714 research outputs found
Stochastic collocation on unstructured multivariate meshes
Collocation has become a standard tool for approximation of parameterized
systems in the uncertainty quantification (UQ) community. Techniques for
least-squares regularization, compressive sampling recovery, and interpolatory
reconstruction are becoming standard tools used in a variety of applications.
Selection of a collocation mesh is frequently a challenge, but methods that
construct geometrically "unstructured" collocation meshes have shown great
potential due to attractive theoretical properties and direct, simple
generation and implementation. We investigate properties of these meshes,
presenting stability and accuracy results that can be used as guides for
generating stochastic collocation grids in multiple dimensions.Comment: 29 pages, 6 figure
Adaptive stochastic Galerkin FEM for lognormal coefficients in hierarchical tensor representations
Stochastic Galerkin methods for non-affine coefficient representations are
known to cause major difficulties from theoretical and numerical points of
view. In this work, an adaptive Galerkin FE method for linear parametric PDEs
with lognormal coefficients discretized in Hermite chaos polynomials is
derived. It employs problem-adapted function spaces to ensure solvability of
the variational formulation. The inherently high computational complexity of
the parametric operator is made tractable by using hierarchical tensor
representations. For this, a new tensor train format of the lognormal
coefficient is derived and verified numerically. The central novelty is the
derivation of a reliable residual-based a posteriori error estimator. This can
be regarded as a unique feature of stochastic Galerkin methods. It allows for
an adaptive algorithm to steer the refinements of the physical mesh and the
anisotropic Wiener chaos polynomial degrees. For the evaluation of the error
estimator to become feasible, a numerically efficient tensor format
discretization is developed. Benchmark examples with unbounded lognormal
coefficient fields illustrate the performance of the proposed Galerkin
discretization and the fully adaptive algorithm
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