This ESI TP will gather at ESI leading researchers from applied mathematics, scientific computing and high-dimensional, computational statistics around the emerging area of numerical uncertainty quantification (UQ for short) in engineering and in the sciences. The TP will concentrate on mathematical foundations and underpinnings of novel computational strategies for the efficient numerical approximation of PDEs with uncertain inputs, as well as on the analysis of statistical methodologies for high-dimensional statistical data resulting from such PDE simulations. Both forward and inverse problems will be considered.

Upon placing (prior) probability measures on input parameter spaces, randomized (sampling) approximations can be employed to sample from the parametric solution manifolds: the proposed thematic program will, therefore, have one focus on Monte Carlo and quasi-Monte Carlo methods for high-dimensional random inputs, with particular attention to multilevel strategies. Other algorithmic techniques to be considered will include adaptive ("stochastic") collocation and Galerkin methods, in particular combined with Model Order Reduction (MOR), Reduced Basis Methods (RBM), low-rank approximations in tensor formats and compressed sensing based algorithms.

Another focus will be statistical modelling of large-scale (spatially or temporally) heterogeneous data for use as inputs of random PDEs. Regression and least squares based methodologies from high-dimensional statistics will be analyzed in the particular case of noisy responses of PDE outputs, and one workshop will be dedicated to kernel and machine learning based approximations of input-output maps for PDEs with highdimensional inputs as well as to new directions at the intersection of UQ and machine learning in general. While engineering models such as diffusion, acoustic, elastic and electromagnetic wave propagation and viscous flow will be foundational applications, extensions to kinetic and more general, integrodifferential equations with random input data will be considered.

Application areas will include computational directions in life sciences, medicine, geosciences, quantum chemistry, nanotechnology, computational mechanics and aerospace engineering.