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A multilevel Monte Carlo method for stochastic elliptic partial differential equations
(2012)Presentation -
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Multiple Point Evaluation on Combined Tensor Product Supports
(2012)We consider the multiple point evaluation problem for an ndimensional space of functions [−1, 1[d#$ R spanned by d-variate basis functions that are the restrictions of simple (say linear) functions to tensor product domains. For arbitrary evaluation points this task is faced in the context of (semi-)Lagrangian schemes using adaptive sparse tensor approximation spaces for boundary value problems in moderately high dimensions. We devise a ...Presentation -
Loss of regularity for Kolmogorov equations
(2012)The celebrated H ormander condition is a sucient (and nearly necessary) condition for a second-order linear Kolmogorov partial dierential equation (PDE) with smooth coecients to be hypoelliptic. As a consequence, the solutions of Kolmogorov PDEs are smooth at all positive times if the coecients of the PDE are smooth and satisfy H ormander's condition even if the initial function is only continuous but not dierentiable. First-order linear ...Presentation -
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Predicting causal effects in high-dimensional settings
(2012)Understanding cause‐effect relationships between variables is of great interest in many fields of science. An ambitious but highly desirable goal is to infer causal effects from observational data obtained by observing a system of interest without subjecting it to interventions. This would allow to circumvent severe experimental constraints or to substantially lower experimental costs. Our main motivation to study this goal comes from ...Presentation -