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Fully Discrete Multiscale Galerkin BEM
(1995)SAM Research ReportWe analyze multiscale Galerkin methods for strongly elliptic boundary integral equations of order zero on closed surfaces in $R^3$. Piecewise polynomial, discontinuous multiwavelet bases of any polynomial degree are constructed explicitly. We show that optimal convergence rates in the boundary energy norm and in certain negative norms can be achieved with "compressed" stiffness matrices containing $O(N({\log N})^2)$ nonvanishing entries ...Report -
Wavelet-discretizations of parabolic integro-differential equations
(2001)SAM Research ReportWe consider parabolic problems u + Au = f in (0,T)x Ω, T < ∞, where Ω ⊂ Rd is a bounded domain and A is a strongly elliptic, classical pseudo-differential operator of order ρ ∈ [0,2] in H ρ/2 (Ω). We use a θ-scheme for time discretization and a Galerkin method with N degrees of freedom for space discretization. The full Galerkin matrix for A can be replaced with a sparse matrix using a wavelet basis, and the linear systems for each time ...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 -
Exponential convergence of $hp$ quadrature for integral operators with Gevrey kernels
(2009)SAM Research ReportGalerkin discretizations of integral equations in $\mathbb{R}^d$ require the evaluation of integrals $ I = \int_{S^{{(1)}}}$ $ \int_{S^{{(2)}}}$ $g(x,y)dydx$ where $S^{{(1)}}$, $S^{{(2)}}$ are $d$-simplices and $g$ has a singularity at $x=y$. We assume that g is Gevrey smooth for $x \neq y$ and satisfies bounds for the derivatives which allow algebraic singularities at $x = y$. This holds for kernel functions commonly occuring in integral ...Report -
Boundary Element Methods for Maxwell Equations in Lipschitz Domains
(2001)SAM Research ReportWe consider the Maxwell equations in a domain with Lipschitz boundary and the boundary integral operator A occuring in the Calderon projector. We prove an inf-sup condition for A using a Hodge decomposition. We apply this to two types of boundary value problems: the exterior scattering problem by a perfectly conducting body, and the dielectric problem with two different materials in the interior and exterior domain. In both cases we obtain ...Report -
Exponential convergence of $hp$ quadrature for integral operators with Gevrey kernels
(2009)SAM Research ReportGalerkin discretizations of integral equations in $\mathbb{R}^d$ require the evaluation of integrals $ I = \int_{S^{{(1)}}}$ $ \int_{S^{{(2)}}}$ $g(x,y)dydx$ where $S^{{(1)}}$, $S^{{(2)}}$ are $d$-simplices and $g$ has a singularity at $x=y$. We assume that g is Gevrey smooth for $x \neq y$ and satisfies bounds for the derivatives which allow algebraic singularities at $x = y$. This holds for kernel functions commonly occuring in integral ...Report