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Hierarchical tensor methods for PDEs with stochastic parameters
Keywords: Hierarchical Tucker; Stochastic PDEs; Low Rank Tensors
Tue, 10:45--11:10
  • Grasedyck, Lars (IGPM, RWTH Aachen, Germany)
  • Ballani, Jonas (Max Planck Institut für Mathematik in den Naturwissenschaften, Leipzig)
  • Kluge, Melanie (IGPM, RWTH Aachen)

We consider the problem to solve a (stochastic) parameter dependent equation \[A(\omega)u(\omega) = b(\omega),\qquad \omega\in\Omega \] for systems $A$ governed by partial differential operators that depend on $\omega$. Our aim is to calculate quantities of interest (mean, variance, maximum etc.) of the set of solutions. One way to solve such a problem is by expansion of the system, the right-hand side as well as the solution in independent stochastic variables $\omega_1,\ldots,\omega_p$, and then solve the arising large-scale deterministic problem \[A(\omega_1,\ldots,\omega_p)u(\omega_1,\ldots,\omega_p) = b(\omega_1,\ldots,\omega_p).\] An alternative approach is to use (quasi or multilevel) Monte Carlo (MC) methods which require just a simple sampling ($M$ simulations), but these are only useful for certain quantities of interest (e.g. the mean). We will present a new approach based on hierarchical Tucker (HT) representations of tensors. This method is based on standard PDE solvers for deterministic systems. The set of solutions is approximated in a low rank (HT) tensor format that allows for many parameters (thousands).