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Projekte

Stochastic Partial Differential Equations

In this project we are concerned with the well posedness of some hyperbolic or parabolic stochastic partial differential equation. Besides the question of existence and uniqueness of a solution, we develop numerical methods for the simulation of paths and moments of solutions. The equations considered play a central role in financial mathematics and physics.

Transport equations with Gaussian noise Transport equations with Gaussian noise

Partial Differential Equations with Stochastic Data

Different random fields used as coefficient Different random fields used as coefficient

In this project we consider different types of partial differential equations with different types of uncertainty. More precisely we look at existence and uniqueness of solutions and numerical methods to simulate the solutions for:

  • Statistical solutions of the Navier-Stokes equations
  • Elliptic equations with stochastic coefficients
  • Hyperbolic equations with stochastic coefficients

Cooperations

  • Stochastic Control Problems:

    Many models in banking and finance are governed by stochastic differential equations and are subject to some (stochastic) control. This leads to stochastic control problems and impuls control problems. In some instances closed form solutions can be found in terms of ordinary differntial equations, often, however, the solution is given by a partial differential equations and has to be approximated numerically. We study various models and their solutions as well as suitable approximations.

    This is a joint project with University of Zurich and the SNB.

  • Reduced Order Models:

    In this joint projekt with Bernard Hassdonk (IANS) we study some time-parallel implementations for reduced order models with applications to uncertainty quantification. The time-parallel implementation has the advantage that the problems can be solved in real-time.

  • Multilevel Monte Carlo methods for transport in porous media:

    In this joint work with Wolfgang Novak (IWS) we study different simulation techniques for transport in porous media, such as the Particle Tracking Random Walk, combined with multilevel Monte Carlo methods. The simulation of transport in purous media is in combination with standard Monte Carlo methods too costly. The adaptation of a mulitlevel technique to these problems reduces the computational complexity considerably.

  • Stochastic inclusions:

    In this joint project with Bastian Harrach (IMNG) and Aalto University, Finland, the inclusion detection problem of electrical impedance tomography with stochastic conductivities is studied. The conductivities are not always know for all possible tissues and substances, therefore, a stochastic model is considered, whereas the shape of the inclusions is deterministic.

Publications

A list of publications can be found here