Search

The replacement of a nonlinear parameter-to-observable mapping with a linear (affine) approximation is often carried out to reduce the computational costs associated with solving large-scale inverse problems governed by partial differential equations (PDEs). In the case of a linear parameter-to-observable mapping with normally distributed additive noise and a Gaussian prior measure on the parameters, the posterior is Gaussian. However, substituting an accurate model for a (possibly well justified) linear surrogate model can give misleading results if the induced model approximation error is not accounted for. To account for the errors, the Bayesian approximation error (BAE) approach can be utilised, in which the first and second order statistics of the errors are computed via sampling. The most common linear approximation is carried out via linear Taylor expansion, which requires the computation of (Fréchet) derivatives of the parameter-to-observable mapping with respect to the parameters of interest. In this paper, we prove that the (approximate) posterior measure obtained by replacing the nonlinear parameter-to-observable mapping with a linear approximation is in fact independent of the choice of the linear approximation when the BAE approach is employed. Thus, somewhat non-intuitively, employing the zero-model as the linear approximation gives the same approximate posterior as any other choice of linear approximations of the parameter-to-observable model. The independence of the linear approximation is demonstrated mathematically and illustrated with two numerical PDE-based problems: an inverse scattering type problem and an inverse conductivity type problem. [ABSTRACT FROM AUTHOR]

Uncertainty quantification of groundwater (GW) aquifer parameters is critical for efficient management and sustainable extraction of GW resources. These uncertainties are introduced by the data, model, and prior information on the parameters. Here, we develop a Bayesian inversion framework that uses Interferometric Synthetic Aperture Radar (InSAR) surface deformation data to infer the laterally heterogeneous permeability of a transient linear poroelastic model of a confined GW aquifer. The Bayesian solution of this inverse problem takes the form of a posterior probability density of the permeability. Exploring this posterior using classical Markov chain Monte Carlo (MCMC) methods is computationally prohibitive due to the large dimension of the discretized permeability field and the expense of solving the poroelastic forward problem. However, in many partial differential equation (PDE)‐based Bayesian inversion problems, the data are only informative in a few directions in parameter space. For the poroelasticity problem, we prove this property theoretically for a one‐dimensional problem and demonstrate it numerically for a three‐dimensional aquifer model. We design a generalized preconditioned Crank‐Nicolson (gpCN) MCMC method that exploits this intrinsic low dimensionality by using a low‐rank‐based Laplace approximation of the posterior as a proposal, which we build scalably. The feasibility of our approach is demonstrated through a real GW aquifer test in Nevada. The inherently two‐dimensional nature of InSAR surface deformation data informs a sufficient number of modes of the permeability field to allow detection of major structures within the aquifer, significantly reducing the uncertainty in the pressure and the displacement quantities of interest. Key Points: Using Interferometric Synthetic Aperture Radar data reduces the uncertainty in selected quantities of interest compared to using prior knowledge onlyThe preconditioned Crank‐Nicolson (pCN) Markov chain Monte Carlo (MCMC) method is extended to exploit posterior curvature and allow better chain mixingWe demonstrate the intrinsic low dimensionality of the poroelastic inverse problem that is, critical for the success of the MCMC method [ABSTRACT FROM AUTHOR]

In this work we develop a new hierarchical multilevel approach to generate Gaussian random field realizations in an algorithmically scalable manner that is well suited to incorporating into multilevel Markov chain Monte Carlo (MCMC) algorithms. This approach builds off of other partial differential equation (PDE) approaches for generating Gaussian random field realizations; in particular, a single field realization may be formed by solving a reaction-diffusion PDE with a spatial white noise source function as the right-hand side. While these approaches have been explored to accelerate forward uncertainty quantification tasks, e.g., multilevel Monte Carlo, the previous constructions are not directly applicable to multilevel MCMC frameworks which build finescale random fields in a hierarchical fashion from coarse-scale random fields. Our new hierarchical multilevel method relies on a hierarchical decomposition of the white noise source function in L² which allows us to form Gaussian random field realizations across multiple levels of discretization in a way that fits into multilevel MCMC algorithmic frameworks. After presenting our main theoretical results and numerical scaling results to showcase the utility of this new hierarchical PDE method for generating Gaussian random field realizations, this method is tested on a four-level MCMC algorithm to explore its feasibility. [ABSTRACT FROM AUTHOR]

We present an extensible software framework, hIPPYlib, for solution of large-scale deterministic and Bayesian inverse problems governed by partial differential equations (PDEs) with (possibly) infinite-dimensional parameter fields (which are high-dimensional after discretization). hIPPYlib overcomes the prohibitively expensive nature of Bayesian inversion for this class of problems by implementing state-of-the-art scalable algorithms for PDE-based inverse problems that exploit the structure of the underlying operators, notably the Hessian of the log-posterior. The key property of the algorithms implemented in hIPPYlib is that the solution of the inverse problem is computed at a cost, measured in linearized forward PDE solves, that is independent of the parameter dimension. The mean of the posterior is approximated by the MAP point, which is found by minimizing the negative log-posterior with an inexact matrix-free Newton-CG method. The posterior covariance is approximated by the inverse of the Hessian of the negative log posterior evaluated at the MAP point. The construction of the posterior covariance is made tractable by invoking a low-rank approximation of the Hessian of the log-likelihood. Scalable tools for sample generation are also discussed. hIPPYlib makes all of these advanced algorithms easily accessible to domain scientists and provides an environment that expedites the development of new algorithms. [ABSTRACT FROM AUTHOR]

We present a scalable method for PDE‐constrained optimization under high‐dimensional uncertainty. The method is based on a functional Taylor approximation of the cost functional with respect to (w.r.t.) the random parameter field. We consider a risk‐averse optimization formulation via a mean‐variance measure for the control objective. A quadratic Taylor approximation of the control objective gives rise to a function involving the trace of the Hessian of the objective w.r.t. the random parameter, preconditioned by the covariance operator. We apply a randomized SVD algorithm to estimate the trace, which is demonstrated to be scalable w.r.t. the parameter dimension, i.e., the number of PDE solves is independent of the parameter dimension and depends only on the decay of the eigenvalues of the covariance‐preconditioned Hessian (which is typically dimension‐independent). We solve the resulting generalized eigenproblem‐constrained optimization problem by a quasi‐Newton method. We apply our method to a RANS turbulence model with an infinite‐dimensional random field representing model inadequacy, and demonstrate scalability on discretized problems with parameter dimensions up to one million. [ABSTRACT FROM AUTHOR]

Many-query problems – arising from, e.g., uncertainty quantification, Bayesian inversion, Bayesian optimal experimental design, and optimization under uncertainty – require numerous evaluations of a parameter-to-output map. These evaluations become prohibitive if this parametric map is high-dimensional and involves expensive solution of partial differential equations (PDEs). To tackle this challenge, we propose to construct surrogates for high-dimensional PDE-governed parametric maps in the form of derivative-informed projected neural networks (DIPNets) that parsimoniously capture the geometry and intrinsic low-dimensionality of these maps. Specifically, we compute Jacobians of these PDE-based maps, and project the high-dimensional parameters onto a low-dimensional derivative-informed active subspace; we also project the possibly high-dimensional outputs onto their principal subspace. This exploits the fact that many high-dimensional PDE-governed parametric maps can be well-approximated in low-dimensional parameter and output subspaces. We use the projection basis vectors in the active subspace as well as the principal output subspace to construct the weights for the first and last layers of the neural network, respectively. This frees us to train the weights in only the low-dimensional layers of the neural network. The architecture of the resulting neural network then captures, to first order, the low-dimensional structure and geometry of the parametric map. We demonstrate that the proposed projected neural network achieves greater generalization accuracy than a full neural network, especially in the limited training data regime afforded by expensive PDE-based parametric maps. Moreover, we show that the number of degrees of freedom of the inner layers of the projected network is independent of the parameter and output dimensions, and high accuracy can be achieved with weight dimension independent of the discretization dimension. • We present neural networks that preserve low dimensional geometry of parametric maps. • These derivative informed networks (DIPNets) generalize well for few training data. • DIPNets outperform conventional strategies, including other projected networks. • Useful for UQ where repeated evaluation of expensive PDE-based maps is bottleneck. [ABSTRACT FROM AUTHOR]

• Multifidelity method that enables estimation of failure probabilities for expensive-to- evaluate models. • For turbulent jet application, it reduces the CPU time to compute the biasing density by 65%. • The new approach draws from information fusion, importance sampling and multifidelity modeling. • The fused estimator is optimal in that it has minimal variance among all possible combinations of the estimators. This paper develops a multifidelity method that enables estimation of failure probabilities for expensive-to-evaluate models via information fusion and importance sampling. The presented general fusion method combines multiple probability estimators with the goal of variance reduction. We use low-fidelity models to derive biasing densities for importance sampling and then fuse the importance sampling estimators such that the fused multifidelity estimator is unbiased and has mean-squared error lower than or equal to that of any of the importance sampling estimators alone. By fusing all available estimators, the method circumvents the challenging problem of selecting the best biasing density and using only that density for sampling. A rigorous analysis shows that the fused estimator is optimal in the sense that it has minimal variance amongst all possible combinations of the estimators. The asymptotic behavior of the proposed method is demonstrated on a convection-diffusion-reaction partial differential equation model for which 105 samples can be afforded. To illustrate the proposed method at scale, we consider a model of a free plane jet and quantify how uncertainties at the flow inlet propagate to a quantity of interest related to turbulent mixing. Compared to an importance sampling estimator that uses the high-fidelity model alone, our multifidelity estimator reduces the required CPU time by 65% while achieving a similar coefficient of variation. [ABSTRACT FROM AUTHOR]