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| 2012 | Dr. rer. nat., RWTH Aachen | | 2012 - 2016 | Postdoc at RWTH Aachen, TU Berlin, UPMC Paris 6 (France) | | Since 2016 | Professor (W2, Bonn Junior Fellow), University of Bonn |
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My research focuses on the numerical analysis of high-dimensional partial differential equations. Such problems arise, for instance, in quantum physics and in the deterministic treatment of uncertainty quantification. I am especially interested in understanding the computational complexity of nonlinear approximation methods such as low-rank tensor decompositions, which can exploit particular structural features beyond classical smoothness. Results in this direction include solvers of near-optimal complexity with adaptive discretizations [4,5], iterative solvers with quasi-optimal rank bounds based on soft thresholding [6], and low-rank approximability of parametric PDEs [7,3]. Another approach that is well established for problems with stochastic coefficients are sparse tensor product polynomial expansions. In [8,9,1], we have obtained new results that demonstrate the dependence of convergence rates on the type of parametrisation of the given random fields.
These recent results are an example of the central role that choices of coordinates, or choices of basis expansions for function spaces, often play in the treatment of high-dimensional problems. In the case of differential equations with stochastic coefficients, I pursue questions in this direction that are crucial for highly irregular coefficients, where also challenging problems concerning numerical solvers need to be addressed. In the case of low-rank tensor methods, in many cases one needs to achieve a tradeoff between preserving separable structures and accommodating the topologies prescribed by the mapping properties of the considered operators. Building on the developments in [5], I study such issues in particular in the context of second quantised formulations of quantum-physical models.
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Research Area J
Here I contribute my expertise on adaptive solvers for sparse and low-rank approximations of high-dimensional problems, as well as on related approximability questions.
Since starting in Bonn in September 2016, we have obtained new results on the deterministic numerical treatment of random PDEs, in particular on fully discrete sparse approximations [1]. We have also considered related questions on expansions of Gaussian random fields [2] and on low-rank methods for multi-parametric PDEs [3]. |
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[ 1] Markus Bachmayr, Albert Cohen, Dinh Dũng, Christoph Schwab
Fully discrete approximation of parametric and stochastic elliptic PDEs
SIAM J. Numer. Anal. , 55: (5): 2151--2186
2017
DOI: 10.1137/17M111626X[ 2] M. Bachmayr, A. Cohen, G. Migliorati
Representations of Gaussian random fields and approximation of elliptic PDEs with lognormal coefficients
J. Fourier Anal. Appl.
2017
DOI: DOI 10.1007/s00041-017-9539-5[ 3] M. Bachmayr, A. Cohen, W. Dahmen
Parametric PDEs: Sparse or low-rank approximations?
IMA J. Numer. Anal.
2017
DOI: 10.1093/imanum/drx052[ 4] Markus Bachmayr, Wolfgang Dahmen
Adaptive near-optimal rank tensor approximation for high-dimensional operator equations
Found. Comput. Math. , 15: (4): 839--898
2015
DOI: 10.1007/s10208-013-9187-3[ 5] Markus Bachmayr, Wolfgang Dahmen
Adaptive low-rank methods: problems on Sobolev spaces
SIAM J. Numer. Anal. , 54: (2): 744--796
2016
DOI: 10.1137/140978223[ 6] Markus Bachmayr, Reinhold Schneider
Iterative methods based on soft thresholding of hierarchical tensors
Found. Comput. Math. , 17: (4): 1037--1083
2017
DOI: 10.1007/s10208-016-9314-z[ 7] Markus Bachmayr, Albert Cohen
Kolmogorov widths and low-rank approximations of parametric elliptic PDEs
Math. Comp. , 86: (304): 701--724
2017
DOI: 10.1090/mcom/3132[ 8] Markus Bachmayr, Albert Cohen, Giovanni Migliorati
Sparse polynomial approximation of parametric elliptic PDEs. Part I: Affine coefficients
ESAIM Math. Model. Numer. Anal. , 51: (1): 321--339
2017
DOI: 10.1051/m2an/2016045[ 9] Markus Bachmayr, Albert Cohen, Ronald DeVore, Giovanni Migliorati
Sparse polynomial approximation of parametric elliptic PDEs. Part II: Lognormal coefficients
ESAIM Math. Model. Numer. Anal. , 51: (1): 341--363
2017
DOI: 10.1051/m2an/2016051[ 10] Markus Bachmayr, Huajie Chen, Reinhold Schneider
Error estimates for Hermite and even-tempered Gaussian approximations in quantum chemistry
Numer. Math. , 128: (1): 137--165
2014
DOI: 10.1007/s00211-014-0605-5 |
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| 2007 | Erwin Wenzl Preis | | 2013 | John Todd Award, Oberwolfach Research Institute for Mathematics (MFO) | | 2014 | Borchers-Plakette, RWTH Aachen |
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