ERC Consolidator Grant

About Project

© University of 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

Research Associate
Computational Mathematics
Research Associate
Computational Mathematics
Chair
Computational Mathematics

Events

Talks

Upcoming:

The 29th Biennial Numerical Analysis Conference 2021, Department of Mathematics and Statistics at the University of Strathclyde, 22. - 25. June 2021

European Conference on Numerical Mathematics and Advanced Applications (ENUMATH 2021), Lisbon, 20. 24. Septermber 2023
 

 

Given talks:

 

[8]

 

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

[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).

[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. (invited talk)

[5]

 

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

[4]

 

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

[3]

 

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

[2]

 

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

[1]

 

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

Publications

Submitted articles:

 

[11]

 

F. Kröpfl, R. Maier and D. Peterseim. Operator compression with deep neural networks. ArXiv e-prints, 2021.

arXiv ]

 

[10]

 

 

M. Hauck and D. Peterseim. Multi-resolution Localized Orthogonal Decomposition for Helmholtz problems. ArXiv e-prints, 2021.

arXiv ]

 

[9]

 

 

R. Altmann, P. Henning and D. Peterseim. Localization and delocalization of ground states of Bose-Einstein condensates under disorder. ArXiv e-prints, 2020.
arXiv ]
 

 

Refereed articles:

 

[8]

 

 

R. Altmann, P. Henning and D. Peterseim. The J-method for the Gross-Pitaevskii eigenvalue problem. Accepted for publication in Numerische Mathematik, 2021.
arXiv ]
 

[7]

 

R. Altmann, P. Henning and D. Peterseim. Numerical homogenization beyond scale separation. To appear in Acta Numerica, 2021.  

[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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