Research

Interests

I work on the qualitative description of the dynamic behavior of complex systems. This combines methods of partial differential equations and stochastic analysis, but also numerical analysis.

The questions arising in this context are related to statistical mechanics, Markov processes, machine learning, and variation methods. The equations under consideration usually have a physical or engineering background, such as a mathematical battery model, droplet formation in chemical-physical reaction kinetics or consensus formation among interacting agents. At the interface to stochastic analysis I am interested in particle systems and their many-particle limits mostly motivated from statistical physics with particular attention to models that can show phase transitions. Under the stochastic influence of the particle model, metastable behavior usually occurs in such models.

Keywords

  • Metastability in molecular dynamics and statistical mechanics
  • phase-transitions, nucleation and coarsening
  • variational methods for evolution equations
    • entropy method for the longtime behaviour (functional inequality: spectral gap and log-Sobolev)
    • gradient flows and their limits (variational convergence, structural properties, behavior of synthetic curvature and geodesic convexity)
    • quantifications of approximation errors (numerical schemes)
    • dimension reduction of multi-scale dynamics (physical and chemical systems, machine learning training)
    • discrete and nonlocal dynamics (on graphs and data-sets)
    • convergence rate analysis of numerical schemes

Publications

Preprints submitted

  1. Anna Shalova, André Schlichting, Mark Peletier. Singular-limit analysis of gradient descent with noise injection. [ arXiv | pdf | html ]
  2. Chun Yin Lam, André Schlichting. Variational convergence of exchange-driven stochastic particle systems in the thermodynamic limit. [ arXiv | pdf ]
  3. Ilka Budde, André Schlichting, David Ing, Sandra Schimmelpfennig, Joelle M-J Romac, Sandip M Swain, Rodger A. Liddle, Angela Stevens, Albrecht Schwab, Zoltán Pethő. Piezo1-induced durotaxis of pancreatic stellate cells depends on TRPC1 and TRPV4 channels. [ bioRxiv | pdf-main, pdf-supp-info, pdf-supp-num ]
  4. Daniel Matthes, Eva-Maria Rott, Giuseppe Savaré, André Schlichting. A structure preserving discretization for the Derrida-Lebowitz-Speer-Spohn equation based on diffusive transport. [ arXiv | pdf ]
  5. Antonio Esposito, Georg Heinze, André Schlichting. Graph-to-local limit for the nonlocal interaction equation. [ arXiv | pdf ]
  6. Anastasiia Hraivoronska, André Schlichting, Oliver Tse. Variational convergence of the Scharfetter-Gummel scheme to the aggregation-diffusion equation and vanishing diffusion limit. [ arXiv | pdf ]
  7. Martin Burger, Franca Hoffmann, Daniel Matthes, Matthias Erbar, André Schlichting. Covariance-modulated optimal transport and gradient flows. [ arXiv | pdf ]

Peer reviewed

  1. Antonio Esposito, Rishabh S. Gvalani, André Schlichting, Markus Schmidtchen, On a novel gradient flow structure for the aggregation equation. accepted at Calc. Var. & PDE. [ arXiv | pdf ]
  2. Antonio Esposito, Francesco S. Patacchini, André Schlichting. On a Class of Nonlocal Continuity Equations on Graphs. Eur. J. Appl. Math, 35(1), 2024, pp. 109-126. [ doi | arXiv | pdf ]
  3. Víctor Navarro-Fernández, André Schlichting. Error estimates for a finite volume scheme for advection-diffusion equations with rough coefficients.
    ESAIM: M2AN, Vol. 57, No. 4, 2023. [ doi | arXiv | pdf ]
  4. Víctor Navarro-Fernández, André Schlichting, Christian Seis. Optimal stability estimates and a new uniqueness result for advection-diffusion equations. Pure appl. anal. Vol. 4 (2022), No. 3, 571–596. [ doi | link | arXiv | pdf ]
  5. Mark Peletier, André Schlichting. Cosh gradient systems and tilting. Nonlinear Analysis Volume 231, June 2023, 113094. [ doi | arXiv | pdf ]
  6. Georg Menz, André Schlichting, Wenpin Tang, Tianqi Wu. Ergodicity of the infinite swapping algorithm at low temperature. Stoch. Process. their Appl. 151, 2022. [ doi | arXiv | pdf ]
  7. Barbara Niethammer, Robert L. Pego, André Schlichting, Juan J. L. Velázquez. Oscillations in a Becker-Döring model with injection and depletion. SIAM Journal on Applied Mathematics. 82(4), 2022. [ doi | arXiv | pdf ]
  8. André Schlichting, Christian Seis. The Scharfetter-Gummel scheme for aggregation-diffusion equations. IMA Journal of Numerical Analysis. Volume 42, Issue 3, July 2022, Pages 2361–2402. [ doi | free access | arXiv | pdf ]
  9. Constantin Eichenberg, André Schlichting. Self-similar behavior of the exchange-driven growth model with product kernel. Commun. Partial. Differ. Equ. 46(3), 2021. [ doi | free access | arXiv | pdf ]
  10. Antonio Esposito, Francesco S. Patacchini, André Schlichting, Dejan Slepčev. Nonlocal-interaction equation on graphs: gradient flow structure and continuum limit. Archive for Rational Mechanics and Analysis, 240, 699–760(2021) [ doi | arXiv | pdf ]
  11. Rishabh S. Gvalani, André Schlichting. Barriers of the McKean-Vlasov energy via a mountain pass theorem in the space of probability measures. Journal for Functional Analysis, 279(11), 108720 (2020). [ doi | arXiv | pdf ]
  12. Matthias Erbar, Max Fathi, André Schlichting, Entropic curvature and convergence to equilibrium for mean-field dynamics on discrete spaces. ALEA – Latin American Journal of Probability and Mathematical Statistics, 17, 445-471 (2020). [ doi | arXiv | pdf ]
  13. André Schlichting. The exchange-driven growth model: basic properties and longtime behavior. Journal Nonlinear Science, 30, 793–830 (2020) [ link | doi | arXiv | pdf ]
  14. José A. Carrillo, Rishabh S. Gvalani, Grigorios A. Pavliotis, André Schlichting. Long-time behaviour and phase transitions for the McKean–Vlasov equation on the torus. Archive for Rational Mechanics and Analysis, 235, 635–690 (2020) [ link | doi | arXiv | pdf ]
  15. André Schlichting, Martin Slowik.  Poincaré and logarithmic Sobolev constants for metastable Markov chains via capacitary inequalities. Annals of Applied Probability, 29(6), 3438-3488 (2019) [ arXiv | doi | pdf ]
  16. André Schlichting. Poincaré and Log–Sobolev Inequalities for Mixtures. Entropy 2019, 21(1). [ doi | arXiv | pdf ]
  17. Joseph G. Conlon, André Schlichting. A non-local problem for the Fokker-Planck equation related to the Becker-Döring model. Discret. Contin. Dyn. Syst. 39-4, 2019. [ doi | arXiv | pdf ]
  18. Manh Hong Duong, Agnes Lamacz, Mark A. Peletier, André Schlichting, Upanshu Sharma. Quantification of coarse-graining error in Langevin and overdamped Langevin dynamics. Nonlinearity 31(10), 2018. [ doi |  arXiv | pdf ]
  19. André Schlichting. Macroscopic limit of the Becker-Döring equation via gradient flows. ESAIM:COCV 25(22), 2019. [ doi | arXiv | pdf ]
  20. André Schlichting, Christian Seis.  Analysis of the implicit upwind finite volume scheme with rough coefficients. Numer. Math., 2017. [ doi | view only | arXiv | pdf ]
  21. Simon Eberle, Barbara Niethammer, André Schlichting. Gradient flow formulation and longtime behaviour of a constrained Fokker-Planck equation. Nonlinear Anal. 158C, 2017). [ doi | journal | arXiv| pdf ]
  22. André Schlichting, Christian Seis. Convergence rates for upwind schemes with rough
    coefficients. SIAM J. Numer. Anal. 55 (2), 812–840, 2017. [ doi | arXiv |pdf ]
  23. Matthias Erbar, Max Fathi, Vaios Laschos, André Schlichting. Gradient flow structure for McKean-Vlasov equations on discrete spaces. Discret. Contin. Dyn. Syst. 36, 2016. [ doi | arXiv | pdf ]
  24. Georg Menz, André Schlichting. Poincaré and logarithmic Sobolev inequalities by decomposition of the energy landscape. Ann. Probab. 42(5), 2014. [ link | arXiv | pdf ]
  25. André. Schlichting. Time discretisation for a class of singular phase field models. Adv. Math. Sci. Appl. 55 (2009) 665–700. [ link | pdf ]

Proceedings

  1. Chun Yin Lam, André Schlichting. Thermodynamic limit of stochastic particle systems via EDP convergence. Oberwolfach Report 57/2023. [ link | pdf ]
  2. Antonio Esposito, Georg Heinze, Jan-Frederik Pietschmann, André Schlichting. Graph-to-local limit for a multi-species nonlocal cross-interaction system. Proceedings in Applied Mathematics & Mechanics, 2023. [ doi | arXiv | pdf ]
  3. José A. Carrillo, Rishabh Gvalani, Grigorios A. Pavliotis, André Schlichting, Christian Seis. Phase transitions and a mountain pass theorem in the space of probability measures, Oberwolfach Report  10/2021.  [ link | pdf ]
  4. Antonio Esposito, Francesco Patacchini, Dejan Slepčev, André Schlichting, Christian Seis. Nonlocal equations on discrete spaces inspired by numerical schemes, Oberwolfach Report 29/2020. [ link | pdf ]
  5. Joseph G. Conlon, André Schlichting. A non-local Fokker-Planck equation related to nucleation and coarsening, Oberwolfach Report 6/2018. [ link | pdf ]
  6. A. Schlichting, M. Slowik. Capacitary inequalities in discrete setting and application to metastable Markov chains, Oberwolfach Report 35/2015. [ link | pdf ]
  7. E. Boissard, N. Gozlan, J. Lehec, C. Léonard, G. Menz, A. Schlichting.
    Some recent developments in functional inequalities. Journées MAS 2012. ESAIM: Proc. 44 338-354 (2014) [ link | pdf ]
  8. A. Schlichting and W. Sproessig. Norm estimations of the modified Teodorescu transform with application to a multidimensional equation of airy type.
    AIP Conference Proceedings, 1048(1):701-705, September 2008. [ doi | pdf ]

Thesis

  1. André Schlichting. Phase Transitions in Interacting Systems, Universität Bonn, Habilitation thesis, 2020. [ pdf ]
  2. André Schlichting. The Eyring-Kramers formula for Poincaré and logarithmic Sobolev inequalities, Universität Leipzig, PhD thesis, 2012. [ Qucosa | pdf ]
  3. A. Schlichting. Solvability, approximation and estimates for a class of singular
    phase field models. Master’s thesis, University of Mining and Technology Freiberg and University of Pavia, Diploma thesis, 2008. [ pdf ]