Profile
Profile

Prof. Dr. Andreas Eberle

E-Mail: eberle(at)uni-bonn.de
Telefon: +49 228 73 3415
Homepage: https://wt.iam.uni-bonn.de/eberle/home/
Raum: 3.045
Standort: Mathematics Center
Institute: Institute for Applied Mathematics
Forschungsbereiche: Research Area G
Research Area J (until 10/2012)
Former Research Area L
Geburtsdatum: 19.Jun 1968
Mathscinet-Number: 363836

Academic Career

1998

PhD, University of Bielefeld

1998 - 1999

Postdoc, Paul Sabatier University (Toulouse III), France

1999 - 2000

DFG research grant, University of California, San Diego, CA, USA

2000 - 2001

Teaching Assistant, University of Bielefeld

2001 - 2003

Lecturer, University of Oxford and Worcester College, UK

2009 - 2016

Head of Examination Board for Bachelor and Master Studies, Bonn

Since 2003

Professor (C3), University of Bonn

Research Profile

My research is based on the combination of methods from probability theory and other branches of mathematics, including differential equations and functional analysis, numerical analysis, geometry, and mathematical physics. A current focus is on coupling methods for stochastic processes on continuous state spaces. Here, a common goal is to quantify stability properties and convergence to equilibrium, for example for stochastic differential equations, systems with mean-field interactions, processes with high and infinite dimensional state spaces, numerical approximations, and both Markov Chain Monte Carlo and sequential Monte Carlo methods. An important tool is an approach developed in recent years that is based on contraction properties for combinations of reflection couplings and other couplings in specifically adjusted Kantorovich distances. Both the underlying metric and the coupling are adapted carefully to the corresponding problem, thus providing quantitative non-asymptotic bounds that are often relatively precise. The approach has first been applied successfully to non-degenerate diffusion processes. More recently, it has been extended to mean-field systems and nonlinear equations with weak interactions, and variants have been applied to numerical approximations and a class of MCMC methods.

Markov Chain Monte Carlo methods are the source of a variety of non-trivial mathematical problems. One example of current interest is the observation that often non-reversible processes seem to approach equilibrium faster than the more standard reversible ones. The question how to implement non-reversible processes in MCMC in the most effective way is still widely open. This is complemented by a much more incomplete mathematical understanding of the long time behavior of non-reversible Markov processes compared to reversible ones. Coupling methods are not based on reversibility. Therefore, they might help to clarify these important questions. First steps in this direction are made in current work in progress which shows that a similar coupling approach as described above yields qualitatively new bounds for convergence to equilibrium of (kinetic) Langevin equations. A goal of my future research is to extend these results to related Monte Carlo methods, and also to other stochastic systems with degenerate noise. Another important question, arising for example in the study of sequential Monte Carlo methods, is how to quantify the deviation of a mean-field approximation from a corresponding nonlinear SDE. Coupling methods might help to gain new insight. More generally, coupling approaches are natural for deriving long-time stable bounds for the difference between two different stochastic dynamics. First steps in this direction are done in current work in progress on sticky couplings.

Research Projects and Activities

Procope-Project: Quantitative convergence rates for diffusions by coupling methods

Contribution to Research Areas

Research Area G
My central research topics are related to couplings of Markov processes on continuous state spaces in both discrete and continuous time, long-time stability of time-dependent stochastic systems and convergence to equilibrium of time homogeneous stochastic processes, spectral gaps and functional inequalities, and stochastic equations in high dimensions and on function spaces. Many problems are motivated by applications to Markov Chain Monte Carlo methods, sequential Monte Carlo methods and systems with mean-field interactions, nonlinear stochastic differential equations, and stochastic systems with degenerate noise.

Selected Publications

[1] A. Eberle, A. Guillin, R. Zimmer
Couplings and quantitative contraction rates for Langevin dynamics
ArXiv e-prints
2017
[2] A. Eberle, R. Zimmer
Sticky couplings of multidimensional diffusions with different drifts
ArXiv e-prints
2016
[3] A. Eberle, A. Guillin, R. Zimmer
Quantitative Harris type theorems for diffusions and McKean-Vlasov processes
ArXiv e-prints
2016
[4] Andreas Eberle
Reflection couplings and contraction rates for diffusions
Probab. Theory Related Fields
,
166: (3-4): 851--886
2016
DOI: 10.1007/s00440-015-0673-1
[5] Andreas Eberle
Error bounds for Metropolis-Hastings algorithms applied to perturbations of Gaussian measures in high dimensions
Ann. Appl. Probab. , 24: (1): 337--377
2014
DOI: 10.1214/13-AAP926
[6] Andreas Eberle, Carlo Marinelli
Quantitative approximations of evolving probability measures and sequential Markov chain Monte Carlo methods
Probab. Theory Related Fields , 155: (3-4): 665--701
2013
DOI: 10.1007/s00440-012-0410-y
[7] Andreas Eberle, Carlo Marinelli
Lp estimates for Feynman-Kac propagators with time-dependent reference measures
J. Math. Anal. Appl. , 365: (1): 120--134
2010
DOI: 10.1016/j.jmaa.2009.10.019
[8] Andreas Eberle
Local spectral gaps on loop spaces
J. Math. Pures Appl. (9) , 82: (3): 313--365
2003
DOI: 10.1016/S0021-7824(03)00003-5
[9] Andreas Eberle
Absence of spectral gaps on a class of loop spaces
J. Math. Pures Appl. (9) , 81: (10): 915--955
2002
DOI: 10.1016/S0021-7824(02)01260-6
[10] Andreas Eberle
Uniqueness and non-uniqueness of semigroups generated by singular diffusion operators
of Lecture Notes in Mathematics : viii+262
Publisher: Springer-Verlag, Berlin
1999
ISBN: 3-540-66628-1
DOI: 10.1007/BFb0103045

Publication List

Editorships

• Annals of Applied Probability (Associate Editor, since 2014)

Selected Invited Lectures

2000

Stochastic analysis and applications, Lisbon, Portugal

2002

Stochastic analysis, Beijing, China

2004

Stochastic partial differential equations and applications, Levico, Italy

2011

Filtering, MCMC, ABC, Lille, France

2016

Computational statistics and molecular simulation, Paris, France

Selected PhD students

Nikolaus Schweizer (2012): “Non-asymptotic Error Bounds for Sequential MCMC Methods”,
now Postdoc, Saarland University

Daniel Gruhlke (2014): “Convergence of multilevel MCMC methods on path spaces”

Raphael Zimmer (current PhD student)

Mateusz Majka (current PhD student)

Supervised Theses

  • Diplom theses: 30, currently 9
  • PhD theses: 2, currently 2
Download Profile