Teaching
Publications
Preprints
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Convergence rates for the Adam optimizer
joint work with Arnulf Jentzen
arXiv:2407.21078
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Non-convergence of Adam and other adaptive stochastic gradient descent optimization methods for non-vanishing learning rates
joint work with Robin Graeber and Arnulf Jentzen
arXiv:2407.08100
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Learning rate adaptive stochastic gradient descent optimization methods: numerical simulations for deep learning methods for partial differential equations and convergence analyses
joint work with Arnulf Jentzen and Adrian Riekert
arXiv:2406.14340
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Traces left by random walks on large random graphs: local limits
arXiv:2403.14787
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On the existence of optimal shallow feedforward networks with ReLU activation
joint work with Sebastian Kassing
J. Mach. Learn. 3, No. 1, 1-22 (2024), DOI: 10.4208/jml.230903
PDF - On the existence of minimizers in shallow residual ReLU neural network optimization landscapes
joint work with Arnulf Jentzen and Sebastian Kassing
accepted by SINUM
Preprint - Quasi-processes for branching Markov chains
joint work with Martin Maiwald
arXiv:2107.06654
Preprint - Convergence of stochastic gradient descent schemes for Lojasiewicz-landscapes
joint work with Sebastian Kassing
J. Mach. Learn. 3, No. 3, 245-281 (2024), DOI: 10.4208/jml.240109
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Artificial neural networks / Stochastic approximation
- Central limit theorems for stochastic gradient descent with averaging for stable manifolds
joint work with Sebastian Kassing
Electron. J. Probab. 28: 1-48 (2023), DOI: 10.1214/23-EJP947
PDF - On minimal representations of shallow ReLU networks
joint work with Sebastian Kassing
Neural Networks 148, 121-128 (2022), DOI 10.1016/j.neunet.2022.01.006
Preprint - Cooling down stochastic differential equations: almost sure convergence
joint work with Sebastian Kassing
Stochastic Process. Appl. 152, 289-311 (2022), DOI 10.1016/j.spa.2022.06.020
Preprint - General multilevel adaptations for stochastic approximation algorithms II: CLTs
Stochastic Process. Appl. 132, 226-260 (2021), DOI 10.1016/j.spa.2020.11.001
Preprint - General multilevel adaptations for stochastic approximation algorithms of Robbins-Monro and Polyak-Ruppert type
joint work with Thomas Müller-Gronbach
Numer. Math. 142, No. 2, 279-328 (2019), DOI 10.1007/s00211-019-01024-y
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Complex networks / Condensation
- The rank-one and the preferential attachment paradigm
In Network Science - An Aerial View, 43-58 (2019) - Local neighbourhoods for first passage percolation on the configuration model
joint work with Marcel Ortgiese
J. Stat. Phys. 137, No. 3-4, 485-501 (2018), DOI 10.1007/s10955-018-2028-7
Preprint - The shape of the emerging condensate in effective models of condensation
joint work with Volker Betz and Peter Mörters
Ann. Henri Poincaré 19, No. 6, 1869-1889 (2018)
Preprint - Non-extensive condensation in reinforced branching processes
joint work with Cécile Mailler and Peter Mörters
Ann. Appl. Probab. 27, No. 4, 2539-2568 (2017)
Preprint - Distances in scale free networks at criticality
joint work with Christian Mönch and Peter Mörters
Electron. J. Probab. 22, No. 77, 38 pp. (2017)
PDF - Preferential attachment with fitness: unfolding the condensate
Electron. J. Probab. 21, No. 3, 1-38 (2016)
PDF - Cycle length distributions in random permutations with diverging cycle weights
joint work with Peter Mörters
Random. Struct. Algor. 46, No. 4, 635-650 (2015)
PDF - Robust analysis of preferential attachment models with fitness
joint work with Marcel Ortgiese
Combin. Probab. Comput. 23, No. 3, 386-411 (2014), DOI 10.1017/S0963548314000157
Preprint - Emergence of condensation in Kingman's model of selection and mutation
joint work with Peter Mörters
Acta Appl. Math. 127, No. 1, 17-26 (2013), DOI 10.1007/s10440-012-9790-3
Preprint - Typical distances in ultrasmall random networks
joint work with Christian Mönch and Peter Mörters
Adv. in Appl. Probab. 44, No. 2, 583-601 (2012)
Preprint - Random networks with sublinear preferential attachment: the giant component
joint work with Peter Mörters
Ann. Probab. 41, No. 1, 329-384 (2013), DOI 10.1214/11-AOP697
Preprint - Random Networks with Concave Preferential Attachment Rule
joint work with Peter Mörters
Jahresber Dtsch Math-Ver 113, No. 1, 21-40 (2011), DOI 10.1365/s13291-010-0011-6 - Random networks with sublinear preferential attachment: Degree evolutions
joint work with Peter Mörters
Electron. J. Probab. 14, 1222-1267 (2009)
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Numerical integration / Quadrature of SDEs
- Multilevel Monte Carlo for Lévy-driven SDEs: central limit theorems for adaptive Euler schemes
joint work with Sangmeng Li
Ann. Appl. Probab. 26, No. 1, 136-185 (2016)
PDF - Quadrature for self-affine distributions on R^d
joint work with Thomas Müller-Gronbach
Found. Comput. Math. 15, No. 6, 1465-1500 (2015), DOI 10.1007/s10208-014-9233-9
Preprint - On the complexity of computing quadrature formulas for SDEs
joint work with Thomas Müller-Gronbach and Klaus Ritter
in Foundations of Computational Mathematics, Budapest 2011 (eds. F. Cucker, T. Krick, A. Pinkus, A. Szanto), LMS Lecture Notes 403, Cambridge University Press, 2012. DOI 10.1017/CBO9781139095402
Preprint - An Euler-type method for the strong approximation of the Cox-Ingersoll-Ross process
joint work with Andreas Neuenkirch and Lukas Szpruch
Proc. R. Soc. A 468, No. 2140, 1105�1115 (2012), DOI 10.1098/rspa.2011.0505
Preprint - Multilevel Monte Carlo algorithms for Lévy-driven SDEs with Gaussian correction
Ann. Appl. Probab. 21, No. 1, 283-311 (2011)
Preprint - A multilevel Monte Carlo algorithm for Lévy driven stochastic differential equations
joint work with Felix Heidenreich
Stochastic Process. Appl. 121, Mo. 7, 1565-1587 (2011)
Preprint - Infinite-dimensional quadrature and approximation of distributions
joint work with Jakob Creutzig, Thomas Müller-Gronbach and Klaus Ritter
Found. Comput. Math. 9, No. 4, 391-429 (2009)
Preprint
Small deviations and approximation theory (quantization)
- Persistence probabilities for an integrated random walk bridge
joint work with Frank Aurzada and Mikhail Lifshits
Probab. Math. Statist. 34, No. 1, 1-22 (2014)
Preprint - Constructive quantization: approximation by empirical measures
joint work with Michael Scheutzow and Reik Schottstedt
Ann. Inst. Henri Poincaré (B) 49, No. 4, 1183-1203 (2013)
Preprint - Universality of the asymptotics of the one-sided exit problem for integrated processes
joint work with Frank Aurzada
Ann. Inst. Henri Poincaré (B) 49, No. 1, 236-251 (2013), DOI 10.1214/11-AIHP427
Preprint - The high resolution vector quantization problem with Orlicz norm distortion
joint work with Christian Vormoor
J. Theor. Probab. 24, No. 2, 517-544 (2011)
Preprint - Small deviations of general Lévy processes
joint work with Frank Aurzada
Ann. Probab. 37, No.5, 2066-2092 (2009)
Preprint - High resolution quantization and entropy coding of jump processes
joint work with Frank Aurzada, Michael Scheutzow and Christian Vormoor
J. Complexity 25, 163-187 (2009)
Preprint - The coding complexity of Lévy processes
joint work with Frank Aurzada
Found. Comput. Math. 9, No. 3, 359-390 (2009)
Preprint - Asymptotic formulae for coding problems and intermediate optimization problems: a review
in Trends in Stochastic Analysis, edited by Jochen Blath, Peter Morters, and Michael Scheutzow, Cambridge University Press, 187-232, 2009
Preprint - The coding complexity of diffusion processes under supremum norm distortion
Stochastic Process. Appl. 118, No.6, 917-937 (2008), doi:10.1016/j.spa.2007.07.003
Preprint - The coding complexity of diffusion processes under L^p[0,1]-norm distortion
Stochastic Process. Appl. 118, No.6, 938-951 (2008), doi:10.1016/j.spa.2007.07.002
Preprint - High-resolution quantization and entropy coding for fractional Brownian motion
joint work with Michael Scheutzow
Electron. J. Probab. 11, 700-722 (2006)
Preprint - Probabilities of randomly centered small balls and quantization in Banach spaces
joint work with Mikhail A. Lifshits
Ann. Probab. 33, No.4, 1397-1421 (2005)
PDF - Asymptotic behavior of the distortion-rate function for Gaussian processes in Banach spaces
Bulletin des Sciences Mathématiques 129, No.10, 791-803 (2005) - Small ball probabilities around random centers of Gaussian measures and applications to quantization
J. Theor. Probab. 16, No.2, 427-449 (2003) - On the link between small ball probabilities and the quantization problem for Gaussian measures on Banach spaces
joint work with Franz Fehringer, Anis Matoussi and Michael Scheutzow
J. Theor. Probab. 16, No.1, 249-265 (2003)
Stochastic analysis
- Real self-similar processes started from the origin
joint work with Leif Döring and Andreas E. Kyprianou
Ann. Probab. 45, No. 3, 1952-2003 (2017)
Preprint - A support theorem and a large deviation principle for Kunita flows
joint work with Georgi Dimitroff
Stochastics and Dynamics 12, No. 3 (2012)
Preprint - Rough paths analysis of general Banach space-valued Wiener processes
J. Funct. Anal. 258, No. 9, 2910-2936 (2010)
Preprint - Enlargement of filtrations and continuous Girsanov-type embeddings
joint work with Stefan Ankirchner and Peter Imkeller
in Séminaire de Probabilités XL (ed. Catherine Donati-Martin, Michel Émery, Alain Rouault, and Christophe Stricker), LNM1899, Springer, 2007 - The Shannon information of filtrations and the additional logarithmic utility of insiders
joint work with Stefan Ankirchner and Peter Imkeller
Ann. Probab. 34, No.2, 743-778 (2006)
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PhD Thesis
- High resolution coding of stochastic processes and small ball probabilities
Ph.D. Dissertation (2003)
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