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Sparse wavelet methods for option pricing under stochastic volatility
(2004)Research ReportReport -
Fourier mode analysis of layers in shallow shell deformations
(1999)SAM Research ReportWe investigate here the length scales of the boundary or interior layer effects in shell deformation. Quantitative information on the layers is obtained by considering two (simplified) `shallow' shell models corresponding to the `classical' three-field (Love-Koiter-Novozhilov), resp. five-field (Reissner-Naghdi) shell models. We start by analysing the layers as functions of the thickness of the shell, while keeping the other geometric ...Report -
Generalized FEM for Homogenization Problems
(2001)SAM Research ReportWe introduce the concept of generalized Finite Element Method (gFEM) for the numerical treatment of homogenization problems. These problems are characterized by highly oscillatory periodic (or patchwise periodic) pattern in the coefficients of the differential equation and their solutions exhibit a multiple scale behavior: a macroscopic behavior superposed with local characteristics at micro length scales. The gFEM is based on two-scale ...Report -
Wavelet Galerkin pricing of American options on Lévy driven assets
(2003)SAM Research ReportThe price of an American style contract on assets driven by Lévy processes with infinite jump activity is expressed as solution of a parabolic variational integro-differential inequality (PIDI). A Galerkin discretization in logarithmic price using a wavelet basis is presented with compression of the moment matrix of the jump part of the price process' Dynkin operator. An iterative solver with wavelet preconditioning for the resulting large ...Report -
Fast deterministic pricing of options on Lévy driven assets
(2002)SAM Research ReportArbitrage-free prices $u$ of European contracts on risky assets whose logreturns are modelled by Lévy processes satisfy a parabolic parabolic partial integrodifferential equation (PIDE) $\partial_t u + {\mathcal{A}}[u] = 0$. This PIDE is localized to bounded domains and the error due to this localization is estimated. The localized PIDE is discretized by the $\theta$-scheme in time and a wavelet Galerkin method with $N$ degrees of freedom ...Report -
Homogenization via p-FEM for Problems with Microstructure
(1999)SAM Research ReportA new class of $p$ version FEM for elliptic problems with microstructure is developed. Based on arguments from the theory of $n$-widths, the existence of subspaces with favourable approximation properties for solution sets of PDEs is deduced. The construction of such subspaces is addressed for problems with (patch-wise) periodic microstructure. Families of adapted spectral shape functions are exhibited which give exponential convergence ...Report -
Generalized p-FEM in Homogenization
(1999)SAM Research ReportA new finite element method for elliptic problems with locally periodic microstructure of length $\varepsilon >0$ is developed and analyzed. It is shown that the method converges, as $\varepsilon \rightarrow 0$, to the solution of the homogenized problem with optimal order in $\varepsilon$ and exponentially in the number of degrees of freedom independent of $\varepsilon > 0$. The computational work of the method is bounded independently ...Report -
Analysis of membrane locking in hp FEM for a cylindrical shell
(1997)SAM Research ReportIn this paper we analyze the performance of the hp-Finite Element Method for a cylindrical shell problem. Our theoretical investigations show that the hp approximation converges exponentially, provided that appropriate boundary layer elements are used. The numerical results illustrate the robustness and exponential convergence properties of the hp-Finite Element Method.Report -
Two-Scale FEM for Homogenization Problems
(2001)SAM Research ReportThe convergence of a two-scale FEM for elliptic problems in divergence form with coefficients and geometries oscillating at length scale \e << 1 is analyzed. Full elliptic regularity independent of \e is shown when the solution is viewed as mapping from the slow into the fast scale. Two-scale FE spaces which are able to resolve the \e scale of the solution with work independent of \e and without analytical homogenization are introduced. ...Report