• An adaptive sparse grid algorithm for elliptic PDEs with lognormal diffusion coefficient

      Nobile, Fabio; Tamellini, Lorenzo; Tesei, Francesco; Tempone, Raul (2016-01-06)
    • Adaptive stochastic Galerkin FEM with hierarchical tensor representations

      Eigel, Martin (2016-01-08)
      PDE with stochastic data usually lead to very high-dimensional algebraic problems which easily become unfeasible for numerical computations because of the dense coupling structure of the discretised stochastic operator. Recently, an adaptive stochastic Galerkin FEM based on a residual a posteriori error estimator was presented and the convergence of the adaptive algorithm was shown. While this approach leads to a drastic reduction of the complexity of the problem due to the iterative discovery of the sparsity of the solution, the problem size and structure is still rather limited. To allow for larger and more general problems, we exploit the tensor structure of the parametric problem by representing operator and solution iterates in the tensor train (TT) format. The (successive) compression carried out with these representations can be seen as a generalisation of some other model reduction techniques, e.g. the reduced basis method. We show that this approach facilitates the efficient computation of different error indicators related to the computational mesh, the active polynomial chaos index set, and the TT rank. In particular, the curse of dimension is avoided.
    • Analysis of the stability and accuracy of the discrete least-squares approximation on multivariate polynomial spaces

      Migliorati, Giovanni (2016-01-05)
      We review the main results achieved in the analysis of the stability and accuracy of the discrete leastsquares approximation on multivariate polynomial spaces, with noiseless evaluations at random points, noiseless evaluations at low-discrepancy point sets, and noisy evaluations at random points.
    • Application of QMC methods to PDEs with random coefficients : a survey of analysis and implementation

      Kuo, Frances; Dick, Josef; Le Gia, Thong; Nichols, James; Sloan, Ian; Graham, Ivan; Scheichl, Robert; Nuyens, Dirk; Schwab, Christoph (2016-01-05)
      In this talk I will provide a survey of recent research efforts on the application of quasi-Monte Carlo (QMC) methods to PDEs with random coefficients. Such PDE problems occur in the area of uncertainty quantification. In recent years many papers have been written on this topic using a variety of methods. QMC methods are relatively new to this application area. I will consider different models for the randomness (uniform versus lognormal) and contrast different QMC algorithms (single-level versus multilevel, first order versus higher order, deterministic versus randomized). I will give a summary of the QMC error analysis and proof techniques in a unified view, and provide a practical guide to the software for constructing QMC points tailored to the PDE problems.
    • Approximation of quantum observables by molecular dynamics simulations

      Sandberg, Mattias (2016-01-06)
      In this talk I will discuss how to estimate the uncertainty in molecular dynamics simulations. Molecular dynamics is a computational method to study molecular systems in materials science, chemistry, and molecular biology. The wide popularity of molecular dynamics simulations relies on the fact that in many cases it agrees very well with experiments. If we however want the simulation to predict something that has no comparing experiment, we need a mathematical estimate of the accuracy of the computation. In the case of molecular systems with few particles, such studies are made by directly solving the Schrodinger equation. In this talk I will discuss theoretical results on the accuracy between quantum mechanics and molecular dynamics, to be used for systems that are too large to be handled computationally by the Schrodinger equation.
    • Basket call option pricing for CCVG using sparse grids

      Crocce, Fabian; Häppölä, Juho; Iania, Alessandro; Tempone, Raul (2016-01-06)
      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.
    • Bayesian inference and model comparison for metallic fatigue data

      Babuska, Ivo; Sawlan, Zaid A; Scavino, Marco; Szabó, Barma; Tempone, Raul (2016-01-06)
      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.
    • Bayesian Inference for Linear Parabolic PDEs with Noisy Boundary Conditions

      Ruggeri, Fabrizio; Sawlan, Zaid A; Scavino, Marco; Tempone, Raul (2016-01-06)
      In this work we develop a hierarchical Bayesian setting to infer unknown parameters in initial-boundary value problems (IBVPs) for one-dimensional linear parabolic partial differential equations. Noisy boundary data and known initial condition are assumed. We derive the likelihood function associated with the forward problem, given some measurements of the solution field subject to Gaussian noise. Such function is then analytically marginalized using the linearity of the equation. Gaussian priors have been assumed for the time-dependent Dirichlet boundary values. Our approach is applied to synthetic data for the one-dimensional heat equation model, where the thermal diffusivity is the unknown parameter. We show how to infer the thermal diffusivity parameter when its prior distribution is lognormal or modeled by means of a space-dependent stationary lognormal random field. We use the Laplace method to provide approximated Gaussian posterior distributions for the thermal diffusivity. Expected information gains and predictive posterior densities for observable quantities are numerically estimated for different experimental setups.
    • Bayesian inference of the heat transfer properties of a wall using experimental data

      Iglesias, Marco; Sawlan, Zaid A; Scavino, Marco; Tempone, Raul; Wood, Christopher (2016-01-06)
      A hierarchical Bayesian inference method is developed to estimate the thermal resistance and volumetric heat capacity of a wall. We apply our methodology to a real case study where measurements are recorded each minute from two temperature probes and two heat flux sensors placed on both sides of a solid brick wall along a period of almost five days. We model the heat transfer through the wall by means of the one-dimensional heat equation with Dirichlet boundary conditions. The initial/boundary conditions for the temperature are approximated by piecewise linear functions. We assume that temperature and heat flux measurements have independent Gaussian noise and derive the joint likelihood of the wall parameters and the initial/boundary conditions. Under the model assumptions, the boundary conditions are marginalized analytically from the joint likelihood. ApproximatedGaussian posterior distributions for the wall parameters and the initial condition parameter are obtained using the Laplace method, after incorporating the available prior information. The information gain is estimated under different experimental setups, to determine the best allocation of resources.
    • Bayesian optimal experimental design for priors of compact support

      Long, Quan (2016-01-08)
      In this study, we optimize the experimental setup computationally by optimal experimental design (OED) in a Bayesian framework. We approximate the posterior probability density functions (pdf) using truncated Gaussian distributions in order to account for the bounded domain of the uniform prior pdf of the parameters. The underlying Gaussian distribution is obtained in the spirit of the Laplace method, more precisely, the mode is chosen as the maximum a posteriori (MAP) estimate, and the covariance is chosen as the negative inverse of the Hessian of the misfit function at the MAP estimate. The model related entities are obtained from a polynomial surrogate. The optimality, quantified by the information gain measures, can be estimated efficiently by a rejection sampling algorithm against the underlying Gaussian probability distribution, rather than against the true posterior. This approach offers a significant error reduction when the magnitude of the invariants of the posterior covariance are comparable to the size of the bounded domain of the prior. We demonstrate the accuracy and superior computational efficiency of our method for shock-tube experiments aiming to measure the model parameters of a key reaction which is part of the complex kinetic network describing the hydrocarbon oxidation. In the experiments, the initial temperature and fuel concentration are optimized with respect to the expected information gain in the estimation of the parameters of the target reaction rate. We show that the expected information gain surface can change its shape dramatically according to the level of noise introduced into the synthetic data. The information that can be extracted from the data saturates as a logarithmic function of the number of experiments, and few experiments are needed when they are conducted at the optimal experimental design conditions.
    • Bayesian techniques for fatigue life prediction and for inference in linear time dependent PDEs

      Scavino, Marco (2016-01-08)
      In this talk we introduce first the main characteristics of a systematic statistical approach to model calibration, model selection and model ranking when stress-life data are drawn from a collection of records of fatigue experiments. Focusing on Bayesian prediction assessment, we consider fatigue-limit models and random fatigue-limit models under different a priori assumptions. In the second part of the talk, we present a hierarchical Bayesian technique for the inference of the coefficients of time dependent linear PDEs, under the assumption that noisy measurements are available in both the interior of a domain of interest and from boundary conditions. We present a computational technique based on the marginalization of the contribution of the boundary parameters and apply it to inverse heat conduction problems.
    • Book of abstracts with posters, UQAW 2016

      Litvinenko, Alexander; Dia, Ben Mansour (2016-01-06)
      This book contains all talk-abstracts and all posters presented at UQAW 2016
    • Chemical model reduction under uncertainty

      Najm, Habib; Galassi, R. Malpica; Valorani, M. (2016-01-05)
      We outline a strategy for chemical kinetic model reduction under uncertainty. We present highlights of our existing deterministic model reduction strategy, and describe the extension of the formulation to include parametric uncertainty in the detailed mechanism. We discuss the utility of this construction, as applied to hydrocarbon fuel-air kinetics, and the associated use of uncertainty-aware measures of error between predictions from detailed and simplified models.
    • Computable error estimates for FEMs for elliptic PDE with lognormal data

      Hall, Eric; Haakon, Hoel; Sandberg, Mattias; Szepessy, Anders; Tempone, Raul (2016-01-06)
    • Computable error estimates for Monte Carlo finite element approximation of elliptic PDE with lognormal diffusion coefficients

      Hall, Eric; Haakon, Hoel; Sandberg, Mattias; Szepessy, Anders; Tempone, Raul (2016-01-09)
      The Monte Carlo (and Multi-level Monte Carlo) finite element method can be used to approximate observables of solutions to diffusion equations with lognormal distributed diffusion coefficients, e.g. modeling ground water flow. Typical models use lognormal diffusion coefficients with H´ older regularity of order up to 1/2 a.s. This low regularity implies that the high frequency finite element approximation error (i.e. the error from frequencies larger than the mesh frequency) is not negligible and can be larger than the computable low frequency error. We address how the total error can be estimated by the computable error.
    • Computation of Electromagnetic Fields Scattered From Dielectric Objects of Uncertain Shapes Using MLMC

      Litvinenko, Alexander; Haji Ali, Abdul Lateef; Uysal, Ismail Enes; Ulku, Huseyin Arda; Oppelstrup, Jesper; Tempone, Raul; Bagci, Hakan (2016-01-06)
      Simulators capable of computing scattered fields from objects of uncertain shapes are highly useful in electromagnetics and photonics, where device designs are typically subject to fabrication tolerances. Knowledge of statistical variations in scattered fields is useful in ensuring error-free functioning of devices. Oftentimes such simulators use a Monte Carlo (MC) scheme to sample the random domain, where the variables parameterize the uncertainties in the geometry. At each sample, which corresponds to a realization of the geometry, a deterministic electromagnetic solver is executed to compute the scattered fields. However, to obtain accurate statistics of the scattered fields, the number of MC samples has to be large. This significantly increases the total execution time. In this work, to address this challenge, the Multilevel MC (MLMC [1]) scheme is used together with a (deterministic) surface integral equation solver. The MLMC achieves a higher efficiency by balancing the statistical errors due to sampling of the random domain and the numerical errors due to discretization of the geometry at each of these samples. Error balancing results in a smaller number of samples requiring coarser discretizations. Consequently, total execution time is significantly shortened.
    • Computation of High-Frequency Waves with Random Uncertainty

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