European Numerical Mathematics and
Advanced Applications Conference 2019
30th sep - 4th okt 2019, Egmond aan Zee, The Netherlands
10:40   Computing and Model Order Reduction (Part 2)
Chair: Matthias Möller
10:40
25 mins
A nonlinear approximation procedure for parameterized Model Order Reduction
Tommaso Taddei, Angelo Iollo
Abstract: We present a nonlinear approximation procedure for parameterized Model Order Reduction; the approach is designed to tackle problems with slowly-decaying Kolmogorov $N$-widths. Given a parameterized equation defined over a domain $\Omega$, we reformulate the problem into a reference configuration through the vehicle of a parameter-dependent bijective mapping $\Phi:\Omega \to \Omega$: the mapping $\Phi$ is here designed to make the mapped solution field $z \circ \Phi$ more amenable for linear approximations. In this work, we propose a general (i.e., independent of the underlying equation) registration procedure for the construction of $\Phi$ based on a snapshot set. We present a theoretical result to prove the mathematical rigor of the registration procedure. We further present numerical results for several problems, to demonstrate the effectivity of our approach for several applications.
11:05
25 mins
Computational estimation of the transformation from small scale to large scale chaos during the evolution of self-organizing patterns
Loreta Saunoriene, Mantas Landauskas, Minvydas Ragulskis
Abstract: 1 page A4 page Abstract is uploaded as *.pdf file Pattern formation and self-organization in physical, chemical, biological, robotic, and cognitive systems has been object of investigations for a long time. Chemical oscillations, thermal convection of fluids, crystallization, and ecological systems are only several examples of self-organization. In this research we focus on the formation of spatiotemporal patterns in Beddington-de-Angelis-type predator-prey model with self- and cross-diffusion [1,2]. If preys and predators populations are placed in the randomly perturbed stationary state and are evolved for a certain number of iterations according to the equations of the model, system develops into a steady or time-dependent state. Different sets of parameters of the model correspond to the different types of self-organizing patterns: patterns of stripes, patterns of spots, patterns of stripes and spots, or even patterns of spiral waves [1]. The series of evolving self-organizing patterns demonstrates the evolution of the self-organization of preys or predators at a predetermined set of parameters. Such an evolution starts from the equilibrium point perturbed by small random noise, which corresponds to a small-scale spatial chaos. During the evolution, randomly looking image gradually develops into irregular striped pattern. The pattern of stripes in the fully developed image can be interpreted as a large scale spatial chaos. Note that different random initial conditions result in a different patterns in the fully developed image. The change in complexity of self-organization during the evolution can be evaluated by Shannon entropy [3], which can be interpreted as a measure of randomness of the digital image. In our research, we compute Shannon entropy for the evolving self-organizing pattern at every 10th time-step. Shannon entropy describes the transformation from small scale to large scale chaos, but also it could reveal how the process of self-organization is influenced by the parameters of the analyzed system. Moreover, it is interesting to observe how the complexity of the evolution of self-organized patterns depends on the magnitude of initial random perturbation. It is interesting, that patterns evolving in a Beddington-de-Angelis-type predator-prey model with self- and cross-diffusion can be employed for the secure communication. Steganographic algorithm presented in Ref. [2] is based on the fact that very small differences between initial perturbations result in minor differences in the evolving pattern. First, the initial random distribution of preys is constructed and then the density of preys for a certain number of time-steps is computed. Next, the initial distribution is perturbed adding dot skeleton mask of the secret information and obtained pattern is evolved for a certain number of iterations [2]. Although images obtained in the first and in the second steps look identical, the difference between both of them reveals the hidden information. In this application, it is important to detect the number of time-steps at which almost fully developed pattern is obtained [2, 4]. As it was described in the previous paragraph, Shannon entropy estimates the complexity of the analysed pattern and in such a way enables us to track the transformation from small scale to large scale chaos during the evolution of self-organization. Therefore, Shannon entropy can be applied for the determination of the optimal number of iterations, required for the formation of fully developed patterns, what is a valuable result allowing to improve steganographic communications techniques based on the models of self-organizing patterns. Acknowledgements This research was supported by the Research, Development and Innovation Fund of Kaunas University of Technology (DDetect, grant No. PP-91M/19). References 1. W. Wang, Y. Lin, L. Zhang, F. Rao, Y. Tan. Complex patterns in a predator– prey model with self and cross-diffusion, Communications in Nonlinear Science and Numerical Simulation. 2011, vol. 16, 2006–2015. 2. L. Saunoriene, M. Ragulskis. A secure steganographic communication algorithm based on self-organizing patterns. Physical Review E. 2011, vol. 84(5), 056213. 3. M. Borda. Fundamentals in Information Theory and Coding. Springer, 2011. 4. T. Telksnys, Z. Navickas, M. Ragulskis, M. Vaidelys. The order of a 2-sequence and the complexity of digital images. Advances in Complex Systems. 2016, vol. 19(4), 1650010.
11:30
25 mins
Approximating eigenvectors with fixed-point arithmetic: a step towards secure spectral clustering
Lisa Steverink, Thijs Veugen, Martin van Gijzen
Abstract: We investigate the adaptation of the spectral clustering algorithm to the privacy preserving domain. Spectral clustering is a data mining technique that divides points according to a measure of connectivity. When the matrix data are privacy sensitive, cryptographic techniques can be applied to protect the data. A pivotal part of spectral clustering is the partial eigendecomposition of the graph Laplacian. Two numerical algorithms are used to approximate the eigenvectors of the Laplacian: the Lanczos algorithm and the QR algorithm. Many cryptographic techniques are designed to work with positive integers, whereas the numerical algorithms are generally applied in the real domain. To overcome this problem, the Lanczos algorithm and the QR algorithm were adapted to be performed with fixed-point arithmetic. Square roots were eliminated and floating-point computations were transformed to fixed-point computations. The effect of these adaptations on the accuracy and stability of the algorithms was investigated by testing the spectral clustering algorithm on three datasets. The performance of the original and the adapted algorithms was similar. At least four eigenvectors that correspond to the smallest eigenvalues can be approximated with high accuracy. The spectral clustering algorithm performed equally well for 2 and 5 clusters. For 10 clusters, the loss of orthogonality affected the accuracy of spectral clustering in both domains.