78 research outputs found
A Duality Approach to Error Estimation for Variational Inequalities
Motivated by problems in contact mechanics, we propose a duality approach for
computing approximations and associated a posteriori error bounds to solutions
of variational inequalities of the first kind. The proposed approach improves
upon existing methods introduced in the context of the reduced basis method in
two ways. First, it provides sharp a posteriori error bounds which mimic the
rate of convergence of the RB approximation. Second, it enables a full
offline-online computational decomposition in which the online cost is
completely independent of the dimension of the original (high-dimensional)
problem. Numerical results comparing the performance of the proposed and
existing approaches illustrate the superiority of the duality approach in cases
where the dimension of the full problem is high.Comment: 21 pages, 8 figure
Uncertainty quantification for basin-scale geothermal conduction models
Geothermal energy plays an important role in the energy transition by providing a renewable energy source with a low CO2 footprint. For this reason, this paper uses state-of-the-art simulations for geothermal applications, enabling predictions for a responsible usage of this earth’s resource. Especially in complex simulations, it is still common practice to provide a single deterministic outcome although it is widely recognized that the characterization of the subsurface is associated with partly high uncertainties. Therefore, often a probabilistic approach would be preferable, as a way to quantify and communicate uncertainties, but is infeasible due to long simulation times. We present here a method to generate full state predictions based on a reduced basis method that significantly reduces simulation time, thus enabling studies that require a large number of simulations, such as probabilistic simulations and inverse approaches. We implemented this approach in an existing simulation framework and showcase the application in a geothermal study, where we generate 2D and 3D predictive uncertainty maps. These maps allow a detailed model insight, identifying regions with both high temperatures and low uncertainties. Due to the flexible implementation, the methods are transferable to other geophysical simulations, where both the state and the uncertainty are important.</p
Learning constitutive models from microstructural simulations via a non-intrusive reduced basis method
In order to optimally design materials, it is crucial to understand the
structure-property relations in the material by analyzing the effect of
microstructure parameters on the macroscopic properties. In computational
homogenization, the microstructure is thus explicitly modeled inside the
macrostructure, leading to a coupled two-scale formulation. Unfortunately, the
high computational costs of such multiscale simulations often render the
solution of design, optimization, or inverse problems infeasible. To address
this issue, we propose in this work a non-intrusive reduced basis method to
construct inexpensive surrogates for parametrized microscale problems; the
method is specifically well-suited for multiscale simulations since the coupled
simulation is decoupled into two independent problems: (1) solving the
microscopic problem for different (loading or material) parameters and learning
a surrogate model from the data; and (2) solving the macroscopic problem with
the learned material model. The proposed method has three key features. First,
the microscopic stress field can be fully recovered. Second, the method is able
to accurately predict the stress field for a wide range of material parameters;
furthermore, the derivatives of the effective stress with respect to the
material parameters are available and can be readily utilized in solving
optimization problems. Finally, it is more data efficient, i.e. requiring less
training data, as compared to directly performing a regression on the effective
stress. For the microstructures in the two test problems considered, the mean
approximation error of the effective stress is as low as 0.1% despite using a
relatively small training dataset. Embedded into the macroscopic problem, the
reduced order model leads to an online speed up of approximately three orders
of magnitude while maintaining a high accuracy as compared to the FE
solver
A reduced order model for geometrically parameterized two-scale simulations of elasto-plastic microstructures under large deformations
In recent years, there has been a growing interest in understanding complex
microstructures and their effect on macroscopic properties. In general, it is
difficult to derive an effective constitutive law for such microstructures with
reasonable accuracy and meaningful parameters. One numerical approach to bridge
the scales is computational homogenization, in which a microscopic problem is
solved at every macroscopic point, essentially replacing the effective
constitutive model. Such approaches are, however, computationally expensive and
typically infeasible in multi-query contexts such as optimization and material
design. To render these analyses tractable, surrogate models that can
accurately approximate and accelerate the microscopic problem over a large
design space of shapes, material and loading parameters are required. In
previous works, such models were constructed in a data-driven manner using
methods such as Neural Networks (NN) or Gaussian Process Regression (GPR).
However, these approaches currently suffer from issues, such as need for large
amounts of training data, lack of physics, and considerable extrapolation
errors. In this work, we develop a reduced order model based on Proper
Orthogonal Decomposition (POD), Empirical Cubature Method (ECM) and a
geometrical transformation method with the following key features: (i) large
shape variations of the microstructure are captured, (ii) only relatively small
amounts of training data are necessary, and (iii) highly non-linear
history-dependent behaviors are treated. The proposed framework is tested and
examined in two numerical examples, involving two scales and large geometrical
variations. In both cases, high speed-ups and accuracies are achieved while
observing good extrapolation behavior
A Certified Trust Region Reduced Basis Approach to PDE-Constrained Optimization
Parameter optimization problems constrained by partial differential equations (PDEs) appear in many science and engineering applications. Solving these optimization problems may require a prohibitively large number of computationally expensive PDE solves, especially if the dimension of the design space is large. It is therefore advantageous to replace expensive high-dimensional PDE solvers (e.g., finite element) with lower-dimensional surrogate models. In this paper, the reduced basis (RB) model reduction method is used in conjunction with a trust region optimization framework to accelerate PDE-constrained parameter optimization. Novel a posteriori error bounds on the RB cost and cost gradient for quadratic cost functionals (e.g., least squares) are presented and used to guarantee convergence to the optimum of the high-fidelity model. The proposed certified RB trust region approach uses high-fidelity solves to update the RB model only if the approximation is no longer sufficiently accurate, reducing the number of full-fidelity solves required. We consider problems governed by elliptic and parabolic PDEs and present numerical results for a thermal fin model problem in which we are able to reduce the number of full solves necessary for the optimization by up to 86%. Key words: model reduction, optimization, trust region methods, partial differential equations, reduced basis methods, error bounds, parametrized systemsFulbright U.S. Student ProgramNational Science Foundation (U.S.). Graduate Research Fellowship ProgramHertz FoundationUnited States. Department of Energy. Office of Advanced Scientific Computing Research (Award DEFG02-08ER2585)United States. Department of Energy. Office of Advanced Scientific Computing Research (Award DE-SC0009297
Choosing observation operators to mitigate model error in Bayesian inverse problems
In statistical inference, a discrepancy between the parameter-to-observable
map that generates the data and the parameter-to-observable map that is used
for inference can lead to misspecified likelihoods and thus to incorrect
estimates. In many inverse problems, the parameter-to-observable map is the
composition of a linear state-to-observable map called an `observation
operator' and a possibly nonlinear parameter-to-state map called the `model'.
We consider such Bayesian inverse problems where the discrepancy in the
parameter-to-observable map is due to the use of an approximate model that
differs from the best model, i.e. to nonzero `model error'. Multiple approaches
have been proposed to address such discrepancies, each leading to a specific
posterior. We show how to use local Lipschitz stability estimates of posteriors
with respect to likelihood perturbations to bound the Kullback--Leibler
divergence of the posterior of each approach with respect to the posterior
associated to the best model. Our bounds lead to criteria for choosing
observation operators that mitigate the effect of model error for Bayesian
inverse problems of this type. We illustrate the feasibility of one such
criterion on an advection-diffusion-reaction PDE inverse problem, and use this
example to discuss the importance and challenges of model error-aware
inference.Comment: 33 pages, 5 figure
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