Haji Ali, Abdul Lateef; Nobile, Fabio; Tamellini, Lorenzo; Tempone, Raul(2016-01-06)[Poster]
In this work we introduce the Multi-Index Stochastic Collocation method (MISC) for computing statistics of the solution of a PDE with random data. MISC is a combination technique based on mixed differences of spatial approximations and quadratures over the space of random data. We propose an optimization procedure to select the most effective mixed differences to include in the MISC estimator: such optimization is a crucial step and allows us to build a method that, provided with sufficient solution regularity, is potentially more effective than other multi-level collocation methods already available in literature. We then provide a complexity analysis that assumes decay rates of product type for such mixed differences, showing that in the optimal case the convergence rate of MISC is only dictated by the convergence of the deterministic solver applied to a one dimensional problem. We show the effectiveness of MISC with some computational tests, comparing it with other related methods available in the literature, such as the Multi-Index and Multilevel Monte Carlo, Multilevel Stochastic Collocation, Quasi Optimal Stochastic Collocation and Sparse Composite Collocation methods.
Markovian pure jump processes can model many phenomena, e.g. chemical reactions at molecular level, protein transcription and translation, spread of epidemics diseases in small populations and in wireless communication networks among many others. In this work we present a novel hybrid algorithm for simulating individual trajectories which adaptively switches between the SSA and the Chernoff tauleap methods. This allows us to: (a) control the global exit probability of any simulated trajectory, (b) obtain accurate and computable estimates for the expected value of any smooth observable of the process with minimal computational work.
Haji Ali, Abdul Lateef; Nobile, Fabio; Tempone, Raul(2016-01-06)[Poster]
We propose and analyze a novel Multi-Index Monte Carlo (MIMC) method for weak approximation of stochastic models that are described in terms of differential equations either driven by random measures or with random coefficients. The MIMC method is both a stochastic version of the combination technique introduced by Zenger, Griebel and collaborators and an extension of the Multilevel Monte Carlo (MLMC) method first described by Heinrich and Giles. Inspired by Giles s seminal work, instead of using first-order differences as in MLMC, we use in MIMC high-order mixed differences to reduce the variance of the hierarchical differences dramatically. Under standard assumptions on the convergence rates of the weak error, variance and work per sample, the optimal index set turns out to be of Total Degree (TD) type. When using such sets, MIMC yields new and improved complexity results, which are natural generalizations of Giles s MLMC analysis, and which increase the domain of problem parameters for which we achieve the optimal convergence, O(TOL-2).
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  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 . 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.
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, ): 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  several payoff functions are considered for a rather general set of models, whose characteristic function is assumed to be known.  presents the framework and theoretical approach for the error analysis, and establishes polynomial convergence rates for approximations of the option prices.  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 . 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.
We present a stochastic multilevel global-local algorithm  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.
Here, we study a crowd model proposed by R. Hughes in  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.
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 . 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.
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