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AuthorTempone, Raul (28)Alouini, Mohamed-Slim (11)Bagci, Hakan (7)Moraes, Alvaro (6)Nobile, Fabio (6)View MoreDepartmentComputer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division (54)Applied Mathematics and Computational Science Program (35)Electrical Engineering Program (19)Physical Sciences and Engineering (PSE) Division (7)Biological and Environmental Sciences and Engineering (BESE) Division (3)View MoreJournalEuropean Microscopy Congress 2016: Proceedings (1)PublisherWiley (1)SubjectWireless (11)Bayesian (9)Sampling (9)CEM (7)SDE (6)View MoreTypePoster (60)Year (Issue Date)

2016 (60)

Item AvailabilityOpen Access (59)Metadata Only (1)

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Computation of High-Frequency Waves with Random Uncertainty

Malenova, Gabriela; Motamed, Mohammad; Runborg, Olof; Tempone, Raul (2016-01-06) [Poster]

We consider the forward propagation of uncertainty in high-frequency waves, described by the second order wave equation with highly oscillatory initial data. The main sources of uncertainty are the wave speed and/or the initial phase and amplitude, described by a finite number of random variables with known joint probability distribution. We propose a stochastic spectral asymptotic method [1] for computing the statistics of uncertain output quantities of interest (QoIs), which are often linear or nonlinear functionals of the wave solution and its spatial/temporal derivatives. The numerical scheme combines two techniques: a high-frequency method based on Gaussian beams [2, 3], a sparse stochastic collocation method [4]. The fast spectral convergence of the proposed method depends crucially on the presence of high stochastic regularity of the QoI independent of the wave frequency. In general, the high-frequency wave solutions to parametric hyperbolic equations are highly oscillatory and non-smooth in both physical and stochastic spaces. Consequently, the stochastic regularity of the QoI, which is a functional of the wave solution, may in principle below and depend on frequency. In the present work, we provide theoretical arguments and numerical evidence that physically motivated QoIs based on local averages of |uE|2 are smooth, with derivatives in the stochastic space uniformly bounded in E, where uE and E denote the highly oscillatory wave solution and the short wavelength, respectively. This observable related regularity makes the proposed approach more efficient than current asymptotic approaches based on Monte Carlo sampling techniques.

Tight Error Bounds for Fourier Methods for Option Pricing for Exponential Levy Processes

Flores, Fabian Crocce; Häppölä, Juho; Keissling, Jonas; Tempone, Raul (2016-01-06) [Poster]

Prices of European options whose underlying asset is driven by the L´evy process are solutions to partial integrodifferential Equations (PIDEs) that generalise the Black-Scholes equation by incorporating a non-local integral term to account for the discontinuities in the asset price. The Levy -Khintchine formula provides an explicit representation of the characteristic function of a L´evy process (cf, [6]): One can derive an exact expression for the Fourier transform of the solution of the relevant PIDE. The rapid rate of convergence of the trapezoid quadrature and the speedup provide efficient methods for evaluationg option prices, possibly for a range of parameter configurations simultaneously. A couple of works have been devoted to the error analysis and parameter selection for these transform-based methods. In [5] several payoff functions are considered for a rather general set of models, whose characteristic function is assumed to be known. [4] presents the framework and theoretical approach for the error analysis, and establishes polynomial convergence rates for approximations of the option prices. [1] presents FT-related methods with curved integration contour. The classical flat FT-methods have been, on the other hand, extended for option pricing problems beyond the European framework [3]. We present a methodology for studying and bounding the error committed when using FT methods to compute option prices. We also provide a systematic way of choosing the parameters of the numerical method, minimising the error bound and guaranteeing adherence to a pre-described error tolerance. We focus on exponential L´evy processes that may be of either diffusive or pure jump in type. Our contribution is to derive a tight error bound for a Fourier transform method when pricing options under risk-neutral Levy dynamics. We present a simplified bound that separates the contributions of the payoff and of the process in an easily processed and extensible product form that is independent of the asymptotic behaviour of the option price at extreme prices and at strike parameters. We also provide a proof for the existence of optimal parameters of the numerical computation that minimise the presented error bound.

A Stochastic Multiscale Method for the Elastic Wave Equations Arising from Fiber Composites

Babuska, Ivo; Motamed, Mohammad; Tempone, Raul (2016-01-06) [Poster]

We present a stochastic multilevel global-local algorithm [1] for computing elastic waves propagating in fiber-reinforced polymer composites, where the material properties and the size and distribution of fibers in the polymer matrix may be random. The method aims at approximating statistical moments of some given quantities of interest, such as stresses, in regions of relatively small size, e.g. hot spots or zones that are deemed vulnerable to failure. For a fiber-reinforced cross-plied laminate, we introduce three problems: 1) macro; 2) meso; and 3) micro problems, corresponding to the three natural length scales: 1) the sizes of plate; 2) the tickles of plies; and 3) and the diameter of fibers. The algorithm uses a homogenized global solution to construct a local approximation that captures the microscale features of the problem. We perform numerical experiments to show the applicability and efficiency of the method.

Some Numerical Aspects on Crowd Motion - The Hughes Model

Gomes, Diogo A.; Machado Velho, Roberto (2016-01-06) [Poster]

Here, we study a crowd model proposed by R. Hughes in [5] and we describe a numerical approach to solve it. This model comprises a Fokker-Planck equation coupled with an Eikonal equation with Dirichlet or Neumann data. First, we establish a priori estimates for the solution. Second, we study radial solutions and identify a shock formation mechanism. Third, we illustrate the existence of congestion, the breakdown of the model, and the trend to the equilibrium. Finally, we propose a new numerical method and consider two numerical examples.

Basket call option pricing for CCVG using sparse grids

Flores, Fabian Crocce; Häppölä, Juho; Iania, Alessandro; Tempone, Raul (2016-01-06) [Poster]

The use of processes with jumps to overcome the shortcomings of the classical Black and Scholes when modelling stock prices has became very popular. One of the best-known models is the Common Clock Variance Gamma model (CCVG), introduced by Madan and Seneta in the 1990 [3]. We propose a method to price European basket call options modelled by the CCVG. The method could be extended to other model obtained by the subordination of a multidimensional Brownian motion and to more general options. To simplify the expositions we consider calls under the CCVG.

A new smoothing technique for European basket options

Bayer, Christian; Siebenmorgen, Markus; Tempone, Raul (2016-01-06) [Poster]

Computable error estimates for FEMs for elliptic PDE with lognormal data

Hall, Eric; Haakon, Hoel; Sandberg, Mattias; Szepessy, Anders; Tempone, Raul (2016-01-06) [Poster]

Flow in Random Microstructures: a Multilevel Monte Carlo Approach

Icardi, Matteo; Tempone, Raul (2016-01-06) [Poster]

In this work we are interested in the fast estimation of effective parameters of random heterogeneous materials using Multilevel Monte Carlo (MLMC). MLMC is an efficient and flexible solution for the propagation of uncertainties in complex models, where an explicit parametrisation of the input randomness is not available or too expensive. We propose a general-purpose algorithm and computational code for the solution of Partial Differential Equations (PDEs) on random heterogeneous materials. We make use of the key idea of MLMC, based on different discretization levels, extending it in a more general context, making use of a hierarchy of physical resolution scales, solvers, models and other numerical/geometrical discretisation parameters. Modifications of the classical MLMC estimators are proposed to further reduce variance in cases where analytical convergence rates and asymptotic regimes are not available. Spheres, ellipsoids and general convex-shaped grains are placed randomly in the domain with different placing/packing algorithms and the effective properties of the heterogeneous medium are computed. These are, for example, effective diffusivities, conductivities, and reaction rates. The implementation of the Monte-Carlo estimators, the statistical samples and each single solver is done efficiently in parallel. The method is tested and applied for pore-scale simulations of random sphere packings.

Time-optimal path planning in uncertain flow fields using ensemble method

Wang, Tong; Le Maitre, Olivier; Hoteit, Ibrahim; Knio, Omar (2016-01-06) [Poster]

An ensemble-based approach is developed to conduct time-optimal path planning in unsteady ocean currents under uncertainty. We focus our attention on two-dimensional steady and unsteady uncertain flows, and adopt a sampling methodology that is well suited to operational forecasts, where a set deterministic predictions is used to model and quantify uncertainty in the predictions. In the operational setting, much about dynamics, topography and forcing of the ocean environment is uncertain, and as a result a single path produced by a model simulation has limited utility. To overcome this limitation, we rely on a finitesize ensemble of deterministic forecasts to quantify the impact of variability in the dynamics. The uncertainty of flow field is parametrized using a finite number of independent canonical random variables with known densities, and the ensemble is generated by sampling these variables. For each the resulting realizations of the uncertain current field, we predict the optimal path by solving a boundary value problem (BVP), based on the Pontryagin maximum principle. A family of backward-in-time trajectories starting at the end position is used to generate suitable initial values for the BVP solver. This allows us to examine and analyze the performance of sampling strategy, and develop insight into extensions dealing with regional or general circulation models. In particular, the ensemble method enables us to perform a statistical analysis of travel times, and consequently develop a path planning approach that accounts for these statistics. The proposed methodology is tested for a number of scenarios. We first validate our algorithms by reproducing simple canonical solutions, and then demonstrate our approach in more complex flow fields, including idealized, steady and unsteady double-gyre flows.

Secure Broadcasting with Uncertain Channel State Information

Hyadi, Amal; Rezki, Zouheir; Khisti, Ashish; Alouini, Mohamed-Slim (2016-01-06) [Poster]

We investigate the problem of secure broadcasting over fast fading channels with imperfect main channel state information (CSI) at the transmitter. In particular, we analyze the effect of the noisy estimation of the main CSI on the throughput of a broadcast channel where the transmission is intended for multiple legitimate receivers in the presence of an eavesdropper. Besides, we consider the realistic case where the transmitter is only aware of the statistics of the eavesdropper s CSI and not of its channel s realizations. First, we discuss the common message transmission case where the source broadcasts the same information to all the receivers, and we provide an upper and a lower bounds on the ergodic secrecy capacity. For this case, we show that the secrecy rate is limited by the legitimate receiver having, on average, the worst main channel link and we prove that a non-zero secrecy rate can still be achieved even when the CSI at the transmitter is noisy. Then, we look at the independent messages case where the transmitter broadcasts multiple messages to the receivers, and each intended user is interested in an independent message. For this case, we present an expression for the achievable secrecy sum-rate and an upper bound on the secrecy sum-capacity and we show that, in the limit of large number of legitimate receivers K, our achievable secrecy sum-rate follows the scaling law log((1-a ) log(K)), where is the estimation error variance of the main CSI. The special cases of high SNR, perfect and no-main CSI are also analyzed. Analytical derivations and numerical results are presented to illustrate the obtained expressions for the case of independent and identically distributed Rayleigh fading channels.

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