Uncertainty Quantification for Linear Hyperbolic Equations with Stochastic Process or Random Field Coefficients

In this paper hyperbolic partial differential equations with random coefficients are discussed. Such random partial differential equations appear for instance in traffic flow problems as well as in many physical processes in random media. Two types of models are presented: The first has a time-dependent coefficient modeled by the Ornstein--Uhlenbeck process. The second has a random field coefficient with a given covariance in space. For the former a formula for the exact solution in terms of moments is derived. In both cases stable numerical schemes are introduced to solve these random partial differential equations. Simulation results including convergence studies conclude the theoretical findings

Milstein Approximation for Advection-Diffusion Equations Driven by Multiplicative Noncontinuous Martingale Noises

In this paper, the strong approximation of a stochastic partial differential equation, whose differential operator is of advection-diffusion type and which is driven by a multiplicative, infinite dimensional, càdlàg, square integrable martingale, is presented. A finite dimensional projection of the infinite dimensional equation, for example a Galerkin projection, with nonequidistant time stepping is used. Error estimates for the discretized equation are derived inL 2 and almost sure senses. Besides space and time discretizations, noise approximations are also provided, where the Milstein double stochastic integral is approximated in such a way that the overall complexity is not increased compared to an Euler-Maruyama approximation. Finally, simulations complete the pape

Stochastic Partial Differential Equations : Approximations and Applications

For many people the behaviour of stock prices may appear to be unpredictable. The price dynamics seem to exhibit no regularity. Although it might be hard to believe, mathematicians and physisists have managed to explain this behaviour via functions whose characteristics match those of the observed phenomena. In mathematics we model such curves with stochastic equations (driven by stochastic processes). They describe chaotic behaviour and can be used to produce computer simulations. The (standard) theory is quite well known and established. However, when one studies more complex financial markets and products, the complexity of the stochastic equations increases considerably. As an extension to the text-book theory, one could devise models in more than one dimension. Eventually this would lead to the notion of stochastic equations taking values in some function space (stochastic partial differential equations) or random fields. The simulation of stochastic partial differential equations is the main contribution of this work. We show convergence of discretizations as the simulation becomes more precise. We introduce as well possible applications like forward pricing in energy markets, or hedging against weather risk due to temperature uncertainty. A Finite Element Method is used for the discretization. This is a well established numerical method for deterministic problems. When we deal with stochastic equations, however, the world is not smooth and thus the problems become more daunting. In this work we introduce Finite Element Methods for stochastic partial differntial equations driven by different noise processes

Multilevel Monte Carlo method for parabolic stochastic partial differential equations

We analyze the convergence and complexity of multilevel Monte Carlo discretizations of a class of abstract stochastic, parabolic equations driven by square integrable martingales. We show under low regularity assumptions on the solution that the judicious combination of low order Galerkin discretizations in space and an Euler-Maruyama discretization in time yields mean square convergence of order one in space and of order1/2 in time to the expected value of the mild solution. The complexity of the multilevel estimator is shown to scale log-linearly with respect to the corresponding work to generate a single path of the solution on the finest mesh, resp. of the corresponding deterministic parabolic problem on the finest mes

Multi-level Monte Carlo Finite Element method for elliptic PDEs with stochastic coefficients

In Monte Carlo methods quadrupling the sample size halves the error. In simulations of stochastic partial differential equations (SPDEs), the total work is the sample size times the solution cost of an instance of the partial differential equation. A Multi-level Monte Carlo method is introduced which allows, in certain cases, to reduce the overall work to that of the discretization of one instance of the deterministic PDE. The model problem is an elliptic equation with stochastic coefficients. Multi-level Monte Carlo errors and work estimates are given both for the mean of the solutions and for higher moments. The overall complexity of computing mean fields as well as k-point correlations of the random solution is proved to be of log-linear complexity in the number of unknowns of a single Multi-level solve of the deterministic elliptic problem. Numerical examples complete the theoretical analysi

Comparison of data-driven uncertainty quantification methods for a carbon dioxide storage benchmark scenario

A variety of methods is available to quantify uncertainties arising with\-in the modeling of flow and transport in carbon dioxide storage, but there is a lack of thorough comparisons. Usually, raw data from such storage sites can hardly be described by theoretical statistical distributions since only very limited data is available. Hence, exact information on distribution shapes for all uncertain parameters is very rare in realistic applications. We discuss and compare four different methods tested for data-driven uncertainty quantification based on a benchmark scenario of carbon dioxide storage. In the benchmark, for which we provide data and code, carbon dioxide is injected into a saline aquifer modeled by the nonlinear capillarity-free fractional flow formulation for two incompressible fluid phases, namely carbon dioxide and brine. To cover different aspects of uncertainty quantification, we incorporate various sources of uncertainty such as uncertainty of boundary conditions, of conceptual model definitions and of material properties. We consider recent versions of the following non-intrusive and intrusive uncertainty quantification methods: arbitary polynomial chaos, spatially adaptive sparse grids, kernel-based greedy interpolation and hybrid stochastic Galerkin. The performance of each approach is demonstrated assessing expectation value and standard deviation of the carbon dioxide saturation against a reference statistic based on Monte Carlo sampling. We compare the convergence of all methods reporting on accuracy with respect to the number of model runs and resolution. Finally we offer suggestions about the methods' advantages and disadvantages that can guide the modeler for uncertainty quantification in carbon dioxide storage and beyond

A laboratory device for mesuring the method of eddy currents the diffusion coefficient of hydrogen in metals hydrogenation

A method is proposed for measuring the diffusion coefficient of hydrogen in a membrane made of a titanium alloy VT1-0 based on the use of eddy currents and an installation for its implementation is described. An eddy current sensor is located on the membrane, which is the wall of the electrolytic cell. The diffusion process is measured under the conditions of formation of titanium hydrides. The formation of titanium hydrides during hydrogenation is monitored over time by measuring the diffraction spectra. The results of the measurement of the diffusion coefficient of hydrogen by the proposed method are given in the conditions of formation of titanium hydrides in the membrane. The results can be used to determine the diffusion coefficients of hydrogen in various structural materials, in space technology, nuclear power engineering, in products subjected to hydrogenation during operation