## What is it?

The reading course is a medium-term (typically one semester or so) group study of a specific topic, based on periodic sessions. The course sessions last two hours and are scheduled bi-weekly (see the calendar below for further details).

A paper or a chapter from a book is discussed during each meeting and presented by one of the attendees. The topics are chosen among relevant trends in Numerical Analysis and Linear Algebra and are based on requests from the audience.

Anybody is welcome to join, and Ph.D. and master’s students are particularly encouraged to participate.

The reading course takes place at the Department of Mathematics. It is currently coordinated by Fabio Durastante, Stefano Massei, and Leonardo Robol (if you wish to join the group just let us know!).

## How to join us

Just subscribe to the reading course mailing list. All communications will be delivered there.

## Calendar and Topics

### Fall 2023: Riemannian Optimization

The new edition of the reading course will focus on optimization on smooth manifolds, following chapters in the book Boumal, Nicolas An introduction to optimization on smooth manifolds. Cambridge University Press, Cambridge, 2023. xviii+338 pp. ISBN:978-1-009-16617-1.

**17/11**, **16:00-18:00** **Aula Riunioni**

[1, Chapter 3] Embedded geometry: first order

Speaker: Igor Simunec**24/11**, **16:00-18:00** **Aula Riunioni**

[1, Chapter 4] First-order optimization algorithms

Speaker: Alberto Bucci**4/12**, **14:00-16:00** **Aula Riunioni**

[1, Chapter 5] Embedded geometry: second order

Speaker: Nikita Deniskin**26/1**, **16:00-18:00** **Aula Seminari Ex-Albergo**

[1, Chapter 6] Second-order optimization algorithms

Speaker: Santolo Leveque**9/2, 16:00-18:00 Aula Seminari Ex-Albergo**

[2,3], Stochastic $p$th root approximation of a stochastic matrix: A Riemannian optimization approach

Speaker: Fabio Durastante**23/2, 16.00-18:00 Aula Seminari Ex-Albergo**

Alcune applicazioni: dal quoziente di Rayleigh alla nearest stable matrix

Speaker: Federico Poloni

#### Bibliography

[1] Boumal, Nicolas An introduction to optimization on smooth manifolds. Cambridge University Press, Cambridge, 2023. xviii+338 pp. ISBN:978-1-009-16617-1.

[2] Douik, Ahmed; Hassibi, Babak Manifold optimization over the set of doubly stochastic matrices: a second-order geometry. IEEE Trans. Signal Process.67(2019), no.22, 5761–5774.

[3] Durastante, Fabio; Meini, Beatrice Stochastic $p$th root approximation of a stochastic matrix: A Riemannian optimization approach. arXiv:2307.14040 (to appear on SIMAX)

## Previous editions

This session is concerned with randomized methods for linear systems, eigenvalue problems, least squares, factorizations, low-rank approximation, and trace estimation problems.

**03/10/2022, 14:00--16:00**, Aula Riunioni

*Introduction to randomized low-rank approximation: randomized embeddings* [1, Chapters 6-10].

Speaker: Angelo Casulli.

**17/10/2022, 14:00--16:00**, Aula Riunioni

*Randomized range finder, interpolative decompositions, and Randomized SVD* [1, Chapters 11-14] and [2].

Speaker: Alberto Bucci.

**07/11/2022, 14:00--16:00**, Aula Riunioni

*Randomized least squares: Blendenpik* [3].

Speaker: Chiara Faccio.

**21/11/2022**, **14:00--16:00**, Aula Riunioni

*Randomized methods for eigenvalues and linear systems* [4].

Speaker: Igor Simunec.

**05/12/2022**, **14:00--16:00**, Aula Riunioni

*Hutch++: Optimal stochastic trace estimation* [5].

Speaker: Michele Rinelli.

**20/12/2022, 11:00--13:00**, Aula Magna

*Norm and trace estimation with random rank one vectors* [6].

Speaker: Nikita Deniskin.

#### Bibliography

[1] *Randomized numerical linear algebra: Foundations and algorithms*, Martinsson and Tropp, Acta Numerica, 2020.

[2] *Finding structure with randomness: Probabilistic algorithms for constructing approximate matrix decompositions*, Haiko et. al, SIAM review, 2011.

[3] Blendenpik: Supercharging LAPACK'S least-squares solver, Avron et. al, SISC, 2010.

[4] *Fast & accurate randomized algorithms for linear systems and eigenvalue problems*, Nakatsukasa and Tropp, arXiv, 2021.

[5] *Hutch++: Optimal stochastic trace estimation*, Meyer et. al, SIAM, 2021.

[6] *Norm and trace estimation with random rank one vectors*, Bujanovic and Kressner, SIMAX, 2021.

[7] *Input sparsity time low-rank approximation via Ridge leverage score sampling*, Cohen et. al, SIAM, 2017.

[8] *Random matrix theory*, Edelman and Rao, Acta Numerica, 2005.

The new edition of the reading course will focus on Model Order Reduction, following chapters in the book edited by Peter Benner, Mario Ohlberger, Albert Cohen, and Karen Willcox, *Model Reduction and Approximation* (find it here on the SIAM website).

**21/03, 16:00 -- 18:00**, Aula Riunioni

*An Introduction to LTI systems and Balanced Truncation*

Speaker: Angelo Casulli, SNS.

**04/04, 16:00 -- 18:00**, Aula Riunioni

*Model Order Reduction by Rational Approximation: rational Krylov and IRKA*

Speaker: Alberto Bucci, University of Pisa

**18/04, 16:00 -- 18:00**, Aula Riunioni

*An introduction to Proper Orthogonal Decomposition*

Speaker: Chiara Faccio, SNS

**02/05, 16:00 -- 18:00**, Aula Riunioni

*POD for linear-quadratic control (part 1)*

Speaker: Santolo Leveque, SNS

**08/05, 14:00 -- 16:00**, Aula Riunioni

*POD for linear-quadratic control (part 2)*

Speaker: Santolo Leveque, SNS

**23/05, 16:00 -- 18:00**, Aula Riunioni

*The Loewner framework*

Speaker: Igor Simunec, SNS

**06/06, 16:00 -- 18:00**, Aula Riunioni

Symplectic Model Reduction of Hamiltonian Systems

Speaker: Milo Viviani, SNS

**27/06, 16:00 -- 18:00**, Aula Seminari

An introduction to Reduced Basis Methods for stationary parametrized PDEs

Speaker: Rafael Diaz Fuentes

### Bibliography

[1] Model Reduction and Approximation Theory and Algorithms. Benner, Ohlberger, Cohen, and Wilcox, SIAM, 2017.

[2] Symplectic Model Reduction of Hamiltonian Systems. Peng and Mohseni, SIAM journal on Scientific Computing, 2016.