Kernel-based stochastic collocation for the random two-phase Navier-Stokes equations

In this work, we apply stochastic collocation methods with radial kernel basis functions for an uncertainty quantification of the random incompressible two-phase Navier-Stokes equations. Our approach is non-intrusive and we use the existing fluid dynamics solver NaSt3DGPF to solve the incompressible two-phase Navier-Stokes equation for each given realization. We are able to empirically show that the resulting kernel-based stochastic collocation is highly competitive in this setting and even outperforms some other standard methods

Reconstruction of cracks and material losses by perimeter-like penalizations and phase-field methods: numerical results

We numerically implement the variational approach for reconstruction in the inverse crack and cavity problems developed by one of the authors. The method is based on a suitably adapted free-discontinuity problem. Its main features are the use of phase-field functions to describe the defects to be reconstructed and the use of perimeter-like penalizations to regularize the ill-posed problem. The numerical implementation is based on the solution of the corresponding optimality system by a gradient method. Numerical simulations are presented to show the validity of the method.Comment: 15 pages, 12 figure

On multigrid for anisotropic equations and variational inequalities: pricing multi-dimensional European and American options

Partial differential operators in finance often originate in bounded linear stochastic processes. As a consequence, diffusion over these boundaries is zero and the corresponding coefficients vanish. The choice of parameters and stretched grids lead to additional anisotropies in the discrete equations or inequalities. In this study various block smoothers are tested in numerical experiments for equations of Black–Scholes-type (European options) in several dimensions. For linear complementarity problems, as they arise from optimal stopping time problems (American options), the choice of grid transfer is also crucial to preserve complementarity conditions on all grid levels. We adapt the transfer operators at the free boundary in a suitable way and compare with other strategies including cascadic approaches and full approximation schemes

Chiral quark-soliton model in the Wigner-Seitz approximation

In this paper we study the modification of the properties of the nucleon in the nucleus within the quark-soliton model. This is a covariant, dynamical model, which provides a non-linear representation of the spontaneously broken SU(2)_L X SU(2)_R symmetry of QCD. The effects of the nuclear medium are accounted for by using the Wigner-Seitz approximation and therefore reducing the complex many-body problem to a simpler single-particle problem. We find a minimum in the binding energy at finite density, a change in the isoscalar nucleon radius and a reduction of the in-medium pion decay constant. The latter is consistent with a partial restoration of chiral symmetry at finite density, which is predicted by other models.Comment: 30 pages, 13 figures; uses REVTeX and epsfi

On multigrid for anisotropic equations and variational inequalities: pricing multi-dimensional European and American options

Partial differential operators in finance often originate in bounded linear stochastic processes. As a consequence, diffusion over these boundaries is zero and the corresponding coefficients vanish. The choice of parameters and stretched grids lead to additional anisotropies in the discrete equations or inequalities. In this study various block smoothers are tested in numerical experiments for equations of Black–Scholes-type (European options) in several dimensions. For linear complementarity problems, as they arise from optimal stopping time problems (American options), the choice of grid transfer is also crucial to preserve complementarity conditions on all grid levels. We adapt the transfer operators at the free boundary in a suitable way and compare with other strategies including cascadic approaches and full approximation schemes