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Sparse Tensor Approximations of High-Dimensional and stochastic Partial Differential Equations

Final Report Summary - STAHDPDE (Sparse Tensor Approximations of High-Dimensional and stochastic Partial Differential Equations)

The project investigated the mathematical formulation and numerical approximation of infinite-dimensional, parametric operator equations.

It identified new sparsity conditions on the problem inputs in order for several new classes of numerical methods to overcome the curse of dimensionality; among these methods where Multi-Level Monte-Carlo methods, higher order, Multi-Level Quasi Monte-Carlo methods, and stochastic collocation and Galerkin methods. Several of these we pioneered in the project.

Prototypical numerical implementations of these methods were developed which clarified important algorithmic ingredients which are necessary to realize in computational practice the theoretically demonstrated, dimension-independent convergence rates; among them a massively parallel, Multi-Level Monte-Carlo Finite-Volume solver for computational uncertainty quantification in nonlinear, hyperbolic PDEs, a dimension-adaptive Smolyak quadrature package which was used in several numerical experiments.

Representative applications include computational uncertainty quantification for elliptic, parabolic and hyperbolic partial differential equations. Specific examples include log-Gaussian permeability models in subsurface flow problems, computational assessment of shape uncertainty in acoustic and electromagnetic scattering, and of large systems of biochemical reaction networks, parabolic initial boundary value problems and stochastic modelling with Gaussian random fields on the sphere, which arises for example in climate models.

Additional applications include novel sparse, deterministic algorithms for the numerical evaluation of Bayesian inverse problems in all of the above-mentioned problem classes, i.e. for Bayesian estimation subject to given observation data that is corrupted by Gaussian observation noise.