# Topic

Natural or engineered materials often contain two or more key constituents, arranged in a heterogeneous structure varying at different scales. Such materials are desirable because their macroscopic properties can be superior to the properties of the individual constituents. It is even possible to explicitly design them for a particular purpose by changing the composition of the constituents.

An example are carbon fibre composites for lightweight structures and vehicles. The mathematical modelling of such heterogeneous or composite materials typically leads to PDEs with highly oscillating coefficients. Direct numerical solution of such problems with traditional methods, such as finite elements is computationally expensive. Just to compute the correct qualitative behaviour, the mesh resolution would need to be sufficiently high to capture all the fine scale variation.

In this seminar, we will study multiscale numerical methods that address various aspects of this challenge. This includes the generation of low-dimensional yet high-quality approximation spaces, acceleration of fine-scale solutions and efficient uncertainty quantification for multiscale problems.

Each student will present a key publication on multiscale numerical methods with an aim to cover most aspects of the field.

## Prerequisites:

Basic knowledge of partial differential equations, Sobolev spaces and Finite Element methods is required. For the uncertainty quantification topics, basics of probability theory are needed.

# Registration and Schedule

**First meeting**: April 22, 2022, 13:00. Room: Mathematikon, SR 0.200Since we use

**MÜSLI**for email communication, please register at: https://muesli.mathi.uni-heidelberg.de/lecture/view/1517Schedule: Two talks each on May 19, June 2, June 9, 9:00 - 11:00.

# Contact

## Main contact:

Linus Seelinger: linus.seelinger at iwr.uni-heidelberg.de

## Secondary contact:

Robert Scheichl: r.scheichl at uni-heidelberg.de

# Structure

As part of this seminar, you will

- develop a solid understanding of an individual topic,
- give a talk on that topic to the other students,
- write a report on that topic,
- and participate in peer assessment of other presentations.

## Talks

The length of each talk is 35 min. + 10 min. for questions and discussion.

**Before your talk you should meet with one of us** to discuss your presentation.

## Reports

Reports will be due **4 weeks after your seminar talk**. The page limit for the report
is around 10 pages.

Your report should have a clear structure, and you should correctly reference sources used.

## Peer assessment

You will be responsible for assessing one other presentation using the same criteria we apply (Assessment guide), and your assessment will enter in the presenters’ grade. The quality of your assessment will in turn be part of your own grade.

Hint: Asking questions is good!

# Grades

Your seminar contribution will be graded based on

- talk (60%), depending on content, structure and quality of presentation,
- report (30%), depending on content, structure and quality of presentation,
- peer assessment (10%), depending on consistency and quality of your critique / feedback.

For peer assessment, please use the assessment form. You can directly fill in the PDF form, this is supported by most PDF readers. Use the assessment guide as a reference for how many points to award.

## Talk guidelines

- Give a general overview of the topic, focus on one or two specific points in detail.
- Make the topic accessible to others (e.g. carefully introduce notation, do not assume too much prior knowledge).
- Show key theoretical aspects and possibly conduct numerical experiments (focus may vary depending on topic and preference).
- Keep a clear structure, carefully select what to show and what to omit. The talk should have an introduction with an overview of the talk, a central part containing the content, and some conclusions at the end. The central part should in itself have a clear structure.
- Make the talk interesting to hear (i.e. “tell a story”, show to the audience why your topic is interesting and useful).
- Give an easy to follow presentation (e.g. do not overload slides, do not talk too slowly or too fast, give conclusions to sections of your talk to wrap them up).

## Report guidelines

- Cover the same content as in your talk. The points above mostly apply to the report as well.
- You should go into more detail. For example, omitting technical details may be fine in a talk for clarity of presentation; the report however should be technically correct.
- Properly cite sources you use.
- Write in a concise, yet accessible style. Relevant points should be written out in an easy to follow way. On the other hand redundancy should be avoided since it distracts from the content.
- As with the talk your report should have a clear structure. Do not forget to add conclusions.
- There is no formal page requirement or limit, but around 10 pages (without appendices, e.g., containing code) should be okay. Of course that depends on how many figures you add to your report.

# Papers (not exclusive)

## Multiscale Methods

- G. Allaire and R. Brizzi, A multiscale finite element method for numerical homogenization, Multiscale Model. Simul. (SIAM), 2005.
- I. Babuska, R. Lipton, Optimal Local Approximation Spaces for Generalized Finite Element Methods with Application to Multiscale Problems, Multiscale Model. Simul. (SIAM), 2010
- Y. Efendiev, J. Galvis and T.Y. Hou, Generalized multiscale finite element methods (GMsFEM), J. Comput. Phys., 2013
- Y. Efendiev and T.Y. Hou, Multiscale Finite Element Methods: Theory and Applications (Chapters 2 and 6)., Springer, New York, 2009
- T.Y. Hou and P, Liu, Optimal local multi-scale basis functions for linear elliptic equations with rough coefficients, Discr. Contin. Dyn. Sys., 2016
- T.J.R. Hughes and G. Sangalli, Variational multiscale analysis: the fine-scale Green’s function, projection, optimization, localization, stabilized methods, SIAM J. Numer. Anal., 2007.
- R. Kornhuber, D. Peterseim, and H. Yserentant, An analysis of a class of variational multiscale methods based on subspace decomposition, Math. Comput., 2018.
- A. Malqvist and D. Peterseim, Localization of elliptic multiscale problems, Math. Comput., 2014.
- P. Ming and P. Zhang, Analysis of the heterogeneous multiscale method for elliptic homogenization problems, J. American Math. Soc., 2005.
- C. Ma and R. Scheichl, Error estimates for fully discrete generalized FEMs with locally optimal spectral approximations, 2021 (preprint)

## Preconditioners for Multiscale Problems

- V. Dolean, F. Nataf, R. Scheichl et al., Analysis of a Two-level Schwarz Method with Coarse Spaces Based on Local Dirichlet-to-Neumann Maps, Computational Methods in Applied Mathematics, 2012
- N. Spillane, V. Dolean, P. Hauret et al., Abstract robust coarse spaces for systems of PDEs via generalized eigenproblems in the overlaps, Numerische Mathematik, 2014

## Uncertainty Quantification for Multiscale Problems

- M. Giles, Multilevel Monte Carlo Path Simulation, Oper. Res., 2008
- T. Dodwell, C. Ketelsen, R. Scheichl et al., Multilevel Markov Chain Monte Carlo, SIAM Review, 2019