T9: Multiscale processes and effective behaviour

Many processes in physics, engineering and life sciences involve multiple spatial and temporal scales, where the underlying geometry and dynamics on the smaller scales typically influence the emerging structures on the coarser ones. A unifying theme running through this research topic is to identify the relevant spatial and temporal scales governing the processes under examination. This is achieved, e.g., by establishing sharp scaling laws, by rigorously deriving effective scale-free theories and by developing novel approximation algorithms which balance various parameters arising in multiscale methods.

In optimisation and design problems, multiple scales can either emerge as optimal structures, or they can be a priori present in the system that one wants to control. In the first case, the main challenge is to characterise emergence of multiple scales and to develop numerical methods that can identify the highly complex optimisers. In the second case, the challenge is to analytically or numerically approximate the macroscopic system behaviour with sufficiently high efficiency such that it can form part of each step in an iterative optimisation.

  • Mathematical fields

    • Applied analysis and theory of PDEs
    • Stochastic analysis
    • Optimisation and calculus of variation
    • Numerical analysis, machine learning and scientific computing

  • Collaborations with other Topics

  • Selected publications and preprints

    since 2019

    $\bullet $ Michael Kartmann, Tim Keil, Mario Ohlberger, Stefan Volkwein, and Barbara Kaltenbacher. Adaptive reduced basis trust region methods for parameter identification problems. Computational Science and Engineering, September 2024. doi:10.1007/s44207-024-00002-z.

    $\bullet $ M. Holler, A. Schlüter, and B. Wirth. Dimension reduction, exact recovery, and error estimates for sparse reconstruction in phase space. Applied and Computational Harmonic Analysis, 70:101631, May 2024. doi:10.1016/j.acha.2024.101631.

    $\bullet $ Tim Keil and Mario Ohlberger. A relaxed localized trust-region reduced basis approach for optimization of multiscale problems. ESAIM: Mathematical Modelling and Numerical Analysis, 58(1):79–105, January 2024. doi:10.1051/m2an/2023089.

    $\bullet $ Christian Engwer, Mario Ohlberger, and Lukas Renelt. Model order reduction of an ultraweak and optimally stable variational formulation for parametrized reactive transport problems. arXiv e-prints, October 2023. arXiv:2310.19674.

    $\bullet $ Tim Keil and Stephan Rave. An online efficient two-scale reduced basis approach for the localized orthogonal decomposition. SIAM Journal on Scientific Computing, 45(4):A1491–A1518, August 2023. doi:10.1137/21M1460016.

    $\bullet $ Lucia Scardia, Konstantinos Zemas, and Caterina Ida Zeppieri. Homogenisation of nonlinear Dirichlet problems in randomly perforated domains under minimal assumptions on the size of perforations. arXiv e-prints, July 2023. arXiv:2307.11605.

    $\bullet $ Gianluca Favre, Marlies Pirner, and Christian Schmeiser. Hypocoercivity and reaction-diffusion limit for a nonlinear generation-recombination model. Archive for Rational Mechanics and Analysis, 247(4):72, July 2023. doi:10.1007/s00205-023-01902-8.

    $\bullet $ Christian Seis. Bounds on the rate of enhanced dissipation. Comm. Math. Phys., 399(3):2071–2081, May 2023. doi:10.1007/s00220-022-04588-3.

    $\bullet $ Stephan Luckhaus and Angela Stevens. Kermack and McKendrick models on a two-scale network and connections to the Boltzmann equations. In Jean-Michel Morel and Bernard Teissier, editors, Mathematics Going Forward, Lecture Notes in Mathematics, pages 417–427. Springer International Publishing, January 2023. doi:10.1007/978-3-031-12244-6_29.

    $\bullet $ Rodrigo Bazaes, Alexander Mielke, and Chiranjib Mukherjee. Stochastic homogenization for Hamilton-Jacobi-Bellman equations on continuum percolation clusters. arXiv e-prints, August 2022. arXiv:2208.07269.

    $\bullet $ Julius Lohmann, Bernhard Schmitzer, and Benedikt Wirth. Formulation of branched transport as geometry optimization. J. Math. Pures Appl., 163:739–779, July 2022. doi:10.1016/j.matpur.2022.05.021.

    $\bullet $ Jonas Potthoff and Benedikt Wirth. Optimal fine-scale structures in compliance minimization for a uniaxial load in three space dimensions. ESAIM: Control Optim. Calc. Var., 28:27, May 2022. doi:10.1051/cocv/2022023.

    $\bullet $ Martin Hutzenthaler, Arnulf Jentzen, Thomas Kruse, Tuan Anh Nguyen, and Philippe von Wurstemberger. Overcoming the curse of dimensionality in the numerical approximation of semilinear parabolic partial differential equations. Proc. A., 476(2244):20190630, 25, December 2020. doi:10.1098/rspa.2019.0630.

    $\bullet $ Christian Engwer, Sandra May, Andreas Nüßing, and Florian Streitbürger. A stabilized DG cut cell method for discretizing the linear transport equation. SIAM J. Sci. Comp., 42(6):A3677–A3703, January 2020. doi:10.1137/19M1268318.

    $\bullet $ Filippo Cagnetti, Gianni Dal Maso, Lucia Scardia, and Caterina Ida Zeppieri. Stochastic homogenisation of free-discontinuity problems. Arch. Ration. Mech. An., 233(2):935–974, March 2019. doi:10.1007/s00205-019-01372-x.

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  • Recent publications and preprints

    since 2023

    $\bullet $ Marlies Pirner. A consistent kinetic Fokker-Planck model for gas mixtures. Journal of Statistical Physics, 191(11):Paper No. 143, 18, October 2024. doi:10.1007/s10955-024-03361-1.

    $\bullet $ Michael Kartmann, Tim Keil, Mario Ohlberger, Stefan Volkwein, and Barbara Kaltenbacher. Adaptive reduced basis trust region methods for parameter identification problems. Computational Science and Engineering, September 2024. doi:10.1007/s44207-024-00002-z.

    $\bullet $ Cyrill B. Muratov, Theresa M. Simon, and Valeriy V. Slastikov. Existence of higher degree minimizers in the magnetic skyrmion problem. arXiv e-prints, September 2024. arXiv:2409.07205.

    $\bullet $ M. Holler, A. Schlüter, and B. Wirth. Dimension reduction, exact recovery, and error estimates for sparse reconstruction in phase space. Applied and Computational Harmonic Analysis, 70:101631, May 2024. doi:10.1016/j.acha.2024.101631.

    $\bullet $ Tim Keil and Mario Ohlberger. A relaxed localized trust-region reduced basis approach for optimization of multiscale problems. ESAIM: Mathematical Modelling and Numerical Analysis, 58(1):79–105, January 2024. doi:10.1051/m2an/2023089.

    $\bullet $ Christian Engwer, Mario Ohlberger, and Lukas Renelt. Model order reduction of an ultraweak and optimally stable variational formulation for parametrized reactive transport problems. arXiv e-prints, October 2023. arXiv:2310.19674.

    $\bullet $ Tim Keil and Stephan Rave. An online efficient two-scale reduced basis approach for the localized orthogonal decomposition. SIAM Journal on Scientific Computing, 45(4):A1491–A1518, August 2023. doi:10.1137/21M1460016.

    $\bullet $ Lucia Scardia, Konstantinos Zemas, and Caterina Ida Zeppieri. Homogenisation of nonlinear Dirichlet problems in randomly perforated domains under minimal assumptions on the size of perforations. arXiv e-prints, July 2023. arXiv:2307.11605.

    $\bullet $ Gianluca Favre, Marlies Pirner, and Christian Schmeiser. Hypocoercivity and reaction-diffusion limit for a nonlinear generation-recombination model. Archive for Rational Mechanics and Analysis, 247(4):72, July 2023. doi:10.1007/s00205-023-01902-8.

    $\bullet $ Martin Hutzenthaler, Arnulf Jentzen, Thomas Kruse, and Tuan Anh Nguyen. Overcoming the curse of dimensionality in the numerical approximation of backward stochastic differential equations. J. Numer. Math., 31(2):1–28, July 2023. doi:10.1515/jnma-2021-0111.

    $\bullet $ Christian Seis. Bounds on the rate of enhanced dissipation. Comm. Math. Phys., 399(3):2071–2081, May 2023. doi:10.1007/s00220-022-04588-3.

    $\bullet $ Christian Beck, Arnulf Jentzen, Konrad Kleinberg, and Thomas Kruse. Nonlinear Monte Carlo methods with polynomial runtime for Bellman equations of discrete time high-dimensional stochastic optimal control problems. arXiv e-prints, March 2023. arXiv:2303.03390.

    $\bullet $ Stephan Luckhaus and Angela Stevens. Kermack and McKendrick models on a two-scale network and connections to the Boltzmann equations. In Jean-Michel Morel and Bernard Teissier, editors, Mathematics Going Forward, Lecture Notes in Mathematics, pages 417–427. Springer International Publishing, January 2023. doi:10.1007/978-3-031-12244-6_29.

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    further publications

Back to research programme

Determination and optimisation of structures

© MM/vl

Investigators: Jentzen, Simon, Wirth

In this unit, we illustrate how our strong track record in the area of optimisation, numerical complexity reduction and multiscale modelling are used to tackle these problems. We also propose approximation algorithms that solve optimal control problems for Hamilton–Jacobi–Bellman equations without the so-called curse of dimensionality, thus allowing a very large number of control variables.

Homogenisation and multiscale methods

© MM/vl

Investigators: Mukherjee, Ohlberger, Rave, Wirth, Zeppieri

In this unit, we consider multiscale problems which can be recast into the theory of non-linear homogenisation. Concretely, we study the convergence rate to the homogenised limit of systems of non-linear elliptic PDEs defined in randomly perforated media where perforations overlap with high probability. Moreover, motivated by applications to combustion and propagation of fronts in random environments, we study homogenisation of viscous Hamilton-Jacobi-Bellman (HJB) equations on continuum percolation clusters, which bring fundamental challenges due to the inherent non-stationarity, non-ellipticity and lack of global Lipschitz properties. Model order reduction (MOR) methods will also be developed and analysed for efficient approximation of dynamic multiscale problems and related PDE constrained optimisation and inverse problems.

Mixing and equilibration

© MM/vl

Investigators: Engwer, Ohlberger, Pirner, Seis, Stevens

Here we study multiscale problems through the lens of mixing and equilibration. Our incentive is guided by pivotal questions in the realm of fluid dynamics such as identifying the maximal rates of mixing and studying enhanced dissipation within certain fluid flows. Our investigation also extends to treating kinetic equations and developing numerical methods to minimise numerical dissipation and address transport and conservation laws.