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Tensor-based multivariate function approximation: methods benchmarking and comparison
(June 2025)
Tensor-based multivariate function approximation: methods benchmarking and comparison
(June 2025)
We suggest a benchmark of some tensor-based (or data-driven) multivariate function approximation.
Reference
A.C. Antoulas, I-V. Gosea, C. Poussot-Vassal and P. Vuillemin, "Tensor-based multivariate function approximation: methods benchmarking and comparison".
>> arXiv
The figure above shows the result of a simulation of a wave guide composed of three regions with different permittivity, in response to an electric wave injected from the left. The figure compares the magnetic (H, top) and electric (E, bottom) fields from the simulation produced by the expert simulator (obtained in 10 minutes) and the reduced model (obtained in 1 second).
Construction of a simplified model of a wave guide simulator
(December 2024)
Construction of a simplified model of a wave guide simulator
(December 2024)
The Loewner framework, extended to port Hamiltoninan structures, is used to construct a simplified / reduced order dynamical model from data collected directly from a Maxwell's equation driven simulator. The reduced model recovers first the structure, second the input/output behavior and third, enable the full state (approximate) reconstruction.
Reference
M. Gouzien, C. Poussot-Vassal, G. Haine and D. Matignon, "Port-Hamiltonian reduced order modelling of the 2D Maxwell equations", in journal for Computation and Mathematics in Electrical and Electronic Engineering (COMPEL).
>> ISAE Open Science .
flop comparison. Cascaded n-D Loewner worst-case upper bounds for varying number of variables n, while the full n-D Loewner is O(N^3) (black dashed); comparison with O(N^2) and O(N log(N)) references are shown in dash-dotted and dotted black lines.
The multivariate Loewner framework :
Taming the curse of dimensionality and the Kolmogorov superposition theorem
(May 2024)
The multivariate Loewner framework :
Taming the curse of dimensionality and the Kolmogorov superposition theorem
(May 2024)
We propose a generalized realization form for rational functions in n-variables (for any "n"), which are described in the Lagrange basis;
We show that the n-dimensional Loewner matrix can be written as the solution of a series of cascaded Sylvester equations;
We demonstrate that the required variables to be determined, i.e. the barycentric coefficients, can be computed using a sequence of small-scale 1-dimensional Loewner matrices instead of the large-scale (NxN) n-dimensional one, therefore drastically reducing the both computational effort and memory needs, and improving accuracy;
We show that this decomposition achieves variables decoupling; thus connecting the Loewner framework for rational interpolation of multivariate functions and the Kolmogorov Superposition Theorem (KST), restricted to rational functions. The result is the formulation of KST for the special case of rational functions;
Connections with KAN neural nets follows (detailed in future work).
Reference
A. C. Antoulas, I. V. Gosea and C. Poussot-Vassal, "On the Loewner framework, the Kolmogorov superposition theorem, and the curse of dimensionality", in SIAM Review (Research Spotlight)
>>arXiv
>> GitHub code
>> YouTube video
Comparison of the prime counting function (thin red) with the Riemann formulae including non-trivial zeros approximated by the Loewner approach (solid blue). As the number of non-trivial (harmonics) zeros increases, the step shape is revealed.
Riemann zeta function zeros and prime counting function approximation
(January 2021)
Riemann zeta function zeros and prime counting function approximation
(January 2021)
The realisation landmark of Mayo and Antoulas, through the lens of the modified Loewner framework is used to approximate the non-trivial zeros of the famous Riemann zeta function.
These approximated zeros are then used to approximate the Riemann prime counting function, as illustrated in the right frame.
Reference
C. Poussot-Vassal, I.V. Gosea, P. Vuillemin and A.C. Antoulas "Loewner framework for Riemann zeta function non-trivial zeros and prime counting function approximation", still in preparation.
Horizontal cross section of pollutants concentration from four sources (red stars) with an eastward wind. Original data from complex simulator Large Eddy Simulation (LES, left), and nonlinear Reduced Order Model (ROM, right). The LES solution runs in 5600 hours on a cluster while the ROM in seconds over a standard laptop.
Non-intrusive nonlinear reduced order modelling of a pollutant dispersion
(December 2020)
Non-intrusive nonlinear reduced order modelling of a pollutant dispersion
(December 2020)
The non-intrusive MII for mixed interpolatory and inference procedure aims at constructing a nonlinear reduced order model (ROM) from time-domain input-output data issued from any complex simulator or measurements.
The proposed process allows to construct a nonlinear ROM that accurately reproduce the complex simulation and that can be used for prediction, analysis. This ROM is a dynamical model of an appropriate structure.
Reference
C. Poussot-Vassal, T. Sabatier, C. Sarrat and P. Vuillemin, "Mixed interpolatory and inference non-intrusive reduced order modeling with application to pollutants dispersion"
>> arXiv
Top: Illustration of the considered closed-loop (the photo is the top view of the PFA). Bottom: performance in signal tracking obtained on the experimental tech bench.
Data-driven control of a pulsed fluidic actuator
(February 2020)
Data-driven control of a pulsed fluidic actuator
(February 2020)
Illustration of the data-driven control design, applied on a pulsed fluidic actuator (PFA). PFA are typical on/off actuators that blow air in order to modify the pressure in a flow setup. They are typically used to control fluidic phenomena. The design is done using the Loewner-Data Driven Control (L-DDC) rationale.
Video
Video illustrating the closed-loop performances obtained on the experimental setup of the controlled PFA. This loop can be considered as the inner-loop of a future flow control setup. Sound is the noise of the blowed on/off air :)
Bottom: the output signal (solid red) tracking the reference one (dashed black) thanks to the control sequence (dotted blue)
Top: the pulsed actuator activity (white) as on-off signal modulation of the signal provided by the L-DDC controller.
Reference
C. Poussot-Vassal, P. Kergus, F. Kerhervé and D. Sipp "Interpolatory-based data-driven pulsed fluidic actuator control design and experimental validation", in IEEE transaction in Control Systems Technology
>> arXiv
Open cavity geometry. The flow U goes from left to right. The actuator is a blowing one modifying the pressure located on B (yellow) and the sensor measure the pressure variation is on C. The feedback loops C to B (see video).
Model reduction and control of fluid phenomena
(June 2017)
Model reduction and control of fluid phenomena
(June 2017)
Model reduction and control of a complex fluid phenomena (here the oscillator open cavity geometry). This has been done by data-driven model approximation done with MDSPACK and an unconventional iteratively designed controller.
Video
Illustration of the feedback controller performance in attenuating the fluid oscillations. Top : with control; bottom : without control.
Reference
C. Poussot-Vassal, C. Leclercq and D. Sipp, "Structured linear fractional parametric controller Hinf design and its applications", in Proceedings of the European Control Conference (ECC), Limassol, Cyprus, June, 2018, pp. 2629-2634.
C. Poussot-Vassal and D. Sipp, "Parametric reduced order dynamical model construction of a fluid flow control problem", in Proceedings of the Workshop on Linear Parameter Varying Systems (LPVS), Grenoble, France, October, 2015, pp. 133-138.
Falcon 7X s/n 001 during ground vibration test (Dassault-Aviation colleagues and me)
Vibration control demonstrator applied on a Dassault-Aviation Business jet
>> Ground vibration tests (September 2015)
>> Flight tests (September 2017)
Vibration control demonstrator applied on a Dassault-Aviation Business jet
>> Ground vibration tests (September 2015)
>> Flight tests (September 2017)
The vibration attenuation achieved on a Dassaut-Aviation Business Jet Falcon 7X has been done. To this aim,
the MDSPACK has been extensively used to approximate the faithful but very complex dynamical models of Dassault-Aviation (obtained from finite elements methods), and
a structured H-infinity controller has been designed to attenuate the vibration over a frequency-limited range.
Video
The video shows the effect of the feedback control design to attenuate the vibrations at the pilot cabin level.
Video (interview)
The interview deals with the importance of vibration and load control in small aircraft.
References
C. Poussot-Vassal, C. Roos, P. Vuillemin, O. Cantinaud and J-P. Lacoste, "Chapter 11: Control-oriented Aeroelastic BizJet Low-order LFT modeling", Control-oriented modelling and identification: theory and practice (ISBN: 978-1-84919-614-7), M. Lovera Eds. January, 2015, IET - Control Engineering Series 80 (Hardcover).
C. Meyer, G. Broux, J. Prodigue, O. Cantinaud and C. Poussot-Vassal, "Demonstration of innovative vibration control on a Falcon Business Jet", in Proceedings of the International Forum on Aeroelasticity and Structural Dynamics (IFASD), Como, Italy, June, 2017.
Experimental set-up in the ONERA S3Ch WT: aeroelastic airofoil (foreground) and gust generator (background)
Active gust load alleviation on an Aeroelastic model
(May 2015)
Active gust load alleviation on an Aeroelastic model
(May 2015)
In the Onera wind tunnel facility, both trans- and sub-sonic configurations to attenuate gust load, have been obtained thanks to
an advanced frequency-domain identification procedure based on data-driven model approximation performed with the MDSPACK,
followed by an active closed-loop control done in the structured H-infinity framework.
Video
Illustration of the loads attenuation of when the feedback control loop is activated (at 19 seconds). The video is shot in high speed.
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