MATHICSE Technical Report: Non-intrusive double-greedy parametric model reduction by interpolation of frequency-domain rational surrogates

We propose a model order reduction approach for non-intrusive surrogate modeling of parametric dynamical systems. The reduced model over the whole parameter space is built by combining surrogates in frequency only, built at few selected values of the parameters. This, in particular, requires matching the respective poles by solving an optimization problem. We detail how to treat unbalanced cases, where the surrogates to be combined have a different number of poles. If the frequency surrogates are constructed by minimal rational interpolation, frequency and parameters can both be sampled in a greedy fashion, by employing a fully non-intrusive "look-ahead" strategy. We explain how our proposed technique can be applied even in a high-dimensional setting, by employing locally-refined sparse grids to weaken the curse of dimensionality. Numerical examples are used to showcase the effectiveness of the method, and to highlight some of its limitations in dealing with unbalanced matching, as well as with a large number of parameters.

Non-intrusive double-greedy parametric model reduction by interpolation of frequency-domain rational surrogates

Fabio Nobile, Davide Pradovera

We propose a model order reduction approach for non-intrusive surrogate modeling of parametric dynamical systems. The reduced model over the whole parameter space is built by combining surrogates in frequency only, built at few selected values of the parameters. This, in particular, requires matching the respective poles by solving an optimization problem. If the frequency surrogates are constructed by a suitable rational interpolation strategy, frequency and parameters can both be sampled in an adaptive fashion. This, in general, yields frequency surrogates with different numbers of poles, a situation addressed by our proposed algorithm. Moreover, we explain how our method can be applied even in high-dimensional settings, by employing locally-refined sparse grids in parameter space to weaken the curse of dimensionality. Numerical examples are used to showcase the effectiveness of the method, and to highlight some of its limitations in dealing with unbalanced pole matching, as well as with a large number of parameters.

2021

Un modèle factoriel dynamique pour séries temporelles

Andrei Zenide

This work deals with factorial models for multiple time series. Its core content puts it at the interface between statistics and finance. After a brief description of the historical link between the two sciences, it reviews the literature on factorial models that are close to the model introduced in this work called The dynamical factor analysis model for time series. This model makes the hypothesis that the observed time series are influenced by a common factor, difficult to define and impossible to measure. No a priori structure is put on the factor, at each point time the value of the factor is considered as a new parameter that has to be estimated. As a consequence of this fact, the number of parameters is large and it is not possible to provide the usual asimptotic properties of the estimations by letting the number of periods tend to infi- nity. Asymptotic results in our context have been obtained by increasing the number of time series. The model makes the hypothesis that there is a linear dependence between the time series and the factor with coefficients, that are not constant over time, but rather follow a smooth random walk. This mean that the linear structure of the model is evolving slowly from a period to another. All the information which is not contained in the factor and the coefficients is considered as white noise. Using the normal distribution makes the estimation more easy and opens a toolbox of statistical methods that have been developed for this kind of data. The model is a part of the family of state space models and the Kalman filter is an essential ingredient of our estimator. The effort was concentrated on the elaboration of the structure of the model. Its complexity was constrained by the difficulty of estimation. The final shape of the model does not allow an analytical solution of the optimization problem introduced by the maximum likelihood estimation of the parameters. Numerical solutions have been found and compared with the parameters for simulated data. Some others models have been developed as simpler versions of the dynamical factor model for time series. The case where the factor can be observed has been studied and a new method for the estimation has been provided and compared with the existing methods. A second study considers a latent factor model without noise. Two methods for the estimation of the factor have been provided. The last chapter contains a detailed description of the main statistical tools used during this work. The links with the previous chapters are highlighted followed by comments.

EPFL2004

Application of 3D range camera in virtual human-computer interfaces

Huan Du

In this work, we propose new ways to employ 3D-ranging systems in advanced human-computer interfaces and show that they can be used for precise hand-tracking systems aiming at virtual keyboard or mouse applications. We first implement a Structured Light (SL) based 3D-ranging system as the testbed for acquiring depth information. The ranging system uses an off-the-shelf projector combined with an industrial color camera, where the projector projects well-designed line patterns onto the scene and the camera captures the reflected light from the scene and the target. The depth information is recovered from the deformed line pattern and the pre-calibrated projector-camera system. With this system we can capture dense depth maps with moderate resolution and high frame rate as the simulated input to the hand-tracking system. We then explore an approach to 3D hand matching that combines a detailed 3D hand model (skeletal + polygonal model) with a Physical-based Model Fitting technique (PMF). Virtual forces are assigned between the depth measurements and the polygonal vertices of the model. The matching problem is formulated as the minimization of the distances between the measurement and the model surface with the guidance of virtual forces. This can employ 3D information simply and directly onto the parameter space, and yield good results under challenging conditions. To demonstrate the effectiveness of the approach we show that we can track hand motions such as typing or grasping frame by frame. The major limitation of this method is that it requires a good initialization of the hand model and the computation time to achieve convergence with good fitting results is quite long. We therefore investigated more complex hand models and feature-based matching techniques. A three-level hand model (skeletal + primitive + polygonal model) is proposed for the Multi-Level Feature Matching approach (MLFM). The matching is carried out from the bottom to top levels of different hand presentations, and distinctive features for each level extracted from both the measurements and the model are matched with Scaled Conjugate Gradient method (SCG). The matching result from lower level is used as the initial state of the model parameters for higher level matching. This hierarchical structure speeds up the optimization process and enhances the precision of the matching result. We demonstrate our approach with the same hand motion as in the previous method and compare the performance of these two methods. We show that the MLFM method is faster and more robust to hand occlusions than the PMF method. Finally, we present a more powerful approach based on the Back-Constrained Gaussian Process Latent Variable Model (BC-GPLVM) that import prior information of categorized hand motion such as typing. BC-GPLVM provides a low-dimensional embedding of hand motion data, with a density function that gives higher probability to gestures and motions close to normal typing motion. This model is learned from a small set of clusters of vast captured motion data. The prior model is then combined with the previous multi-level models and features to generate a uniform cost function for optimization. We show the tracking result to demonstrate that this method can be used for human-computer interfaces requiring precise 3D hand tracking.