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

Reduced-Basis Approximation of the Viscosity-Parametrized Incompressible Navier-Stokes Equation: Rigorous A Posteriori Error Bounds

We present a technique for the rapid and reliable prediction of linear-functional outputs of elliptic partial differential equations with affine (or approximately affine) parameter dependence. The essential components are (i) rapidly uniformly convergent global reduced-basis approximations — Galerkin projection onto a space WN spanned by solutions of the governing partial differential equation at N selected points in parameter space; (ii) a posteriori error estimation — relaxations of the residual equation that provide inexpensive yet sharp and rigorous bounds for the error in the outputs of interest; and (iii) offline/online computational procedures — stratagems which decouple the generation and projection stages of the approximation process. The operation count for the online stage — in which, given a new parameter value, we calculate the output of interest and associated error bound — depends only on N (typically very small) and the parametric complexity of the problem. In this paper we extend our methodology to the viscosity-parametrized incompressible Navier-Stokes equations. There are two critical new ingredients: first, the now-classical Brezzi-Rappaz-Raviart framework for (here, a posteriori) error analysis of approximations of nonlinear elliptic partial differential equations; and second, offline/online computational procedures for efficient calculation of the "constants" required by the Brezzi-Rappaz-Raviart theory — in particular, rigorous lower and upper bounds for the BabuÅ¡ka inf-sup stability and Sobolev "L⁴-H¹" continuity factors, respectively. Numerical results for a simple square-cavity model problem confirm the rapid convergence of the reduced-basis approximation and the good effectivity of the associated a posteriori error bounds.Singapore-MIT Alliance (SMA

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

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

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$^2$ solver

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