Set-Valued Numerical Analysis

start of the project: 2008

funding institution: The Hermann Minkowski Center for Geometry in Tel Aviv, Israel; DAAD; Bulgarian Academy of Sciences

project members

principal investigator

AD Dr. Robert Baier

project members

Dr. Elza Farkhi

Dr. Gilbert Perria (project member until 09/2010  )

aims of the project

The project leader continues the project which was started 1995 by Prof. Dr. Frank Lempio.

Modelling of differential equations with discontinuous right-hand sides, dynamical systems with uncertainties, non-smooth optimization problems, and optimal control problems lead to differential equations with set-valued right-hand sides, so-called differential inclusions.

Within the research project "Set-Valued Numerical Analysis", differential inclusions are investigated qualitatively and quantitatively, and algorithms for their numerical solution are developed.

The numerical solution of such differential inclusions requires algorithms from set-valued numerical analysis, especially integration techniques for set-valued integrands and interpolation techniques for set-valued maps, based e.g. on set-valued finite element methods, set-valued divided differences and embeddings of convex compact sets. Especially, one focus is laid on the approximation of reachable sets at a given time which consist of all endpoints of admissible trajectories of a differential inclusion.

There are strong connections to the direcet discretization of optimal control problems, e.g. in questions on the (order of) convergence of the optimal control/trajectory of the discrete problems to the corresponding optimal control/trajectory of the continuous problem.

Minisymposia/Sessions

The project members organized the following minisymposia and sessions on set-valued numerics.

Links

Selected publications of the project at the ERef server

Doctoral thesis

Baier, Robert
Mengenwertige Integration und die diskrete Approximation erreichbarer Mengen
Bayreuth, Universität, 1995(dissertation, , )

Article in a journal (not assessed)

Riedl, Wolfgang
Optimization-based subdivision algorithm for reachable sets
in Proceedings in Applied Mathematics and Mechanics volume 14 (2014) issue 1. - page 937-938
doi:10.1002/pamm.201410449 ...
85th Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM), Erlangen 2014, Erlangen, Germany


Article in a journal (assessed)

Baier, Robert; Farkhi, Elza
Regularity of set-valued maps and their selections through set differences. Part 1: Lipschitz c ...
in Serdica Mathematical Journal volume 39 (2013) issue 3-4. - page 365-390
http://www.math.bas.bg/serdica/2013/2013-365-390.p ...
rev.: AMS MathSciNet
revZentralblatt


Baier, Robert; Dellnitz, Michael; Hessel-von Molo, Mirko; Kevrekidis, Yannis G.; Sertl, Stefan
The computation of convex invariant sets via Newton's method
in Journal of Computational Dynamics volume 1 (2014) issue 1. - page 39-69
doi:10.3934/jcd.2014.1.39 ...
rev.: Zentralblatt
revAMS MathSciNet


Baier, Robert; Gerdts, Matthias; Xausa, Ilaria
Approximation of reachable sets using optimal control algorithms
in Numerical Algebra, Control and Optimization volume 3 (2013) issue 3. - page 519-548
doi:10.3934/naco.2013.3.519 ...
rev.: AMS MathSciNet
revZentralblatt


Farkhi, Elza; Donchev, Tzanko D.; Baier, Robert
Existence of Solutions for Nonconvex Differential Inclusions of Monotone Type
in Comptes Rendus de l'Académie Bulgare des Sciences volume 67 (2014) issue 3. - page 323-330
http://www.proceedings.bas.bg/content/2014_3_cnten ...
rev.: AMS MathSciNet
revZentralblatt


Baier, Robert; Farkhi, Elza
Regularity of set-valued maps and their selections through set differences. Part 2: One-sided L ...
in Serdica Mathematical Journal volume 39 (2013) issue 3-4. - page 391-422
http://www.math.bas.bg/serdica/2013/2013-391-422.p ...
rev.: AMS MathSciNet
revZentralblatt


Article in a book (assessed)

Zanon, Mario; Boccia, Andrea; Palma, Vryan Gil; Parenti, Sonja; Xausa, Ilaria
Direct Optimal Control and Model Predictive Control
Optimal Control : Novel Directions and Applications
Cham, Switzerland : Springer, 2017. - page 263-382 . - (Lecture Notes in Mathematics; 2180)
doi:10.1007/978-3-319-60771-9_3 ...
rev.: Zentralblatt Mathematik


Xausa, Ilaria; Baier, Robert; Gerdts, Matthias; Gonter, Mark; Wegwerth, Christian
Avoidance trajectories for driver assistance systems via solvers for optimal control problems
Proceedings on the 20th International Symposium on Mathematical Theory of Networks and Systems (MTNS 2012), July 9-13, 2012, Melbourne, Australia
Melbourne, Australia : Think Business Events, 2012. - page 8
http://www.mtns2012.conference.net.au/Full%20Paper ...


Conference item (not assessed)

Xausa, Ilaria; Baier, Robert; Bokanowski, Olivier; Gerdts, Matthias; Töpfer, Daniel
Software for verification of collision avoidance algorithms via optimal control techniques
2014
https://hal.inria.fr/hal-01024587
Conference on New Trends in Optimal Control (NetCO 2014), Tours, Frankreich


Preprint, postprint, working paper, discussion paper

Baier, Robert; Le, Thuy Thi Thien
Construction of the Minimum Time Function Via Reachable Sets of Linear Control Systems. Part 1: ...
Bayreuth ; Padova, 2015. - 30 page

Baier, Robert; Le, Thuy Thi Thien
Construction of the Minimum Time Function Via Reachable Sets of Linear Control Systems. Part 2: ...
Bayreuth ; Padova, 2015. - 16 page

Xausa, Ilaria; Baier, Robert; Bokanowski, Olivier; Gerdts, Matthias
Computation of safety regions for driver assistance systems by using a Hamilton-Jacobi approach
Paris, France, 2014. - 22 page
https://hal.inria.fr/hal-01123490

publications

Please go to the list of publications of this project.

Chair -

|  University of Bayreuth -