4,866 research outputs found
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
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
Numerical analysis for time-dependent advection-diffusion problems with random discontinuous coefficients
Subsurface flows are commonly modeled by advection-diffusion equations.
Insufficient measurements or uncertain material procurement may be accounted
for by random coefficients. To represent, for example, transitions in
heterogeneous media, the parameters of the equation are spatially
discontinuous. Specifically, a scenario with coupled advection- and diffusion
coefficients that are modeled as sums of continuous random fields and
discontinuous jump components are considered. For the numerical approximation
of the solution, an adaptive, pathwise discretization scheme based on a Finite
Element approach is introduced. To stabilize the numerical approximation and
accelerate convergence, the discrete space-time grid is chosen with respect to
the varying discontinuities in each sample of the coefficients, leading to a
stochastic formulation of the Galerkin projection and the Finite Element basis
On properties and applications of Gaussian subordinated L\'evy fields
We consider Gaussian subordinated L\'evy fields (GSLFs) that arise by
subordinating L\'evy processes with positive transformations of Gaussian random
fields on some spatial domain , . The
resulting random fields are distributionally flexible and have in general
discontinuous sample paths. Theoretical investigations of the random fields
include pointwise distributions, possible approximations and their covariance
function. As an application, a random elliptic PDE is considered, where the
constructed random fields occur in the diffusion coefficient. Further, we
present various numerical examples to illustrate our 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, cà dlà g, 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
Multilevel Monte Carlo estimators for elliptic PDEs with L\'evy-type diffusion coefficient
General elliptic equations with spatially discontinuous diffusion
coefficients may be used as a simplified model for subsurface flow in
heterogeneous or fractured porous media. In such a model, data sparsity and
measurement errors are often taken into account by a randomization of the
diffusion coefficient of the elliptic equation which reveals the necessity of
the construction of flexible, spatially discontinuous random fields.
Subordinated Gaussian random fields are random functions on higher dimensional
parameter domains with discontinuous sample paths and great distributional
flexibility. In the present work, we consider a random elliptic partial
differential equation (PDE) where the discontinuous subordinated Gaussian
random fields occur in the diffusion coefficient. Problem specific multilevel
Monte Carlo (MLMC) Finite Element methods are constructed to approximate the
mean of the solution to the random elliptic PDE. We prove a-priori convergence
of a standard MLMC estimator and a modified MLMC - Control Variate estimator
and validate our results in various numerical examples
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