ERC Consolidator Grant

About Project

© Universität Augsburg
CC BY-NC-ND

CC BY-NC-ND
ERC: European Research Council

Computational Random Multiscale Problems

PI: Prof. Dr. Daniel Peterseim

 

Geometrically or statistically heterogeneous microstructures and high physical contrast are the key to astonishing physical phenomena such as invisibility cloaking with metamaterials or the localization of quantum waves in disordered media. Due to the complex experimental observation of such processes, numerical simulation has very high potential for their understanding and control. However, the underlying mathematical models of random partial differential equations are characterized by a complex interplay of effects on many non-separable or even a continuum of characteristic scales. The attempt to resolve them in a direct numerical simulation easily exceeds today's computing resources by multiple orders of magnitude. The simulation of physical phenomena from multiscale models, hence, requires a new generation of computational multiscale methods that accounts for randomness and disorder in a hierarchical and adaptive fashion.


This project concerns the design and numerical analysis of such methods. The main goals are connected to fundamental mathematical and algorithmic challenges at the intersection of multiscale modeling and simulation, uncertainty quantification and computational physics:


(A) Numerical stochastic homogenization beyond stationarity and ergodicity,
(B) Uncertainty quantification in truly high-dimensional parameter space,
(C) Computational multiscale scattering in random heterogeneous media,
(D) Numerical prediction of Anderson localization and quantum phase transitions.

These objectives base upon recent breakthroughs of deterministic numerical homogenization beyond periodicity and scale separation and its deep links to seemingly unrelated theories ranging all the way from domain decomposition to information games and their Bayesian interpretation. It is this surprising nexus of classical and probabilistic numerics that clears the way to the envisioned new computational paradigm for multiscale problems at randomness and disorder.

 

Community Research and Development Information Service (CORDIS) of the European Commission

Members

Wissenschaftlicher Mitarbeiter
Numerische Mathematik
Wissenschaftlicher Mitarbeiter
Numerische Mathematik
Research Assistant
Numerische Mathematik
Lehrstuhlinhaber
Numerische Mathematik

Events

Upcoming:

 

One World Numerical Analysis Seminar, series (every two weeks), online
Jena-Augsburg-Meeting (JAM) on Numerical Analysis, Universität Augsburg, 7.-10. June 2023

 

Previous:

 

Summer School - Uncertainty, Adaptivity, and Machine Learning, Universität Augsburg, 12.-14. September 2022
Workshop on scattering by random heterogeneous media, Universität Augsburg, 13.-15. September 2021

 

Talks

Upcoming:

 

22nd GAMM Seminar on Microstructures, TU Wien, 26.-28. January 2023

The 29th Biennial Numerical Analysis Conference 2021, Department of Mathematics and Statistics at the University of Strathclyde, 27.-30. June 2023

European Conference on Numerical Mathematics and Advanced Applications (ENUMATH), Lisbon, 20.-24. September 2023
 

 

Given talks:

 

[43]

 

P. Freese. Numerical homogenization of dispersive Maxwell systems, 12. December 2022 at Kolloquium für Angewandte Mathematik, Technische Universität Hamburg.

[42]

 

D. Peterseim. Nonlinear eigenvector problems and the simulation of Bose-Einstein condensates, 8. December 2022 at Lothar-Collatz-Kolloquium für Angewandte Mathematik, Universität Hamburg.

[41]

 

P. Freese. A super-localized generalized finite element method, 1. December 2022 at the Research Seminar Numerical Analysis, Universität Jena.

[40]

 

D. Peterseim. Nonlinear eigenvector problems and the simulation of Bose-Einstein condensates, 17. November 2022 at the Research Seminar Numerical Analysis, Universität Jena.

[39]

 

 

M. Hauck. Super-localization of elliptic multiscale problems with an extension to spatial networks, 28. September 2022 at Computational and Applied Mathematics (CAM) seminar, Chalmers University of Technology and University of Gothenburg.

[38]

 

P. Freese. Super-localized orthogonal decomposition for high-frequency Helmholtz problems, 30. August 2022 at CMAM, Vienna.

[37]

 

P. Freese. Super-localization for convection-dominated diffusion problems, 28. July 2022 at SciCADE, Reykjavik.

[36]

 

M. Hauck. Super-localization of the elliptic multiscale problem, 28. July 2022 at the 27th International Domain Decomposition Conference, Prague.

[35]

 

 

F. Bonizzoni. Rational-based MOR methods for Helmholtz frequency response problems with adaptive finite element snapshots, 25. July 2022 at the 15th International Conference on Mathematical and Numerical Aspects of Wave Propagation (WAVES), ENSTA Paris in Palaiseau, France.

[34]

 

P. Freese. Super-localized orthogonal decomposition for convection-dominated diffusion problems, 14. July 2022 at EQUADIFF, Brno.

[33]

 

D. Peterseim. Nonlinear Eigenvector problems and the simulation of Bose-Einstein condensates, 14. July 2022 at EQUADIFF, Brno (invited talk).

[32]

 

P. Freese. Super-localized orthogonal decomposition for convection-dominated diffusion problems, 6. July 2022 at Workshop on Numerical Methods and Analysis in CFD, Weierstrass Institute, Berlin.

[31]

 

 

D. Peterseim. Super-localized numerical homogenization and its links to variational multiscale and isogeometric analysis, 7. June 2022 at the 8th European Congress on Computational Methods in Applied Sciences and Engineering (ECCOMAS), Oslo (invited talk).

[30]

 

D. Peterseim. The localization problem in numerical homogenization, 23. May 2022 at Asymptotics, Operators, and Functionals Seminar, University of Bath (online).

[29]

 

D. Peterseim. Compression of partial differential operators by numerical homogenization, 12. May 2022 at Numerical methods for Compression and Learning, Gran Sasso Science Institute.

[28]

 

 

F. Bonizzoni. Rational-based MOR methods for parametric-in-frequency Helmholtz problems with adapted snapshots, 15. February 2022 at the Conference on Mathematics of Wave Phenomena, Karlsruhe Institute of Technology, Germany.

[27]

 

D. Peterseim. The localization problem in numerical homogenization, 13. January 2022 at Forschungsseminar Numerische Mathematik, Humboldt-Universität zu Berlin (online).

[26]

 

F. Bonizzoni. Model Order Reduction Methods for Time-Harmonic Wave Problems, 1. December 2021 at IWR Kolloquium, Romberg Inaugural Lecture, IWR Heidelberg.

[25]

 

D. Peterseim. On the localization problem in numerical homogenization, 25. November 2021 at Wilhelm Killing Kolloquium, Universität Münster.

[24]

 

 

D. Peterseim. Energy-adaptive Riemannian Optimization on the Stiefel Manifold, 3. November 2021 at Computational and Applied Mathematics (CAM) seminar, Chalmers University of Technology and University of Gothenburg (hybrid).

[23]

 

D. Peterseim. Energy-adaptive Riemannian optimization on the Stiefel manifold, 29. September - 1. October 2021 at Numerical Analysis Workshop, Universität Bielefeld.

[22]

 

D. Peterseim. Super-localization of elliptic multiscale problems, 22. Septermber 2021 at Second French-German Workshop on Multiscale Problems, Paris.

[21]

 

D. Peterseim. Riemannian gradient flows for nonlinear eigenvector problems, 17.-18. September 2021 at FORTH Heraklion, Crete.

[20]

 

 

F. Bonizzoni. Rational-based Model Order Reduction methods for parametric Helmholtz problems, 15. September 2021 at the workshop “Scattering by random heterogeneous media”, University of Augsburg, Germany.

[19]

 

F. Bonizzoni. Shape optimization for a noise reduction problem by non-intrusive parametric reduced modeling, 10. September 2021 at the 18th European Finite Element Fair 2021, Inria Paris, France.

[18]

 

F. Bonizzoni. Shape optimization for a noise reduction problem by non-intrusive parametric reduced modeling, 01. September 2021 at SIMAI 2020+2021, Parma, Italy.

[17]

 

F. Bonizzoni. A greedy method for the tracking of eigensolutions to parametric PDEs, 30. August 2022 at the CMAM 2022, Vienna, Austria.

[16]

 

F. Bonizzoni. Analytical derivation and sparse approximation of the recursive first moment equations for the Lognormal Darcy problem, 14. July 2021 at ICOSAHOM 2020/2021, Vienna, Austria (online).

[15]

 

D. Peterseim. Super-localization of elliptic multiscale problems, 12. July 2021 at workshop about new trends in numerical multiscale methods and beyond, Institut Mittag-Leffler, Stockholm, Sweden (online).

[14]

 

F. Bonizzoni. Shape optimization for a noise reduction problem by non-intrusive parametric reduced modeling, 08. July 2021 at the 6th ECCOMAS Young Investigators Conference, Valencia, Spain (online).

[13]

 

 

F. Bonizzoni. BGG sequences of tensor product finite elements with arbitrary continuity, 22. June 2022 at the Workshop “Hilbert Complexes: Analysis, Applications, and Discretizations”, Oberwolfach, Germany (invited talk).

[12]

 

 

F. Bonizzoni. Rational-based MOR methods for parametric-in-frequency Helmholtz problems with adapted snapshots, 14. June 2022 at the ESI Thematic Programme “Computational Uncertainty Quantification:

Mathematical Foundations, Methodology & Data", Universität Wien (invited talk).

[11]

 

F. Bonizzoni. H1-conforming finite element cochain complexes on Cartesian meshes, 07. June 2022 at the ECCOMAS 2022, Oslo, Norway.

[10]

 

F. Bonizzoni. Interplay between Isogeometric and Stochastic collocation for Uncertainty Quantification of timber beams, 14. April 2022 at the SIAM conference on Uncertainty Quantification 2022, Atlanta, Georgia, USA.

[9]

 

M. Hauck. Multiresolution Localized Orthogonal Decomposition for Helmholtz problems, 25. March 2021 at Sion Young Academics Workshop 2021 (online).

[8]

 

F. Bonizzoni. A reduced order model for the acoustic scattering of an airfoil, 18. March 2021 at the 2021 Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM) (online).

[7]

 

 

D. Peterseim. Introduction to numerical homogenization of PDEs with arbitrary rough coefficients, 16. March 2021 at the 2021 Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM) (online).

[6]

 

D. Peterseim. A priori error analysis of a numerical stochastic homogenization method, 01. March 2021 at

Scaling Cascades in Complex Systems 2021, FU Berlin (online).

[5]

 

D. Peterseim. Three thematic lectures on numerical homogenization, 15.-17. February 2021 at ICTS MATHLEC 2021 (invited lecture series, online).

[4]

 

D. Peterseim. Numerical methods for the nonlinear Schrödinger eigenvalue problem, 10. December 2020 at Analysis-Seminar Augsburg-Munich (invited talk, online).

[3]

 

D. Peterseim. Localized Eigenstates by Domain Decomposition, 8. December 2020 at 26th International Domain Decomposition Conference, Chinese University of Hong Kong (online).

[2]

 

D. Peterseim. Nonlinear eigenvector problems and the simulation of Bose-Einstein condensates, 4. December 2020 at Mathematical Colloquium, RWTH Aachen University (invited talk, online).

[1]

 

M. Hauck. Enriched Galerkin - Subcell enrichment and Application to the shallow water equations, 12. October 2020 at MoST 2020.

Publications

Submitted articles:

 

[23]

 

 

P. Freese, D. Gallistl, D. Peterseim, T. Sprekeler. Computational multiscale methods for nondivergence-form elliptic partial differential equations, ArXiv e-prints, 2022.

[ arXiv ]

 

[22]

 

 

G. Balduzzi, F. Bonizzoni, L. Tamellini. Uncertainty quantification in timber-like beams using sparse grids:

theory and examples with off-the-shelf software utilization, ArXiv e-prints, 2022.

[ arXiv ]

 

[21]

 

 

F. Bonizzoni, M. Hauck, and D. Peterseim. A reduced basis super-localized orthogonal decomposition for reaction-convection-diffusion problems, ArXiv e-prints, 2022.

[ arXiv ]

 

[20]

 

 

P. Freese, M. Hauck, T. Keil and D. Peterseim. A Super-Localized Generalized Finite Element Method, Arxiv e-prints, 2022.

[ arXiv ]

 

[19]

 

 

Z. Dong, M. Hauck and R. Maier. An improved high-order method for elliptic multiscale problems, Arxiv e-prints, 2022.

[ arXiv ]

 

[18]

 

M. Hauck and A.Målqvist. Super-localization of spatial network models, ArXiv e-prints, 2022.

[ arXiv ]

 

[17]

 

 

M. Alghamdi, D. Boffi and F. Bonizzoni. A greedy MOR method for the tracking of eigensolutions to parametric elliptic PDEs. ArXiv e-prints, 2022.

[ arXiv ]

 

[16]

 

 

F. Bonizzoni, P. Freese and D. Peterseim. Super-localized orthogonal decomposition for convection-dominated diffusion problems, ArXiv e-prints, 2022.

[ arXiv ]

 

[15]

 

 

P. Freese, M. Hauck and D. Peterseim. Super-localized Orthogonal Decomposition for high-frequency Helmholtz problems, ArXiv e-prints, 2021.

[ arXiv ]

 

[14]

 

 

F. Bonizzoni, D. Pradovera and M. Ruggeri. Rational-based model order reduction of Helmholtz frequency response problems with adaptive finite element snapshots. ArXiv e-prints, 2021.

[ arXiv ]

 

 

Refereed articles:

 

[13]

 

 

R. Altmann, D. Peterseim and T. Stykel. Energy-adaptive Riemannian optimization on the Stiefel manifold. ESAIM: M2AN, 56(5): 1629-1653, 2022.

[ arXiv | DOI ]

 

[12]

 

 

F. Kröpfl, R. Maier and D. Peterseim. Operator compression with deep neural networks. Adv Cont Discr Mod 2022, 29 (2022).

[ arXiv | DOI ]

 

[11]

 

 

M. Hauck and D. Peterseim. Super-localization of elliptic multiscale problems. Mathematics of Computation, 2022.

[ arXiv | DOI ]

 

[10]

 

 

M. Hauck and D. Peterseim. Multi-resolution Localized Orthogonal Decomposition for Helmholtz problems. SIAM Multiscale Modeling & Simulation, 20(2): 657-684, 2022.

[ DOI ]

 

[9]

 

 

R. Altmann, P. Henning and D. Peterseim. Localization and delocalization of ground states of Bose-Einstein condensates under disorder. Accepted for publication in SIAM J. Appl. Math., 2021.
[ arXiv | DOI ]
 

[8]

 

 

R. Altmann, P. Henning and D. Peterseim. The J-method for the Gross-Pitaevskii eigenvalue problem. Numerische Mathematik, 148(3): 575-610, 2021.
[ arXiv | DOI ]
 

[7]

 

 

R. Altmann, P. Henning and D. Peterseim. Numerical homogenization beyond scale separation. Acta Numerica, pp. 1-86, 2021.

[ DOI ]

 

[6]

 

 

J. Fischer, D. Gallistl and D. Peterseim. A priori error analysis of a numerical stochastic homogenization method. SIAM J. Numer. Anal., 59(2): 660-674, 2021.
[ arXiv | DOI ]
 

 

Monograph:

 

[5]

 

 

A. Målqvist and D. Peterseim. Numerical homogenization by localized orthogonal decomposition. SIAM Spotlights 5, ISBN: 978-1-611976-44-1, 2020.
[ SIAM ]
 

 

Preliminary work

 

[4]

 

 

M. Feischl and D. Peterseim. Sparse compression of expected solution operators. SIAM J. Numer. Anal.,

58(6):3144-3164, 2020.

[ arXiv | DOI ]

[3]

 

 

P. Henning and D. Peterseim. Sobolev gradient flow for the Gross-Pitaevskii eigenvalue problem: global convergence and computational efficiency. SIAM J. Numer. Anal., 58(3):1744–1772, 2020.
[ arXiv | DOI ]

[2]

 

 

R. Altmann, P. Henning, and D. Peterseim. Quantitative Anderson localization of Schrödinger eigenstates under disorder potentials. M3AS Math. Models Methods Appl. Sci., 30(5):917-955, 2020.

[ arXiv | DOI ]

[1]

 

 

D. Peterseim and B. Verfürth. Computational high frequency scattering from high contrast heterogeneous media. Math. Comp., 89:2649-2674, 2020.
[ arXiv | DOI ]
 

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