26,451 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
Coupling the reduced-order model and the generative model for an importance sampling estimator
In this work, we develop an importance sampling estimator by coupling the
reduced-order model and the generative model in a problem setting of
uncertainty quantification. The target is to estimate the probability that the
quantity of interest (QoI) in a complex system is beyond a given threshold. To
avoid the prohibitive cost of sampling a large scale system, the reduced-order
model is usually considered for a trade-off between efficiency and accuracy.
However, the Monte Carlo estimator given by the reduced-order model is biased
due to the error from dimension reduction. To correct the bias, we still need
to sample the fine model. An effective technique to reduce the variance
reduction is importance sampling, where we employ the generative model to
estimate the distribution of the data from the reduced-order model and use it
for the change of measure in the importance sampling estimator. To compensate
the approximation errors of the reduced-order model, more data that induce a
slightly smaller QoI than the threshold need to be included into the training
set. Although the amount of these data can be controlled by a posterior error
estimate, redundant data, which may outnumber the effective data, will be kept
due to the epistemic uncertainty. To deal with this issue, we introduce a
weighted empirical distribution to process the data from the reduced-order
model. The generative model is then trained by minimizing the cross entropy
between it and the weighted empirical distribution. We also introduce a penalty
term into the objective function to deal with the overfitting for more
robustness. Numerical results are presented to demonstrate the effectiveness of
the proposed methodology
Dimension reduction for systems with slow relaxation
We develop reduced, stochastic models for high dimensional, dissipative
dynamical systems that relax very slowly to equilibrium and can encode long
term memory. We present a variety of empirical and first principles approaches
for model reduction, and build a mathematical framework for analyzing the
reduced models. We introduce the notions of universal and asymptotic filters to
characterize `optimal' model reductions for sloppy linear models. We illustrate
our methods by applying them to the practically important problem of modeling
evaporation in oil spills.Comment: 48 Pages, 13 figures. Paper dedicated to the memory of Leo Kadanof
Contracting Nonlinear Observers: Convex Optimization and Learning from Data
A new approach to design of nonlinear observers (state estimators) is
proposed. The main idea is to (i) construct a convex set of dynamical systems
which are contracting observers for a particular system, and (ii) optimize over
this set for one which minimizes a bound on state-estimation error on a
simulated noisy data set. We construct convex sets of continuous-time and
discrete-time observers, as well as contracting sampled-data observers for
continuous-time systems. Convex bounds for learning are constructed using
Lagrangian relaxation. The utility of the proposed methods are verified using
numerical simulation.Comment: conference submissio
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