schlottbom: Low-rank tensor product Richardson iteration for radiative transfer in plane-parallel geometry, arXiv:2403.14229.

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Refereed journal articles

  1. M. Bachmayr and M. Faldum: A space-time adaptive low-rank method for high-dimensional parabolic partial differential equations, to appear in Journal of Complexity, 2024. Preprint: arXiv:2302.01658.
  2. M. Bachmayr, S. Boisserée and L. M. Kreusser: Analysis of nonlinear poroviscoelastic flows with discontinuous porosities, Nonlinearity 36, 7025, 2023. Preprint: arXiv:2302.12166
  3. M. Bachmayr: Low-rank tensor methods for partial differential equations, Acta Numerica 32, pp. 1-121, 2023.
  4. M. Bachmayr, G. Dusson, Ch. Ortner, and J. Thomas: Polynomial approximation of symmetric functions, Mathematics of Computation, 10.1090/mcom/3868. Preprint: arXiv:2109.14771.
  5. M. Bachmayr, A. Nouy, and R. Schneider: Approximation by tree tensor networks in high dimensions: Sobolev and compositional functions, Pure and Applied Functional Analysis 8(2), pp. 405-428, 2023. Preprint: arXiv:2112.01474.
  6. M. Bachmayr and I. Voulis: An adaptive stochastic Galerkin method based on multilevel expansions of random fields: Convergence and optimality, ESAIM:M2AN 56(6), pp. 1955-1992, 2022. Preprint: arXiv:2109:09136.
  7. M. Bachmayr and A. Djurdjevac: Multilevel representations of isotropic Gaussian random fields on the sphere, IMA Journal of Numerical Analysis, 10.1093/imanum/drac034, 2022. Preprint: arXiv:2011.06987.
  8. M. Bachmayr, M. Götte, and M. Pfeffer: Particle number conservation and block structures in matrix product states, Calcolo 59, 24, 2022. Preprint: arXiv:2104.13483.
  9. G. Dusson, M. Bachmayr, G. Csanyi, R. Drautz, S. Etter, C. van der Oord, and Ch. Ortner: Atomic cluster expansion: completeness, efficiency and stability, Journal of Computational Physics 454, 110946, 2022.
  10. M. Bachmayr, H. Eisenmann, E. Kieri, and A. Uschmajew: Existence of dynamical low-rank approximations to parabolic problems, Mathematics of Computation, 90(330), pp 1799-1830, 2021.
  11. M. Bachmayr, I. G. Graham, V. K. Nguyen, and R. Scheichl: Unified analysis of periodization-based sampling methods for Matérn covariances, SIAM J. Numer. Anal., 58(5), pp 2953–2980, 2020.
  12. M. Bachmayr and V. Kazeev: Stability of low-rank tensor representations and structured multilevel preconditioning for elliptic PDEs, Foundations of Computational Mathematics, 20(5), pp 1175-1236, 2020.
    [ Code: TensorTrains.jl, TensorTrainFEM.jl ]
  13. M. Bachmayr and V. K. Nguyen: Identifiability of diffusion coefficients for source terms of non-uniform sign, Inverse Problems & Imaging, 13 (5), pp 1007-1021, 2019.
  14. B. Arras, M. Bachmayr, and A. Cohen: Sequential sampling for optimal weighted least squares approximations in hierarchical spaces, SIAM Journal on Mathematics of Data Science, 1(1), pp 189-207, 2019.
  15. M. Bachmayr, A. Cohen, and W. Dahmen: Parametric PDEs: Sparse or low-rank approximations?, IMA Journal of Numerical Analysis, 38(4), pp 1661-1708, 2018.   [ Editor's Choice 2018 ]
  16. M. Bachmayr, A. Cohen, and G. Migliorati: Representations of Gaussian random fields and approximation of elliptic PDEs with lognormal coefficients, J. Fourier Anal. Appl., 24(3), pp 621-649, 2018.
  17. M. Bachmayr, A. Cohen, D. Dũng, and Ch. Schwab: Fully discrete approximation of parametric and stochastic elliptic PDEs, SIAM J. Numer. Anal., 55, 2151-2186, 2017.
  18. M. Bachmayr, A. Cohen, R. DeVore, and G. Migliorati: Sparse polynomial approximation of parametric elliptic PDEs. Part II: lognormal coefficients, ESAIM Math. Model. Numer. Anal., 51(1), pp 341-363, 2017.
  19. M. Bachmayr, A. Cohen, and G. Migliorati: Sparse polynomial approximation of parametric elliptic PDEs. Part I: affine coefficients, ESAIM Math. Model. Numer. Anal., 51(1), pp 321-339, 2017.
  20. M. Bachmayr and R. Schneider: Iterative methods based on soft thresholding of hierarchical tensors, Found. Comput. Math., 17(4), pp 1037-1083, 2017.
  21. M. Bachmayr and A. Cohen: Kolmogorov widths and low-rank approximations of parametric elliptic PDEs, Mathematics of Computation, 86, pp 701-724, 2017.
  22. M. Bachmayr, R. Schneider, and A. Uschmajew: Tensor networks and hierarchical tensors for the solution of high-dimensional partial differential equations, Found. Comput. Math., 16(6), pp 1423-1472, 2016. (survey article)
  23. M. Bachmayr and W. Dahmen: Adaptive low-rank methods: Problems on Sobolev spaces, SIAM J. Numer. Anal., 54(2), pp 744-796, 2016.
  24. M. Bachmayr and W. Dahmen: Adaptive low-rank methods for problems on Sobolev spaces with error control in L2, ESAIM Math. Model. Numer. Anal., 50(4), pp 1107-1136, 2016.
  25. M. Bachmayr and W. Dahmen: Adaptive near-optimal rank tensor approximation for high-dimensional operator equations, Foundations of Computational Mathematics, 15(4), pp 839-898, 2015.
  26. M. Bachmayr, W. Dahmen, R. DeVore, and L. Grasedyck: Approximation of high-dimensional rank one tensors, Constructive Approximation, 39(2), pp 385-395, 2014, 2014.
  27. M. Bachmayr, H. Chen, and R. Schneider: Error estimates for Hermite and even-tempered Gaussian approximations in quantum chemistry, Numerische Mathematik, 128(1), pp 137-156, 2014.
  28. M. Bachmayr: Integration of products of Gaussians and wavelets with applications to electronic structure calculations, SIAM J. Numer. Anal., 51(5), pp 2491-2513, 2013.
  29. M. Bachmayr: Hyperbolic wavelet discretization of the two-electron Schrödinger equation in an explicitly correlated formulation, ESAIM: M2AN 46(6), pp 1337-1362, 2012.
  30. M. Bachmayr and M. Burger: Iterative total variation schemes for nonlinear inverse problems, Inverse Problems, 25, 105004, 2009.

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Refereed proceedings

  1. M. Bachmayr: Low-rank tensor solvers for high-dimensional parabolic PDEs, in Oberwolfach Report 48/2023, Mathematisches Forschungsinstitut Oberwolfach.
  2. M. Bachmayr: A near-optimal adaptive stochastic Galerkin method based on multilevel expansions of random fields, in Oberwolfach Report 36/2021, Mathematisches Forschungsinstitut Oberwolfach.
  3. M. Bachmayr and W. Dahmen: Adaptive Low-Rank Approximations for Operator Equations: Accuracy Control and Computational Complexity, in Contemporary Mathematics 754, 2020. (survey article, preprint: arXiv:1910.07052)
  4. M. Bachmayr: New notions of numerical conditioning for multilevel tensor representations of differential operators, in Oberwolfach Report 40/2019, Mathematisches Forschungsinstitut Oberwolfach.
  5. M. Bachmayr: Multilevel representations of stationary Gaussian random fields and efficient sampling methods, in Oberwolfach Report 12/2019, Mathematisches Forschungsinstitut Oberwolfach.
  6. M. Bachmayr: Space-parameter-adaptive approximation of affine-parametric elliptic PDEs, in Oberwolfach Report 17/2017, Mathematisches Forschungsinstitut Oberwolfach.
  7. M. Bachmayr: Kolmogorov widths and low-rank approximations of parametric elliptic PDEs, in Oberwolfach Report 2/2015, Mathematisches Forschungsinstitut Oberwolfach.
  8. M. Bachmayr: Adaptivity and preconditioning for high-dimensional elliptic partial differential equations, in Oberwolfach Report 24/2014, Mathematisches Forschungsinstitut Oberwolfach.
  9. M. Bachmayr: Adaptive near-optimal rank tensor approximation for high-dimensional operator equations, in Oberwolfach Report 39/2013, Mathematisches Forschungsinstitut Oberwolfach.
  10. M. Bachmayr: Hyperbolic wavelet discretization of the electronic Schrödinger equation: Explicit correlation and separable approximation of potentials, in Oberwolfach Report 33/2010, Mathematisches Forschunginstitut Oberwolfach.

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Theses


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