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Parabolic molecules
(2012)SAM Research ReportAnisotropic decompositions using representation systems based on parabolic scaling such as curvelets or shearlets have recently attracted significantly increased attention due to the fact that they were shown to provide optimally sparse approximations of functions exhibiting singularities on lower dimensional embedded manifolds. The literature now contains various direct proofs of this fact and of related sparse approximation results. ...Report -
Geometric multiscale decompositions of dynamic low-rank matrices
(2012)SAM Research ReportThe present paper is concerned with the study of manifold-valued multiscale transforms with a focus on the Stiefel manifold. For this speci c geometry we derive several formulas and algorithms for the computation of geometric means which will later enable us to construct multiscale transforms of wavelet type. As an application we study compression of piecewise smooth families of low-rank matrices both for synthetic data and also real-world ...Report -
hp-DG-QTT solution of high-dimensional degenerate diffusion equations
(2012)SAM Research ReportWe consider the discretization of degenerate, time-inhomogeneous Fokker-Planck equations for diffusion problems in high-dimensional domains. Well-posedness of the problem in time-weighted Bochner spaces is established. Analytic regularity of the time-dependence of the solution in countably normed, weighted Sobolev spaces is established. Time discretization by the hp-discontinuous We consider the discretization of degenerate, time-inhomogeneous ...Report -
Multi-level quasi-Monte Carlo finite element methods for a class of elliptic partial differential equations with random coefficients
(2012)SAM Research ReportThis paper is a sequel to our previous work $({Kuo, Schwab, Sloan, SIAM J.\ Numer.\ Anal., 2013})$ where quasi-Monte Carlo (QMC) methods (specifically, randomly shifted lattice rules) are applied to Finite Element (FE) discretizations of elliptic partial differential equations (PDEs) with a random coefficient represented in a countably infinite number of terms. We estimate the expected value of some linear functional of the solution, as ...Report -
High-dimensional adaptive sparse polynomial interpolation and applications to parametric PDEs
(2012)SAM Research ReportWe consider the problem of Lagrange polynomial interpolation in high or countably infinite dimension, motivated by the fast computation of solution to parametric/stochastic PDE’s. In such applications there is a substantial advantage in considering polynomial spaces that are sparse and anisotropic with respect to the different parametric variables. In an adaptive context, the polynomial space is enriched at different stages of the ...Report -
Trends in Resilience: Spotlight on Teaching & Learning Resilience. SKI Factsheet No. 9
(2012)CSS Risk and Resilience ReportsReport -
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Factsheet: Risiko, Verwundbarkeit, Resilienz
(2012)Die heutigen sicherheitspolitischen Debatten und Untersuchungen werden zunehmend von neuen Gefahrenkonzepten geprägt. Während Begrifflichkeiten wie Sicherheit und Bedrohung früher als dominante Leitideen galten, so werden Gefährdungen heute vermehrt anhand der neuen Konzepte Risiko, Verwundbarkeit und Resilienz untersucht. Welchen Mehrwert liefern diese neuen Begriffe in der Sicherheitsanalyse? Dieses Factsheet definiert die Ansätze dieser ...Report -
Numerical solution of scalar conservation laws with random flux functions
(2012)SAM Research ReportWe consider scalar hyperbolic conservation laws in several space dimensions, with a class of random (and parametric) flux functions. We propose a Karhunen-Loève expansion on the state space of the random flux. For random flux functions which are Lipschitz continuous with respect to the state variable, we prove the existence of a unique random entropy solution. Using a Karhunen-Loève spectral decomposition of the random flux into principal ...Report -
Low-rank tensor structure of linear diffusion operators in the TT and QTT formats
(2012)SAM Research ReportWe consider a class of multilevel matrices, which arise from the discretization of linear diffusion operators in a $d$-dimensional hypercube. Under certain assumptions on the structure of the diffusion tensor (motivated by financial models), we derive an explicit representation of such a matrix in the recently introduced Tensor Train (TT) format with the $TT$ ranks bounded from above by $2 + \lfloor \frac{d}{2}\rfloor$. We also show that ...Report