• Multilevel Drift-Implicit Tau-Leap

      Ben Hammouda, Chiheb; Moraes, Alvaro; Tempone, Raul (2016-01-06) [Poster]
      The dynamics of biochemical reactive systems with small copy numbers of one or more reactant molecules is dominated by stochastic effects. For those systems, discrete state-space and stochastic simulation approaches were proved to be more relevant than continuous state-space and deterministic ones. In systems characterized by having simultaneously fast and slowtimescales, the existing discrete space-state stochastic path simulation methods such as the stochastic simulation algorithm (SSA) and the explicit tauleap method can be very slow. Implicit approximations were developed in the literature to improve numerical stability and provide efficient simulation algorithms for those systems. In this work, we propose an efficient Multilevel Monte Carlo method in the spirit of the work by Anderson and Higham (2012) that uses drift-implicit tau-leap approximations at levels where the explicit tauleap method is not applicable due to numerical stability issues. We present numerical examples that illustrate the performance of the proposed method.
    • Bayesian inference and model comparison for metallic fatigue data

      Babuska, Ivo; Sawlan, Zaid A; Scavino, Marco; Szabó, Barma; Tempone, Raul (2016-01-06) [Poster]
      In this work, we present a statistical treatment of stress-life (S-N) data drawn from a collection of records of fatigue experiments that were performed on 75S-T6 aluminum alloys. Our main objective is to predict the fatigue life of materials by providing a systematic approach to model calibration, model selection and model ranking with reference to S-N data. To this purpose, we consider fatigue-limit models and random fatigue-limit models that are specially designed to allow the treatment of the run-outs (right-censored data). We first fit the models to the data by maximum likelihood methods and estimate the quantiles of the life distribution of the alloy specimen. We then compare and rank the models by classical measures of fit based on information criteria. We also consider a Bayesian approach that provides, under the prior distribution of the model parameters selected by the user, their simulation-based posterior distributions.
    • A multilevel adaptive reaction-splitting method for SRNs

      Moraes, Alvaro; Tempone, Raul; Vilanova, Pedro (2016-01-06) [Poster]
      In [5], we present a novel multilevel Monte Carlo method for kinetic simulation of stochastic reaction networks (SRNs) specifically designed for systems in which the set of reaction channels can be adaptively partitioned into two subsets characterized by either high or low activity. To estimate expected values of observables of the system, our method bounds the global computational error to be below a prescribed tolerance, TOL, within a given confidence level. This is achieved with a computational complexity of order O(TOL-2). We also present a novel control variate technique which may dramatically reduce the variance of the coarsest level at a negligible computational cost.
    • Multilevel Hybrid Chernoff Tau-Leap

      Moraes, Alvaro; Tempone, Raul; Vilanova, Pedro (2016-01-06) [Poster]
      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 [6] we present a novel multilevel algorithm for the Chernoff-based hybrid tauleap algorithm. This variance reduction technique 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.
    • Multilevel ensemble Kalman filter

      Chernov, Alexey; Hoel, Haakon; Law, Kody; Nobile, Fabio; Tempone, Raul (2016-01-06) [Poster]
      This work embeds a multilevel Monte Carlo (MLMC) sampling strategy into the Monte Carlo step of the ensemble Kalman filter (EnKF). In terms of computational cost vs. approximation error the asymptotic performance of the multilevel ensemble Kalman filter (MLEnKF) is superior to the EnKF s.
    • An adaptive sparse grid algorithm for elliptic PDEs with lognormal diffusion coefficient

      Nobile, Fabio; Tamellini, Lorenzo; Tesei, Francesco; Tempone, Raul (2016-01-06) [Poster]
    • Quasi-optimal sparse-grid approximations for random elliptic PDEs

      Nobile, Fabio; Tamellini, Lorenzo; Tempone, Raul (2016-01-06) [Poster]
    • Solving inverse problem via non-linear update of PCE coefficients

      Litvinenko, Alexander; Matthies, Hermann G.; Rosic, Bojana V.; Zander, Elmar (2016-01-06) [Poster]
    • Tight Error Bounds for Fourier Methods for Option Pricing for Exponential Levy Processes

      Crocce, Fabian; 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.
    • New Bayesian inference method using two steps of Markov chain Monte Carlo and its application to shock tube experiment data of Furan oxidation

      Kim, Daesang; El Gharamti, Iman; Bisetti, Fabrizio; Farooq, Aamir; Knio, Omar (2016-01-06) [Poster]
      A new Bayesian inference method has been developed and applied to Furan shock tube experimental data for efficient statistical inferences of the Arrhenius parameters of two OH radical consumption reactions. The collected experimental data, which consist of time series signals of OH radical concentrations of 14 shock tube experiments, may require several days for MCMC computations even with the support of a fast surrogate of the combustion simulation model, while the new method reduces it to several hours by splitting the process into two steps of MCMC: the first inference of rate constants and the second inference of the Arrhenius parameters. Each step has low dimensional parameter spaces and the second step does not need the executions of the combustion simulation. Furthermore, the new approach has more flexibility in choosing the ranges of the inference parameters, and the higher speed and flexibility enable the more accurate inferences and the analyses of the propagation of errors in the measured temperatures and the alignment of the experimental time to the inference results.
    • Optimal design of experiments considering noisy control parameters for the inference of Furan combustion reaction rate

      Long, Quan; Kim, Daesang; Bisetti, Fabrizio; Farooq, Aamir; Tempone, Raul; Knio, Omar (2016-01-06) [Poster]
      We carry out the design of experiments for the identification of the reaction parameters in Furan combustion. The lacks of information on the true value of the control parameters, specifically, the initial temperature and the initial TBHP concentration, are considered in the design procedure by errors-invariables models. We use two types of observables. The first is a scaler observable, i.e., half decay time of the [TBHP]. The second is the time history of the concentration.
    • Mathematical Model of Delamination in Composite Materials

      Dia, Ben Mansour; Espath, Lusi F.; Prudhomme, Serge; Selvakumaran, Lakshmi; Tempone, Raul (2016-01-06) [Poster]
    • A Sparsity-Regularized Reconstruction of Two-Dimensional Piecewise Continuous Domains!

      Sandhu, Ali Imran; Desmal, Abdulla; Bagci, Hakan (2016-01-06) [Poster]
    • Multi-Index Stochastic Collocation (MISC) for random elliptic PDEs

      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.
    • Hybrid Chernoff Tau-Leap

      Moraes, Alvaro; Tempone, Raul; Vilanova, Pedro (2016-01-06) [Poster]
      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.