ECTS credits ECTS credits: 6
ECTS Hours Rules/Memories Student's work ECTS: 102 Hours of tutorials: 6 Expository Class: 18 Interactive Classroom: 24 Total: 150
Use languages Spanish, Galician
Type: Ordinary subject Master’s Degree RD 1393/2007 - 822/2021
Center Faculty of Mathematics
Call: Second Semester
Teaching: With teaching
Enrolment: Enrollable | 1st year (Yes)
Al the information is at the M2i web site for this subject:
http://m2i.es/en/docs/modulos/Modelling/AdvancedModelling/ModelReductio…
1. General introduction to the course
Reduced order models, classification and goals; acceleration in numerical
simulations. Projection based models and data-driven models. Large databases
post-processing and description, patterns identification. Spatial patterns and
symmetries. Spatio-temporal periodic and quasi-periodic patterns.
Standing and
traveling waves.
Some examples of scientific and industrial applications.
Patterns formations in non-linear systems: Ginzburg-Landau equation and
thermal convection systems. Benchmark fluid dynamic problems: flow past
cylinders, backward facing step. Fluid dynamics in industrial problems: flows in
urban environments, underground reservoir flows, flight test data and the
analysis of aeroelastic frequencies.
2. Interpolation, proper ortogonal decomposition (POD) and singular value
decomposition (SVD)
POD and SVD, comparing the two methodologies in two-dimensional databases.
Analysis of large databases. Difficulties to extend SVD; canonical decomposition
and tensor rank. Tucker’s method and high order singular value decomposition.
Using these techniques for database compression, interpolation and repair
(Sirovich method).
Reduced order models based on SVD and HOSVD.
Additional
techniques: Kriging interpolation and sampling techniques such as DEIM, Q-
DEIM and LUPOD.
Some examples and applications.
3. Reduced order models based on the projection of a physical model
Reduced order model obtained using physical model projections. Galerkin
projection and some other projection techniques; modelling the non-lineal
terms. Pre-processing reduced order models to solve evolution problems based
on projection techniques. Adaptive reduced order models based on projection
techniques. Some examples and applications.
4. Reduced order models based on the identification of spatio-temporal patterns
Limitations on some techniques based of the Fourier decomposition of a signal,
such as FFT, PSD and Laskar. Dynamic mode decomposition (DMD) and some
extensions such as optimized DMD. Koopman observability theory and DMD.
Spatial and spectral complexities; limitation of the previous methods. Higher
order DMD (HODMD) and iterative HODMD; data extrapolation and cleaning
noisy experimental databases. Spatio-temporal Koopman decomposition
(STKD). Spatio-temporal patterns extraction and identification of standing and
traveling waves. Data-driven reduced order models based on the previous
techniques. Some examples and applications.
BIBLIOGRAFÍA BÁSICA
T.G. Kolda, B.W. Bader; Tensor decompositions and applications. SIAM Review, 51
(2009) pp. 455-500.
J.N. Kutz; Data-driven Modeling & Scientific Computation. Oxford University Press,
2003.
A. Quarteroni, A. Manzoni, F. Negri; Reduced Basis Methods for Partial Differential
Equations. An Introduction. Springer, 2016.
P.J. Schmid; Dynamic mode decomposition of numerical and experimental data. Journal
of Fluid Mechanics, 656 (2010) pp. 5-28.
G. Strang. Introduction to Linear Algebra. Wellesley-Cambridge Press. 5th Edition 2016.
BIBLIOGRAFÍA AUXILIAR
P. Benner, S. Gugercin, K. Willcox; A survey of projection-based model reduction
methods for parametric dynamical systems. SIAM Review 57(4) (2015) 483-531.
T. Bui-Thanh; Proper orthogonal decomposition extensions and their applications in
steady aerodynamics. MSc thesis. Massachusetts Institute of Technology (2003).
A. Chatterjee; An introduction to the proper orthogonal decomposition. Current.
Science, 78 (2000) pp. 808-817
R. Everson, L. Sirovich; Karhunen–Loeve procedure for gappy data J. Opt. Soc. Am. A,
12 (1995), pp. 1657-1664
J.N. Kutz, S.L Brunton, B.W. Brunton, J.L. Proctor; Dynamic Mode Decomposition. SIAM,
2016.
S. LeClainche, J.M. Vega; Analyzing Nonlinear Dynamics via Data-Driven Dynamic Mode
Decomposition-Like Methods. Complexity (2018) article ID 6920783.
B.R. Noack, M. Morzynski, G. Tadmor (Eds); Reduced-Order Modelling for Flow Control.
Springer, 2011.
A. Quarteroni, G. Rozza (Eds.); Reduced Order Methods for Modeling and
Computational Reduction.
Springer, 2014.
Basic and general skills:
CG1 Having the knowledge that provides a basis or opportunity for originality when
developing and/or applying ideas, often within a research context and knowing how to
translate industrial needs in terms of R&D in the field of Industrial Mathematics.
CG2 – Be able to apply the acquired knowledge and abilities to solve problems in new
or unfamiliar environments within broader contexts, including the ability to integrate
multidisciplinary R&D in the business environments.
CG3 – Have the ability to communicate the conclusions reached together with the
knowledge and reasons that support them to specialist and non-specialist audiences in
a clear and unambiguous way.
CG4 – Have the appropriate learning skills to be able to continue studying in a way
that will largely be selfdirected or autonomous and also to be able to successfully
undertake doctoral studies.
Specific skills:
CE1 – Acquire a basic knowledge in an area of Engineering/Applied Science, as a
starting point for an adequate mathematical modelling by using well-established
contexts or in new or unfamiliar environments within broader and multidisciplinary
contexts.
CE2 – Model specific ingredients and make the appropriate simplifications in a model
to facilitate their numerical treatment, maintaining the degree of accuracy, according
to previous requirements.
CE5 – Be able to validate and interpret the obtained results, comparing them
with visualizations, experimental measurements and/or functional requirements of the
corresponding physical engineering system.
Specialization on “Mathematical Modelling”:
CM2: Know how to model elements and complex systems or not very common fields
which lead to well posed formulated problems.
Lectures will combine the essential ideas of the previous techniques with their
practical applications. For such aim, simple examples will be introduced in class and
some small projects will be developed using some tools provided to the students
(using for instance MATLAB or Python codes able to call full models with the idea of
speeding up their resolution). Codes of all the tools and techniques introduced in the
course will be provided as well.
The criterion to evaluate the students in the form ‘continuous evaluation’ will divide
the students in groups up to four people. The students will solve three problems along
the course, related to contents developed in 2, 3 and 4
CRITERIA FOR THE FIRSTT ASSESMENT OPPORTUNITY:
Group report plus oral presentation of the project by one of the team members,
followed by a maximum of fifteen minutes of questions and answers.
The project done during the subject lead the student to study different problems and
look for information for them. This allows to evaluate general skills CG1, CG2 and
CG5 as well as specific skills CE1, CE2, CE5 and CM2. The presentation of the project
allows to evaluate general skill CG4.
CRITERIA FOR THE SECOND ASSESMENT OPPORTUNITY:
Same as in the first assesment opportunity.
Esta materia la imparten los docentes de la UPM:
Fernando Varas Mérida (fernando.varas [at] upm.es (fernando[dot]varas[at]upm[dot]es))
Soledad LeClainche Martínez (soledad.leclainche [at] upm.es (soledad[dot]leclainche[at]upm[dot]es))
y de la UC3M:
Filippo Terragni (fterragn [at] ing.uc3m.es (fterragn[at]ing[dot]uc3m[dot]es))
Ángel G. Velázquez López (angel.velazquez [at] upm.es (angel[dot]velazquez[at]upm[dot]es))
Las clases se impartirán con los sistemas que indique el M2i. La tutorías también se pueden solicitar por Skype o MS Teams.