50 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
Kontextsensitive Modellhierarchien fĂŒr Quantifizierung der höherdimensionalen Unsicherheit
We formulate four novel context-aware algorithms based on model hierarchies aimed to enable an efficient quantification of uncertainty in complex, computationally expensive problems, such as fluid-structure interaction and plasma microinstability simulations. Our results show that our algorithms are more efficient than standard approaches and that they are able to cope with the challenges of quantifying uncertainty in higher-dimensional, complex problems.Wir formulieren vier kontextsensitive Algorithmen auf der Grundlage von Modellhierarchien um eine effiziente Quantifizierung der Unsicherheit bei komplexen, rechenintensiven Problemen zu ermöglichen, wie Fluid-Struktur-Wechselwirkungs- und Plasma-MikroinstabilitÀtssimulationen. Unsere Ergebnisse zeigen, dass unsere Algorithmen effizienter als StandardansÀtze sind und die Herausforderungen der Quantifizierung der Unsicherheit in höherdimensionalen, komplexen Problemen bewÀltigen können
An hpâadaptive multiâelement stochastic collocation method for surrogate modeling with information reâuse
This article introduces an hpâadaptive multiâelement stochastic collocation method, which additionally allows to reâuse existing model evaluations during either hâ or pârefinement. The collocation method is based on weighted Leja nodes. After hârefinement, local interpolations are stabilized by adding and sorting Leja nodes on each newly created subâelement in a hierarchical manner. For pârefinement, the local polynomial approximations are based on totalâdegree or dimensionâadaptive bases. The method is applied in the context of forward and inverse uncertainty quantification to handle nonâsmooth or strongly localized response surfaces. The performance of the proposed method is assessed in several test cases, also in comparison to competing methods
Numerical Methods for PDE Constrained Optimization with Uncertain Data
Optimization problems governed by partial differential equations (PDEs) arise in many applications in the form of optimal control, optimal design, or parameter identification problems. In most applications, parameters in the governing PDEs are not deterministic, but rather have to be modeled as random variables or, more generally, as random fields. It is crucial to capture and quantify the uncertainty in such problems rather than to simply replace the uncertain coefficients with their mean values. However, treating the uncertainty adequately and in a computationally tractable manner poses many mathematical challenges. The numerical solution of optimization problems governed by stochastic PDEs builds on mathematical subareas, which so far have been largely investigated in separate communities: Stochastic Programming, Numerical Solution of Stochastic PDEs, and PDE Constrained Optimization.
The workshop achieved an impulse towards cross-fertilization of those disciplines which also was the subject of several scientific discussions. It is to be expected that future exchange of ideas between these areas will give rise to new insights and powerful new numerical methods