Project B07

This work considers the inclusion detection problem of electrical impedance tomography with stochastic conductivities. It is shown that a conductivity anomaly with a random conductivity can be identified by applying the factorization method or the monotonicity method to the mean value of the corresponding Neumann-to-Dirichlet map provided that the anomaly has high enough contrast in the sense of expectation. The theoretical results are complemented by numerical examples in two spatial dimensions.

In this paper approximation methods for infinite-dimensional Lévy processes, also called (time-dependent) Lévy fields, are introduced. For square integrable fields beyond the Gaussian case, it is no longer given that the one-dimensional distributions in the spectral representation with respect to the covariance operator are independent. When simulated via a Karhunen–Loève expansion a set of dependent but uncorrelated one-dimensional Lévy processes has to be generated. The dependence structure among the one-dimensional processes ensures that the resulting field exhibits the correct point-wise marginal distributions. To approximate the respective (one-dimensional) Lévy-measures, a numerical method, called discrete Fourier inversion, is developed. For this method, Lp-convergence rates can be obtained and, under certain regularity assumptions, mean square and Lp-convergence of the approximated field is proved. Further, a class of (time-dependent) Lévy fields is introduced, where the point-wise marginal distributions are dependent but uncorrelated subordinated Wiener processes. For this specific class one may derive point-wise marginal distributions in closed form. Numerical examples, which include hyperbolic and normal-inverse Gaussian fields, demonstrate the efficiency of the approach.

Continuous sedimentation processes in a clarifier-thickener unit can be described by a scalar nonlinear conservation law whose flux density function is discontinuous with respect to the spatial position. In the applications of this model, which include mineral processing and wastewater treatment, the rate and composition of the feed flow cannot be given deterministically. Efficient numerical simulation is required to quantify the effect of uncertainty in these control parameters in terms of the response of the clarifier-thickener system. Thus, the problem at hand is one of uncertainty quantification for nonlinear hyperbolic problems with several random perturbations. The presented hybrid stochastic Galerkin method is devised so as to extend the polynomial chaos approximation by multiresolution discretization in the stochastic space. This approach leads to a deterministic hyperbolic system, which is partially decoupled and therefore suitable for efficient parallelisation. Stochastic adaptivity reduces the computational effort. Several numerical experiments are presented.

In this paper hyperbolic partial differential equations (PDEs) with random coefficients are discussed. We consider the challenging problem of flux functions with coefficients modeled by spatiotemporal random fields. Those fields are given by correlated Gaussian random fields in space and Ornstein--Uhlenbeck processes in time. The resulting system of equations consists of a stochastic differential equation for each random parameter coupled to the hyperbolic conservation law. We define an appropriate solution concept in this setting and analyze errors and convergence of discretization methods. A novel discretization framework, based on Monte Carlo finite volume methods, is presented for the robust computation of moments of solutions to those random hyperbolic PDEs. We showcase the approach on two examples which appear in applications---the magnetic induction equation and linear acoustics---both with a spatiotemporal random background velocity field.

In this paper an infinite-dimensional approach to model energy forward markets is introduced. Similar to the Heath–Jarrow–Morton framework in interest-rate modelling, a first-order hyperbolic stochastic partial differential equation models the dynamics of the forward price curves. These equations are analysed, and in particular regularity and no-arbitrage conditions in the general situation of stochastic partial differential equations driven by an infinite-dimensional martingale process are studied. Both arithmetic and geometric forward price dynamics are studied, as well as accounting for the delivery period of electricity forward contracts. A stable and convergent numerical approximation in the form of a finite element method for hyperbolic stochastic partial differential equations is introduced and applied to some examples with relevance to energy markets.

We propose Monte Carlo (MC), single level Monte Carlo (SLMC) and multilevel Monte Carlo (MLMC) methods for the numerical approximation of statistical solutions to the viscous, incompressible Navier–Stokes equations (NSE) on a bounded, connected domain D⊂Rd, d=1,2 with no-slip or periodic boundary conditions on the boundary ∂D. The MC convergence rate of order 1/2 is shown to hold independently of the Reynolds number with constant depending only on the mean kinetic energy of the initial velocity ensemble. We discuss the effect of space-time discretizations on the MC convergence. We propose a numerical MLMC estimator, based on finite samples of numerical solutions with finite mean kinetic energy in a suitable function space and give sufficient conditions for mean-square convergence to a (generalized) moment of the statistical solution. We provide in particular error bounds for MLMC approximations of statistical solutions to the viscous Burgers equation in space dimension d=1 and to the viscous, incompressible Navier-Stokes equations in space dimension d=2 which are uniform with respect to the viscosity parameter. For a more detailed presentation and proofs we refer the reader to Barth et al. (Multilevel Monte Carlo Approximations of Statistical Solutions of the Navier-Stokes Equations, 2013).

In this paper Monte Carlo finite element approximations for elliptic homogenization problems with random coefficients, which oscillate on $nınN$ a priori known, separated microscopic length scales, are considered. The convergence of multilevel Monte Carlo finite element discretizations is analyzed. In particular, it is considered that the multilevel finite element discretization resolves the finest physical length scale, but the coarsest finite element mesh does not, so that the so-called resonance case occurs at intermediate multilevel Monte Carlo sampling levels. It is shown that for first order finite elements in two space dimensions, the multilevel Monte Carlo finite element method converges at the same rate as the corresponding single-level Monte Carlo finite element method, despite the majority of samples being underresolved in the multilevel Monte Carlo finite element estimator. It is proved that switching to a hierarchic multiscale finite element method such as the finite element heterogeneous multiscale method to compute the multilevel Monte Carlo finite element estimator, when only meshes are used which underresolve all physical length scales, implies optimal convergence. Specifically, both methods proposed here allow one to obtain estimates of the expectation of the random solution, with accuracy versus work that is identical to the solution of a single deterministic problem. In the case of the finite element heterogeneous multiscale method the estimate is, moreover, robust with respect to the physical length scales. Numerical experiments corroborate our analytical findings.

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 in L 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 paper.

In this work, the approximation of Hilbert-space-valued random variables is combined with the approximation of the expectation by a multilevel Monte Carlo (MLMC) method. The number of samples on the different levels of the multilevel approximation are chosen such that the errors are balanced. The overall work then decreases in the optimal case to O(h −2) if h is the error of the approximation. The MLMC method is applied to functions of solutions of parabolic and hyperbolic stochastic partial differential equations as needed, for example, for option pricing. Simulations complete the paper.

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 analysis.