Profile
Profile

Prof. Dr. Markus Bachmayr

E-mail: bachmayr(at)ins.uni-bonn.de
Phone: +49 228 73 60457
Homepage: http://bachmayr.ins.uni-bonn.de/
Room: 4.003
Location: Computing Center
Institute: Institute for Numerical Simulation
Research Area: Research Area J

Academic Career

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

Research Profile

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 [1], [2], iterative solvers with quasi-optimal rank bounds based on soft thresholding [3], recovery of tensors by point samples [4], and low-rank approximability of parametric PDEs [5]. Another approach that is well established for problems with stochastic coefficients are sparse tensor product polynomial expansions. In [6], [7], 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 intend to pursue questions in this direction that have a flavour of harmonic analysis. Moreover, for highly irregular coefficients, challenging problems concerning numerical solvers need to be addressed, and relations to methods for stochastic homogenisation also remain to be explored. 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 [2], I plan to study such issues in particular in the context of second quantised formulations of quantum-physical models.

Selected Publications

[1] Markus Bachmayr, Wolfgang Dahmen
Adaptive near-optimal rank tensor approximation for high-dimensional operator equations
Found. Comput. Math. , 15: (4): 839--898
2015
ISSN: 1615-3375
DOI: 10.1007/s10208-013-9187-3
[2] Markus Bachmayr, Wolfgang Dahmen
Adaptive low-rank methods: problems on Sobolev spaces
SIAM J. Numer. Anal. , 54: (2): 744--796
2016
ISSN: 0036-1429
DOI: 10.1137/140978223
[3] Markus Bachmayr, Reinhold Schneider
Iterative methods based on soft thresholding of hierarchical tensors
Found. Comput. Math.
2016
DOI: 10.1007/s10208-016-9314-z
[4] Markus Bachmayr, Wolfgang Dahmen, Ronald DeVore, Lars Grasedyck
Approximation of high-dimensional rank one tensors
Constr. Approx.
, 39: (2): 385--395
2014
ISSN: 0176-4276
DOI: 10.1007/s00365-013-9219-x
[6] Markus Bachmayr, Albert Cohen, Giovanni Migliorati
Sparse polynomial approximation of parametric elliptic PDEs. Part I: affine coefficients
to appear
ESAIM Math. Model. Numer. Anal.
2016
DOI: 10.1051/m2an/2016045
[7] Markus Bachmayr, Albert Cohen, Ronald DeVore, Giovanni Migliorati
Sparse polynomial approximation of parametric elliptic PDEs. Part II: lognormal coefficients
to appear
ESAIM Math. Model. Numer. Anal.
2016
DOI: 10.1051/m2an/2016051
[8] 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
ISSN: 0029-599X
DOI: 10.1007/s00211-014-0605-5
[9] Markus Bachmayr
Integration of products of Gaussians and wavelets with applications to electronic structure calculations
SIAM J. Numer. Anal. , 51: (5): 2491--2513
2013
ISSN: 0036-1429
DOI: 10.1137/120874424
[10] Markus Bachmayr, Martin Burger
Iterative total variation schemes for nonlinear inverse problems
Inverse Problems , 25: (10): 105004, 26
2009
ISSN: 0266-5611
DOI: 10.1088/0266-5611/25/10/105004

Publication List

Awards

2007

Erwin Wenzl Preis

2013

John Todd Award, Oberwolfach Research Institute for Mathematics (MFO)

2014

Borchers-Plakette, RWTH Aachen

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