# American Institute of Mathematical Sciences

March  2010, 28(1): 147-159. doi: 10.3934/dcds.2010.28.147

## Wave propagation in random waveguides

 1 Ulsan National Institute of Science and Technology, San 194, Banyeon-ri, Eonyang-eup, Ulju-gun, Ulsan, South Korea 2 Department of Mathematics and Statistics, Arizona State University, Tempe, AZ 85287-1804, United States

Received  October 2009 Revised  February 2010 Published  April 2010

We study uncertainty bounds and statistics of wave solutions through a random waveguide which possesses certain random inhomogeneities. The waveguide is composed of several homogeneous media with random interfaces. The main focus is on two homogeneous media which are layered randomly and periodically in space. Solutions of stochastic and deterministic problems are compared. The waveguide media parameters pertaining to the latter are the averaged values of the random parameters of the former. We investigate the eigenmodes coupling due to random inhomogeneities in media, i.e. random changes of the media parameters. We present an efficient numerical method via Legendre Polynomial Chaos expansion for obtaining output statistics including mean, variance and probability distribution of the wave solutions. Based on the statistical studies, we present uncertainty bounds and quantify the robustness of the solutions with respect to random changes of interfaces.
Citation: Chang-Yeol Jung, Alex Mahalov. Wave propagation in random waveguides. Discrete and Continuous Dynamical Systems, 2010, 28 (1) : 147-159. doi: 10.3934/dcds.2010.28.147
 [1] Theodore Papamarkou, Alexey Lindo, Eric B. Ford. Geometric adaptive Monte Carlo in random environment. Foundations of Data Science, 2021, 3 (2) : 201-224. doi: 10.3934/fods.2021014 [2] Ajay Jasra, Kody J. H. Law, Yaxian Xu. Markov chain simulation for multilevel Monte Carlo. Foundations of Data Science, 2021, 3 (1) : 27-47. doi: 10.3934/fods.2021004 [3] Giacomo Dimarco. The moment guided Monte Carlo method for the Boltzmann equation. Kinetic and Related Models, 2013, 6 (2) : 291-315. doi: 10.3934/krm.2013.6.291 [4] Michael B. Giles, Kristian Debrabant, Andreas Rössler. Analysis of multilevel Monte Carlo path simulation using the Milstein discretisation. Discrete and Continuous Dynamical Systems - B, 2019, 24 (8) : 3881-3903. doi: 10.3934/dcdsb.2018335 [5] Jiakou Wang, Margaret J. Slattery, Meghan Henty Hoskins, Shile Liang, Cheng Dong, Qiang Du. Monte carlo simulation of heterotypic cell aggregation in nonlinear shear flow. Mathematical Biosciences & Engineering, 2006, 3 (4) : 683-696. doi: 10.3934/mbe.2006.3.683 [6] Ludwig Arnold, Igor Chueshov. Cooperative random and stochastic differential equations. Discrete and Continuous Dynamical Systems, 2001, 7 (1) : 1-33. doi: 10.3934/dcds.2001.7.1 [7] Tzong-Yow Lee and Fred Torcaso. Wave propagation in a lattice KPP equation in random media. Electronic Research Announcements, 1997, 3: 121-125. [8] María J. Garrido–Atienza, Kening Lu, Björn Schmalfuss. Random dynamical systems for stochastic partial differential equations driven by a fractional Brownian motion. Discrete and Continuous Dynamical Systems - B, 2010, 14 (2) : 473-493. doi: 10.3934/dcdsb.2010.14.473 [9] Stephen Coombes, Helmut Schmidt, Carlo R. Laing, Nils Svanstedt, John A. Wyller. Waves in random neural media. Discrete and Continuous Dynamical Systems, 2012, 32 (8) : 2951-2970. doi: 10.3934/dcds.2012.32.2951 [10] Jan Lorenz, Stefano Battiston. Systemic risk in a network fragility model analyzed with probability density evolution of persistent random walks. Networks and Heterogeneous Media, 2008, 3 (2) : 185-200. doi: 10.3934/nhm.2008.3.185 [11] Dan Stanescu, Benito Chen-Charpentier. Random coefficient differential equation models for Monod kinetics. Conference Publications, 2009, 2009 (Special) : 719-728. doi: 10.3934/proc.2009.2009.719 [12] Zhiyan Ding, Qin Li. Constrained Ensemble Langevin Monte Carlo. Foundations of Data Science, 2022, 4 (1) : 37-70. doi: 10.3934/fods.2021034 [13] Tomás Caraballo, María J. Garrido-Atienza, Björn Schmalfuss, José Valero. Attractors for a random evolution equation with infinite memory: Theoretical results. Discrete and Continuous Dynamical Systems - B, 2017, 22 (5) : 1779-1800. doi: 10.3934/dcdsb.2017106 [14] Bixiang Wang. Mean-square random invariant manifolds for stochastic differential equations. Discrete and Continuous Dynamical Systems, 2021, 41 (3) : 1449-1468. doi: 10.3934/dcds.2020324 [15] Fuke Wu, Peter E. Kloeden. Mean-square random attractors of stochastic delay differential equations with random delay. Discrete and Continuous Dynamical Systems - B, 2013, 18 (6) : 1715-1734. doi: 10.3934/dcdsb.2013.18.1715 [16] Guillaume Bal, Ian Langmore, Youssef Marzouk. Bayesian inverse problems with Monte Carlo forward models. Inverse Problems and Imaging, 2013, 7 (1) : 81-105. doi: 10.3934/ipi.2013.7.81 [17] Guillaume Bal, Wenjia Jing. Homogenization and corrector theory for linear transport in random media. Discrete and Continuous Dynamical Systems, 2010, 28 (4) : 1311-1343. doi: 10.3934/dcds.2010.28.1311 [18] Patrick W. Dondl, Martin Jesenko. Threshold phenomenon for homogenized fronts in random elastic media. Discrete and Continuous Dynamical Systems - S, 2021, 14 (1) : 353-372. doi: 10.3934/dcdss.2020329 [19] Yanfeng Guo, Jinqiao Duan, Donglong Li. Approximation of random invariant manifolds for a stochastic Swift-Hohenberg equation. Discrete and Continuous Dynamical Systems - S, 2016, 9 (6) : 1701-1715. doi: 10.3934/dcdss.2016071 [20] Patrick W. Dondl, Michael Scheutzow. Positive speed of propagation in a semilinear parabolic interface model with unbounded random coefficients. Networks and Heterogeneous Media, 2012, 7 (1) : 137-150. doi: 10.3934/nhm.2012.7.137

2020 Impact Factor: 1.392