Uncertainty quantification plays an increasingly important role in a wide range of problems in the physical sciences and financial markets. The underlying model may be subject to various uncertainties such as parameter or domain uncertainty, model uncertainty, numerical errors, or some intrinsic stochastic variability of the model. In the latter case, the uncertainty could be either introduced by measuring instruments or is the result of insufficient observations. For realistic simulations in the underlying differential equation model this is reflected via a random operator and/or random data. These parameters are often modelled as space-time Gaussian processes, leading to continuous random functions and thin-tailed, symmetric normal distributions.
Although Gaussian random objects have convenient analytical properties, for several applications, however, it might be favourable to model the stochastic quantities as discontinuous fields or processes which also allow for asymmetric and heavy-tailed distributions. In this minisymposium we bring together researchers whose foci are on stochastic or random partial differential equations which are influenced by discontinuous fields or processes.
14:00
Smoothed Lévy Fields and Random PDEs
Oliver Ernst | TU Chemnitz | Germany
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Oliver Ernst | TU Chemnitz | Germany
In this introductory talk to the topic of the MS, we recall the mathematical foundations of generalized random fields indexed by Schwartz functions based on the Bochner-Minlos theorem. We then show how convolution of such random noise fields with smoothing kernels from the Matérn family can be employed to yield smoothed random fields with continuous realizations and characterize precisely the minimal value of the Matérn smoothness parameter for this to hold. We specialize to generalized random fields with a Lévy distribution and establish that, in our setting, they are uniquely characterized as stationary noise fields. We close with some specific examples, which can be used to model smoothed Lévy diffusion coefficients of stationary diffusion equations.
The presentation is based on joint work with Hanno Gottschalk, Thomas Kalmes, Toni Kowalewitz and Marco Reese.
14:30
Elliptic PDE with Lévy coefficients: Integrability and Convergence of Approximations of Karhunen-Loève Type
Hanno Gottschalk | University of Wuppertal | Germany
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Hanno Gottschalk | University of Wuppertal | Germany
We study the integrability of solutions to elliptic partial differential equations with coefficients given by Lévy random fields in the n-th power of the $H^1$-Sobolev norm and furthermore the convergence of approximations of the Karhunen-Loève type in this sense. To this end, we develop and apply the extreme value theory for smoothed Lévy fields, based on prior results of Talagrand for Gaussian random fields. Combining this with a priori estimates from elliptic boundary value problems, we are also able prove convergence rates for the underlying two-step approximation procedure: In a first step, we remove discrete contributions due to perturbations far away from the region of interest and we then apply the classical KL-mode decomposition on the kernel of the smoothing operator. We also comment on further, statistical applications of our methods. This work is based on joint work with Oliver Ernst, Thomas Kalmes, Toni Kowalewitz and Marco Reese.
15:00
A Fully Discrete Approximation Scheme for a Stochastic Transport Problem with Lévy Noise
Andreas Stein | University of Stuttgart | Germany
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Andreas Stein | University of Stuttgart | Germany
To model the dynamics of interest rate and energy forward markets, linear hyperbolic SPDEs may be utilized. The forward rate is then given as the solution to a transport equation with a time-dependent random source term as driving noise.
To capture temporal discontinuities and allow for heavy-tailed distributions, we consider Hilbert space valued-Lévy processes (or fields) as driving noise. The numerical discretization of the corresponding SPDE involves several difficulties: Low spatial and temporal regularity of the solution to the problem entails slow convergence rates and instabilities for space/time-discretization schemes. Furthermore, the Lévy noise admits values in a possibly infinite-dimensional Hilbert space, hence projections into a finite-dimensional subspace for each discrete point in time are necessary. Finally, unbiased sampling from the resulting finite-dimensional Lévy field may not be possible.
We introduce a fully discrete approximation scheme that addresses the issues above. A discontinuous Galerkin approach for the spatial approximation is coupled with a suitable time stepping scheme to avoid numerical oscillations. Moreover, we approximate the driving noise by truncated Karhunen-Loeve expansions. The latter essentially yields a sum of scaled and uncorrelated one-dimensional Lévy processes, which may be simulated with controlled bias by Fourier inversion techniques.
This is joint work with Andrea Barth (SimTech, University of Stuttgart)
15:30
Subordinated Gaussian Random Fields in Elliptic SPDEs
Robin Merkle | University of Stuttgart | Germany
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Robin Merkle | University of Stuttgart | Germany
In many applications in the field of stochastic modeling, one needs random fields that allow spatial discontinuities. In case of a one-dimensional parameter space, Lévy processes are often convenient as they allow jumps and possess nice stochastic properties. One of the biggest advantages of (standard) Lévy processes is the parametrization given by the Lévy-Khintchine formula. However, in various situations (e.g. microstructure modeling of materials), a one-dimensional parameter space is not sufficient. Classical extensions of Lévy processes on two parameter dimensions suffer from the fact that they do not allow spatial discontinuities. Therefore, we use a new subordination approach to generate Lévy-type discontinuous random fields on a two dimensional spatial parameter domain. A Lévy-Chintschin-type formula is derived which allows a parametrization of the constructed random fields and other important stochastic properties are investigated. Further, we solve elliptic partial differential equations where subordinated (log-)Gaussian random fields appear as discontinuous coefficients using a Multilevel Monte Carlo approach and finite element methods.
This is joint work with Andrea Barth (SimTech, University of Stuttgart)