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The goal of this short course is to review the analysis and the systematic dis-cretization of saddlepoint systems that arise in the weak formulation of certain fluid flow problems by Galekrin methods. To demonstrate the applicability of the main arguments in various situations, we consider the slow flow of water through a channel surrounded by a porous(More)
This paper investigates the stable identification of local volatility surfaces σ(S, t) in the Black-Scholes/Dupire equation from market prices of European Vanilla options. Based on the properties of the parameter-to-solution mapping, which assigns option prices to given volatilities, we show stability and convergence of approximations gained by Tikhonov(More)
We propose and analyse a new finite-element method for convection–diffusion problems based on the combination of a mixed method for the elliptic and a discontinuous Galerkin (DG) method for the hyper-bolic part of the problem. The two methods are made compatible via hybridization and the combination of both is appropriate for the solution of intermediate(More)
Diffuse optical tomography is a functional imaging technique based on propagation and absorption of light in biological tissues. We investigate the fast solution of the governing partial differential equations and the nonlinear inverse reconstruction problem on graphics hardware. In particular, we discuss the discretization by finite element methods, the(More)
In this paper we consider the regularization of the inverse problem of diffuse optical tomography by standard regularization methods with quadratic penalty terms. We therefore investigate in detail the properties of the associated forward operators, and derive continuity and differentiability results, which are based on derivation of W 1,p regularity(More)
In this paper we investigate convergence of Landweber iteration in Hilbert scales for linear and nonlinear inverse problems. As opposed to the usual application of Hilbert scales in the framework of regularization methods, we focus here on the case s ≤ 0, which (for Tikhonov regularization) corresponds to regularization in a weaker norm. In this case, the(More)
This manuscript deals with the hp error analysis of a hybrid discontinuous Galerkin method for incompressible flow. Besides the usual coercivity and boundedness estimates, we establish the inf-sup stability for the discrete incompressibility constraint with a constant, which is only slightly sub-optimal with respect to the polynomial degree. This result(More)
Image reconstruction in fluorescence optical tomography is a three-dimensional nonlinear ill-posed problem governed by a system of partial differential equations. In this paper we demonstrate that a combination of state of the art numerical algorithms and a careful hardware optimized implementation allows to solve this large-scale inverse problem in a few(More)
Correct pricing of options and other financial derivatives is of great importance to financial markets and one of the key subjects of mathematical finance. Usually, parameters specifying the underlying stochastic model are not directly observable, but have to be determined indirectly from observable quantities. The identification of local volatility(More)