27,426 research outputs found

    Numerical Integral Transform Methods for Random Hyperbolic Models with a Finite Degree of Randomness

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    [EN] This paper deals with the construction of numerical solutions of random hyperbolic models with a finite degree of randomness that make manageable the computation of its expectation and variance. The approach is based on the combination of the random Fourier transforms, the random Gaussian quadratures and the Monte Carlo method. The recovery of the solution of the original random partial differential problem throughout the inverse integral transform allows its numerical approximation using Gaussian quadratures involving the evaluation of the solution of the random ordinary differential problem at certain concrete values, which are approximated using Monte Carlo method. Numerical experiments illustrating the numerical convergence of the method are included.This work was partially supported by the Ministerio de Ciencia, Innovacion y Universidades Spanish grant MTM2017-89664-P.Casabán, M.; Company Rossi, R.; Jódar Sánchez, LA. (2019). Numerical Integral Transform Methods for Random Hyperbolic Models with a Finite Degree of Randomness. Mathematics. 7(9):1-21. https://doi.org/10.3390/math7090853S12179Casabán, M.-C., Company, R., Cortés, J.-C., & Jódar, L. (2014). Solving the random diffusion model in an infinite medium: A mean square approach. Applied Mathematical Modelling, 38(24), 5922-5933. doi:10.1016/j.apm.2014.04.063Casabán, M.-C., Cortés, J.-C., & Jódar, L. (2016). Solving random mixed heat problems: A random integral transform approach. Journal of Computational and Applied Mathematics, 291, 5-19. doi:10.1016/j.cam.2014.09.021Casaban, M.-C., Cortes, J.-C., & Jodar, L. (2018). Analytic-Numerical Solution of Random Parabolic Models: A Mean Square Fourier Transform Approach. Mathematical Modelling and Analysis, 23(1), 79-100. doi:10.3846/mma.2018.006Saadatmandi, A., & Dehghan, M. (2010). Numerical solution of hyperbolic telegraph equation using the Chebyshev tau method. Numerical Methods for Partial Differential Equations, 26(1), 239-252. doi:10.1002/num.20442Weston, V. H., & He, S. (1993). Wave splitting of the telegraph equation in R 3 and its application to inverse scattering. Inverse Problems, 9(6), 789-812. doi:10.1088/0266-5611/9/6/013Jordan, P. M., & Puri, A. (1999). Digital signal propagation in dispersive media. Journal of Applied Physics, 85(3), 1273-1282. doi:10.1063/1.369258Banasiak, J., & Mika, J. R. (1998). Singularly perturbed telegraph equations with applications in the random walk theory. Journal of Applied Mathematics and Stochastic Analysis, 11(1), 9-28. doi:10.1155/s1048953398000021Kac, M. (1974). A stochastic model related to the telegrapher’s equation. Rocky Mountain Journal of Mathematics, 4(3), 497-510. doi:10.1216/rmj-1974-4-3-497Iacus, S. M. (2001). Statistical analysis of the inhomogeneous telegrapher’s process. Statistics & Probability Letters, 55(1), 83-88. doi:10.1016/s0167-7152(01)00133-xCasabán, M.-C., Cortés, J.-C., & Jódar, L. (2015). A random Laplace transform method for solving random mixed parabolic differential problems. Applied Mathematics and Computation, 259, 654-667. doi:10.1016/j.amc.2015.02.091Casabán, M.-C., Cortés, J.-C., & Jódar, L. (2018). Solving linear and quadratic random matrix differential equations using: A mean square approach. The non-autonomous case. Journal of Computational and Applied Mathematics, 330, 937-954. doi:10.1016/j.cam.2016.11.049Casabán, M.-C., Cortés, J.-C., & Jódar, L. (2016). Solving linear and quadratic random matrix differential equations: A mean square approach. Applied Mathematical Modelling, 40(21-22), 9362-9377. doi:10.1016/j.apm.2016.06.01

    Hearing the clusters in a graph: A distributed algorithm

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    We propose a novel distributed algorithm to cluster graphs. The algorithm recovers the solution obtained from spectral clustering without the need for expensive eigenvalue/vector computations. We prove that, by propagating waves through the graph, a local fast Fourier transform yields the local component of every eigenvector of the Laplacian matrix, thus providing clustering information. For large graphs, the proposed algorithm is orders of magnitude faster than random walk based approaches. We prove the equivalence of the proposed algorithm to spectral clustering and derive convergence rates. We demonstrate the benefit of using this decentralized clustering algorithm for community detection in social graphs, accelerating distributed estimation in sensor networks and efficient computation of distributed multi-agent search strategies

    Staircase polygons: moments of diagonal lengths and column heights

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    We consider staircase polygons, counted by perimeter and sums of k-th powers of their diagonal lengths, k being a positive integer. We derive limit distributions for these parameters in the limit of large perimeter and compare the results to Monte-Carlo simulations of self-avoiding polygons. We also analyse staircase polygons, counted by width and sums of powers of their column heights, and we apply our methods to related models of directed walks.Comment: 24 pages, 7 figures; to appear in proceedings of Counting Complexity: An International Workshop On Statistical Mechanics And Combinatorics, 10-15 July 2005, Queensland, Australi
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