4,261 research outputs found
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
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
Probabilistic error estimation for non-intrusive reduced models learned from data of systems governed by linear parabolic partial differential equations
This work derives a residual-based a posteriori error estimator for reduced
models learned with non-intrusive model reduction from data of high-dimensional
systems governed by linear parabolic partial differential equations with
control inputs. It is shown that quantities that are necessary for the error
estimator can be either obtained exactly as the solutions of least-squares
problems in a non-intrusive way from data such as initial conditions, control
inputs, and high-dimensional solution trajectories or bounded in a
probabilistic sense. The computational procedure follows an offline/online
decomposition. In the offline (training) phase, the high-dimensional system is
judiciously solved in a black-box fashion to generate data and to set up the
error estimator. In the online phase, the estimator is used to bound the error
of the reduced-model predictions for new initial conditions and new control
inputs without recourse to the high-dimensional system. Numerical results
demonstrate the workflow of the proposed approach from data to reduced models
to certified predictions
The ROMES method for statistical modeling of reduced-order-model error
This work presents a technique for statistically modeling errors introduced
by reduced-order models. The method employs Gaussian-process regression to
construct a mapping from a small number of computationally inexpensive `error
indicators' to a distribution over the true error. The variance of this
distribution can be interpreted as the (epistemic) uncertainty introduced by
the reduced-order model. To model normed errors, the method employs existing
rigorous error bounds and residual norms as indicators; numerical experiments
show that the method leads to a near-optimal expected effectivity in contrast
to typical error bounds. To model errors in general outputs, the method uses
dual-weighted residuals---which are amenable to uncertainty control---as
indicators. Experiments illustrate that correcting the reduced-order-model
output with this surrogate can improve prediction accuracy by an order of
magnitude; this contrasts with existing `multifidelity correction' approaches,
which often fail for reduced-order models and suffer from the curse of
dimensionality. The proposed error surrogates also lead to a notion of
`probabilistic rigor', i.e., the surrogate bounds the error with specified
probability
An error indicator-based adaptive reduced order model for nonlinear structural mechanics -- application to high-pressure turbine blades
The industrial application motivating this work is the fatigue computation of
aircraft engines' high-pressure turbine blades. The material model involves
nonlinear elastoviscoplastic behavior laws, for which the parameters depend on
the temperature. For this application, the temperature loading is not
accurately known and can reach values relatively close to the creep
temperature: important nonlinear effects occur and the solution strongly
depends on the used thermal loading. We consider a nonlinear reduced order
model able to compute, in the exploitation phase, the behavior of the blade for
a new temperature field loading. The sensitivity of the solution to the
temperature makes {the classical unenriched proper orthogonal decomposition
method} fail. In this work, we propose a new error indicator, quantifying the
error made by the reduced order model in computational complexity independent
of the size of the high-fidelity reference model. In our framework, when the
{error indicator} becomes larger than a given tolerance, the reduced order
model is updated using one time step solution of the high-fidelity reference
model. The approach is illustrated on a series of academic test cases and
applied on a setting of industrial complexity involving 5 million degrees of
freedom, where the whole procedure is computed in parallel with distributed
memory
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