Self-Adaptive Reliable Numerical Treatment of Polymorphic Uncertainty by Hierarchical Tensors

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The aim of this project is to develop a fast and reliable self-adaptive simulation tool that can be used for polymorphic uncertainty quantification. The idea is to use model reduction techniques intertwined with tensor compression in order to produce a parametric representation of the high resolution model under consideration. The self-adaptivity is necessary since the compressed model should be used as a black box by researchers that are not specialized in tensors. The model reduction part is responsible for the reduction of the high resolution from the discretisation of the PDE model. The tensor compression part can cope with the many parameters or equivalently high dimensionality from the uncertainty in the model. Both parts combined provide a tool that produces the compressed representation in a complexity that is linear in the number of parameters and linear in the size of the number of unknowns for the PDE discretisation. We consider several practical model problems involving a mixture of uncertainties that arise from parameters in the model, external forces and initial data. We transform this problem into a multiparametric and high-dimensional one where parameters may come from different sources of uncertainty. The reduction of the parametric model gives rise to a compressed hierarchical low rank tensor representation which can be evaluated instantly for any given choice of parameters.