CIME Summerschool

Model order reduction and applications


Aims of the school

The behaviour of processes in several fields like mechanical engineering, geophysics, seismic modeling,climate/weather prediction is usually modeled by dynamical systems. Such models often involve systems of nonlinear partial differential equations. Their approximation by classical discretization techniques like finite difference and finite element methods, leads to high-dimensional systems of ordinary differential equations or difference equations. The number of equations is typically very large and it can easily reach a few million. Even after a linearization this system is therefore at best computationally very expensive, and often it is not feasible to simulate its evolution. The aim of model order reduction(MOR) is to develop reduced order models giving an accurate approximation of the dynamics of the underlying large-scale system, by enabling the reduction process to be implemented as computationally efficient and fast. Today’s computational and experimental paradigms feature complex models along with disparate and, frequently, enormous data sets. This has motivated the development of theoretical and computational strategies for the construction of efficient and robust numerical algorithms that effectively resolve the important features and characteristics of these complex computational models, possibly in real time when needed by the application. Clearly resolving the underlying model is often application-specific and combines mathematical tasks like approximation, prediction, calibration, design, control and optimization. In fact running simulations that fully account for the variability of the complexities of modern scientific models can be a very difficult task due to the curse of dimensionality, chaotic behaviour of dynamics, and/or overwhelming streams of informative data.

Focus and organization
The School will address the state of the art of reduced order methods for modeling and computational reduction of complex parametrized systems, governed by ordinary and/or partial differential equations,with a special emphasis on real time computing techniques and applications in various fields. The lecturers of the school are internationally recognized experts in MOR and other related areas and will present several point of view and techniques to solve demanding problems of increasing complexity.The school will focus on theoretical investigation and applicative algorithm development for reduction in the complexity – the dimension, the degrees of freedom, the data – arising in these models.
The four broad
thrusts of the program are: (1) Mathematics of reduced order models, (2) Algorithms for approximation and complexity reduction, (3) Computational statistics and data-driven techniques, and (4) Application-specific design. The particular topics include classical strategies such as parametric sensitivity analysis and best approximations, as well as mature but active topics like principal component analysis and information-based complexity, and also rising promising topics such as layered neural networks and high-dimensional statistics.

We would like to attract researchers and PhD students working or willing to work on model order reduction, data-driven model calibration and simplification, computations and approximations in high dimensions, and data-based uncertainty quantification. Hopefully investigation and assimilation of complementary approaches will create a productive cross-fertilization and serve as a stronger and more structured link for several diverse research communities.



We plan to organize the school at Hotel S. Michele in Cetraro from June 29 to July 3. 
Venue: Hotel S. Michele, Cetraro (CS)


School Directors

Prof. Maurizio Falcone
Dipartimento di Matematica Università di Roma ”La Sapienza”

Prof. Gianluigi Rozza
Dipartimento di Matematica,



CIME, Firenze

PRIN 2017: Innovative Numerical Methods for Evolutionary Partial Differential Equations and Applications, Project code: 2017KKJP4X

Dipartimento di Matematica, Sapienza Università di Roma

SISSA, Trieste


Teachers and courses


The school program will include five courses. All the confirmed speakers have agreed to write the lecture notes of their courses. The first course is an introduction to the subject. The four remaining courses will be focused on specific techniques and/or applications and are scheduled on 4.5 hours, i.e. 3 blocks of 1.5hours. The final time table is shown below.The school will integrate in its courses several tools coming from mathematical analysis, statistical sciences, numerical analysis, data and computer science. To allow prospective students to be prepared for the courses, a list of suggested readings will be announced on the WEB page of the school several months in advance. Many lectures will also be held in-class, but, due to the pandemia restrictions, all the courses will be delivered on-line using the ZOOM platform in order to allow the participants to follow the school by distance.

Hours Mon, 28 Jun Tue, 29 Jun Wed, 30 Jun Thu, 1 Jul Fri, 2 Jul Sat, 3 Jul
9.00-10.30       Course 2 Course 3 Course 5
11.00-12.30     Course 2 Course 3 Course 5 Course 3
12.30-15.30     Lunch Lunch Lunch Lunch
15.30-17.00   Course 1  Course 1 Course 2 Seminars Course 5
17.30-19.00   Course 1 Course 4 Course 4 Course 4  


Course 1 (4.5 hours, 3 blocks)
Systems-theoretic Optimal Model Reduction of Dynamical Systems
Prof. Serkan Gugercin (Virginia Tech)
Abstract:In this set of lectures, we will focus on optimal model reduction dynamical systems using tools from systems and control theory and rational approximation. We will discuss how to construct input-independent optimal reduced models that are guaranteed to provide high-fidelity approximation sto underlying original dynamics. Both linear and nonlinear dynamics will be studied.


Course 2 (4.5 hours, 3 blocks)
MOR for optimal control problems
Prof. Dr. Michael Hinze (University of Koblez-Landau)

Lecture 1: We will cover the construction of MOR reduced order models for nonlinear PDE systems. The approximation of the nonlinearities is performed with (D)EIM and/or QDEIM. Emphasis will be taken on the choice of the inner product for the basis construction and the treatment of spatially adaptively generated snapshots, and also on reduced basis approximations. Furthermore, error analysis will be sketched.
Lecture 2: This lecture deals with the use of MOR models in optimization with PDE constraints. Emphasis is put on the variational discretization of the controls, which is perfectly tailored to the use of MOR models for the state approximation. In addition we introduce a novel snapshot location procedure for MOR in optimal control. A priori and a posteriori error analysis will be sketched.
Lecture 3: This lecture deals with certification of MOR models in parametrized optimal control, where the emphasis is taken on reliability and also effectivity of the MOR approximation. Concepts from a posteriori finite element analysis are adapted for the construction of a sharp (up to a constant) error bound for the variables involved in the optimization process. Moreover, we sketch convergence of the approach.

Course 3 (4.5 hours, 3 blocks)
Reduced modeling and learning for state estimation
Dr. Olga Mula (CEREMADE, Paris Dauphine)
WEB: mula/index.html
Abstract:This course is devoted to inverse state estimation problems where the goal is to compute a fast reconstruction of the state of a physical system from available measurement observations and the knowledge of a physical PDE model. Due to their ill-posedness, these problems are often addressed withBayesian approaches that consist in searching for the most plausible solution using sampling strategies ofthe posterior density. In view of their high numerical cost, especially in a high dimensional framework, reduced models have recently been proposed as a vehicle to reduce complexity and achieve near realtime in the reconstructions. The course will give an overview of optimal linear and nonlinear strategies combining reduced modeling and statistical learning algorithms


Course 4 (4.5 hours, 3 blocks)
Machine learning methods for reduced order modeling
Prof. J. Nathan Kutz (University of Washington)
Abstract:A major challenge in the study of complex systems is that of producing reduced ordermodels (ROMs) capable of capturing the salient features of the high fidelity model. Emerging methodsfrom machine learning allow us to turn data into models that are not just predictive, but provide insightinto the nature of the underlying dynamical system that generated the data and characterize the high-dimensional dynamics. This problem is made more difficult by the fact that many systems of interestexhibit parametric dependencies and diverse behaviors across multiple time scales. We introduce a numberof data-driven strategies for discovering ROMs and their coordinate embeddings from data. We consider two canonical cases: (i) systems for which we have full measurements of the governing variables, and (ii)systems for which we have incomplete measurements. For systems with full state measurements, we showthat the recent sparse identification of nonlinear dynamical systems (SINDy) method can discover ROM models with relatively little data and introduce a sampling method that allows SINDy to scale efficientlyto problems with multiple time scales and parametric dependencies. Specifically, we can discover distinctgoverning equations at slow and fast scales. For systems with incomplete observations, we show thatusing time-delay embedding coordinates can be used to obtain a linear models, such as dynamic modedecomposition, and Koopman invariant measurement system that captures the dynamics of nonlinearsystems. Together, our approaches provide a suite of mathematical strategies for leveraging and reducing the data required to discover and model nonlinear systems with ROM architectures.


Course 5 (4.5 hours, 3 blocks)
Model Order Reduction: limits and perspectives
Prof. K. Urban (University of Ulm)
Abstract:Classical MOR methods rely on projection, namely a possibly high-dimension model isprojected onto a hopefully small dimensional reduced space. In such a setting, the MOR method is usually linear. In order to get an efficient MOR, the relation of the dimension N of the reduced space and the achievable error is of great importance. In particular, one would want that the error decays fastas N grows. The best possible error in such a framework is the Kolmogorov N-width. Its decay as N grows sets an upper bound for the performance of a linear MOR method. It is well-known that elliptic problems with nice parameter dependence allow for an exponential decay of the KolmogorovN-width and thate.g. Greedy-based MOR in fact allow for the same rate of convergence, i.e., they are optimal. But whathappens for non-elliptic problems? We will start by considering time-dependent problems and show that and how the optimality for MOR methods can be preserved. The situation is different e.g. for transport-dominated and wave-type problems. There, lower bounds for the Kolmogorov N-width are known, i.e.,linear model reduction cannot work in general, which can also be seen in numerical investigations. Hence,nonlinear methods are required. We will introduce some approaches in that direction and also link todynamical systems (Course 1), optimal control (Course 2) and learning methods (Courses 3 and 4).

Further information at

Courses material

Prof. Gugercin: slides
Prof. Hinze: slides
Dr. Mula: slides
Prof. Kutz: slides
Prof Urban: slides1, slides2, slides3
Seminars: N. Beranek, A. Reinhold, M. Feuerle, N. Reich, G. Kirsten, F. Pichi


Access to the ZOOM recordings:

June 29, 2021

June 30, 2021 – Morning

June 30, 2021 –  Afternoon (part 1)

June 30, 2021 –  Afternoon (part 2)

July 1, 2021 – Morning

July 1, 2021 – Afternoon (part 1)

July 1, 2021 –  Afternoon (part 2)

July 2, 2021 – Morning (part 1)

July 2, 2021 – Morning (part2)

July 2, 2021 – Afternoon (part 1)

July 2, 2021 – Afternoon (part 2)

July 3, 2021 – Morning (part 1)

July 3, 2021 – Morning (part 2)

July 3, 2021 – Afternoon (part 1)



A very short list of the main references on model order reduction is:

1. A. C. Antoulas, Approximation of Large-Scale Dynamical Systems, Adv. Des. Control 6, SIAM,Philadelphia, PA, 2005.
2. P. Benner, V. Mehrmann, and D. C. Sorensen, Dimension Reduction of Large-Scale Systems, Lect.Notes Comput. Sci. Eng. 45, Springer-Verlag, Berlin, Heidelberg, 2005.
3. J. Hesthaven, G. Rozza, and B. Stamm, Certified Reduced Basis Methods for Parametrized PartialDifferential Equations, SpringerBriefs in Mathematics, Springer International Publishing, Cham,Switzerland, 2016.
4. Qu, Zu-Qing, Model Order Reduction Techniques with Applications in Finite Element Analysis,Springer, 2004
5. Quarteroni, Alfio, Rozza, Gianluigi (Eds.) , Reduced Order Methods for Modeling and Computa-tional Reduction, Springer, 2014
6. Benner, P., Ohlberger, M., Patera, A., Rozza, G., Urban, K. (Eds.), Model Reduction of ParametrizedSystems, Springer MS and A, Vol. 1
7, 2017.
7. W. H. A. Schilders, H. A. van der Vorst, and J. Rommes, Model Order Reduction: Theory, ResearchAspects and Applications, Springer-Verlag, Berlin, Heidelberg, 2008. 


Financial Support for Young Researchers:

Funding opportunities for the local expenses will be offered to young participants (PhD students, postdocs).

Candidates should apply writing before March 31 via CIME WEB page

A  motivation from the candidate is required and a recommendation letter from the advisor of the candidates to the directors of the school will be appreciated.






Cetraro is a beatiful location on the Tirrenian coast of Calabria.

To reach Cetraro, you can use the following ways:

By train:

The nearest train station is that of Cetraro, on the line Roma-Salerno-Reggio Calabria where do stop only local trains.

Fast trains (eurostars and intercities) stop in Paola, Complete informations on the train schedules can be found (in italian and english) on the web pages of Trenitalia (Italian Railways), at:

If informed in time, the hotel will send a car to pick you up at the Paola station. The shuttle service is not free of charge. The payment and the reservation of this service have to be arranged DIRECTLY with the hotel.

By air:

The nearest Airport is Lamezia Terme.
If informed in time, the hotel will send a car to pick you up at the Lamezia airport. The shuttle service is not free of charge. The payment and the reservation of this service have to be arranged DIRECTLY with the hotel.

By car:

take the freeway Roma-Napoli-Reggio Calabria and exit in Lagonegro Nord, then road N18 (coastal road) southway or exit in Spezzano Terme and follow N283 road to Cetraro. Grand Hotel San Michele is located at GPS N39,32.349 E15,54.196.

Participants are lodged at the Grand Hotel S. Michele; it is a nice Hotel, with a well kept garden and a large swimming pool.