2,677 research outputs found
A bilinear differential forms approach to parametric structured state-space modelling
We use one-variable Loewner techniques to compute polynomial-parametric models for MIMO systems from vector-exponential data gathered at various points in the parameter space. Instrumental in our approach are the connections between vector-exponential modelling via bilinear differential forms and the Loewner framework
A literature survey of low-rank tensor approximation techniques
During the last years, low-rank tensor approximation has been established as
a new tool in scientific computing to address large-scale linear and
multilinear algebra problems, which would be intractable by classical
techniques. This survey attempts to give a literature overview of current
developments in this area, with an emphasis on function-related tensors
A posteriori error estimation and adaptivity in stochastic Galerkin FEM for parametric elliptic PDEs: beyond the affine case
We consider a linear elliptic partial differential equation (PDE) with a
generic uniformly bounded parametric coefficient. The solution to this PDE
problem is approximated in the framework of stochastic Galerkin finite element
methods. We perform a posteriori error analysis of Galerkin approximations and
derive a reliable and efficient estimate for the energy error in these
approximations. Practical versions of this error estimate are discussed and
tested numerically for a model problem with non-affine parametric
representation of the coefficient. Furthermore, we use the error reduction
indicators derived from spatial and parametric error estimators to guide an
adaptive solution algorithm for the given parametric PDE problem. The
performance of the adaptive algorithm is tested numerically for model problems
with two different non-affine parametric representations of the coefficient.Comment: 32 pages, 4 figures, 6 table
A posteriori error estimation and adaptivity in stochastic Galerkin FEM for parametric elliptic PDEs: beyond the affine case
We consider a linear elliptic partial differential equation (PDE) with a
generic uniformly bounded parametric coefficient. The solution to this PDE
problem is approximated in the framework of stochastic Galerkin finite element
methods. We perform a posteriori error analysis of Galerkin approximations and
derive a reliable and efficient estimate for the energy error in these
approximations. Practical versions of this error estimate are discussed and
tested numerically for a model problem with non-affine parametric
representation of the coefficient. Furthermore, we use the error reduction
indicators derived from spatial and parametric error estimators to guide an
adaptive solution algorithm for the given parametric PDE problem. The
performance of the adaptive algorithm is tested numerically for model problems
with two different non-affine parametric representations of the coefficient.Comment: 32 pages, 4 figures, 6 table
Finite element discretization methods for velocity-pressure and stream function formulations of surface Stokes equations
In this paper we study parametric TraceFEM and parametric SurfaceFEM (SFEM)
discretizations of a surface Stokes problem. These methods are applied both to
the Stokes problem in velocity-pressure formulation and in stream function
formulation. A class of higher order methods is presented in a unified
framework. Numerical efficiency aspects of the two formulations are discussed
and a systematic comparison of TraceFEM and SFEM is given. A benchmark problem
is introduced in which a scalar reference quantity is defined and numerically
determined.Comment: 26 page
Low rank surrogates for polymorphic fields with application to fuzzy-stochastic partial differential equations
We consider a general form of fuzzy-stochastic PDEs depending on the interaction of probabilistic
and non-probabilistic ("possibilistic") influences. Such a combined modelling of aleatoric
and epistemic uncertainties for instance can be applied beneficially in an engineering context for
real-world applications, where probabilistic modelling and expert knowledge has to be accounted
for. We examine existence and well-definedness of polymorphic PDEs in appropriate function
spaces. The fuzzy-stochastic dependence is described in a high-dimensional parameter space,
thus easily leading to an exponential complexity in practical computations.
To aleviate this severe obstacle in practise, a compressed low-rank approximation of the problem
formulation and the solution is derived. This is based on the Hierarchical Tucker format which
is constructed with solution samples by a non-intrusive tensor reconstruction algorithm. The performance
of the proposed model order reduction approach is demonstrated with two examples.
One of these is the ubiquitous groundwater flow model with Karhunen-Loeve coefficient field
which is generalized by a fuzzy correlation length
- …