7,101 research outputs found
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
A weighted reduced basis method for parabolic PDEs with random data
This work considers a weighted POD-greedy method to estimate statistical
outputs parabolic PDE problems with parametrized random data. The key idea of
weighted reduced basis methods is to weight the parameter-dependent error
estimate according to a probability measure in the set-up of the reduced space.
The error of stochastic finite element solutions is usually measured in a root
mean square sense regarding their dependence on the stochastic input
parameters. An orthogonal projection of a snapshot set onto a corresponding POD
basis defines an optimum reduced approximation in terms of a Monte Carlo
discretization of the root mean square error. The errors of a weighted
POD-greedy Galerkin solution are compared against an orthogonal projection of
the underlying snapshots onto a POD basis for a numerical example involving
thermal conduction. In particular, it is assessed whether a weighted POD-greedy
solutions is able to come significantly closer to the optimum than a
non-weighted equivalent. Additionally, the performance of a weighted POD-greedy
Galerkin solution is considered with respect to the mean absolute error of an
adjoint-corrected functional of the reduced solution.Comment: 15 pages, 4 figure
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
POD model order reduction with space-adapted snapshots for incompressible flows
We consider model order reduction based on proper orthogonal decomposition
(POD) for unsteady incompressible Navier-Stokes problems, assuming that the
snapshots are given by spatially adapted finite element solutions. We propose
two approaches of deriving stable POD-Galerkin reduced-order models for this
context. In the first approach, the pressure term and the continuity equation
are eliminated by imposing a weak incompressibility constraint with respect to
a pressure reference space. In the second approach, we derive an inf-sup stable
velocity-pressure reduced-order model by enriching the velocity reduced space
with supremizers computed on a velocity reference space. For problems with
inhomogeneous Dirichlet conditions, we show how suitable lifting functions can
be obtained from standard adaptive finite element computations. We provide a
numerical comparison of the considered methods for a regularized lid-driven
cavity problem
- …