Goal-Oriented Inference: Approach, Linear Theory, and Application to Advection Diffusion

Lieberman, Chad; Willcox, Karen

doi:10.1137/110857763

Cited by 15 publications

(25 citation statements)

References 20 publications

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“…Diffusive processes (e.g., the heat equation) tend to be associated with rapid singular value decay, while convection-dominated problems are not. However, the particular inputs and outputs under consideration play a key role-even in the presence of strong convection, an output that corresponds to an integrated quantity (e.g., an average solution over the domain) or a highly localized quantity (e.g., the solution at a particular spatial location) can be characterized by a low-dimensional input-output map [155].…”

Section: Parametric Model Reduction and Surrogate Modelingmentioning

confidence: 99%

A Survey of Projection-Based Model Reduction Methods for Parametric Dynamical Systems

2015

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Abstract. Numerical simulation of large-scale dynamical systems plays a fundamental role in studying a wide range of complex physical phenomena; however, the inherent large-scale nature of the models often leads to unmanageable demands on computational resources. Model reduction aims to reduce this computational burden by generating reduced models that are faster and cheaper to simulate, yet accurately represent the original large-scale system behavior. Model reduction of linear, nonparametric dynamical systems has reached a considerable level of maturity, as reflected by several survey papers and books. However, parametric model reduction has emerged only more recently as an important and vibrant research area, with several recent advances making a survey paper timely. Thus, this paper aims to provide a resource that draws together recent contributions in different communities to survey the state of the art in parametric model reduction methods. Parametric model reduction targets the broad class of problems for which the equations governing the system behavior depend on a set of parameters. Examples include parameterized partial differential equations and large-scale systems of parameterized ordinary differential equations. The goal of parametric model reduction is to generate low-cost but accurate models that characterize system response for different values of the parameters. This paper surveys state-of-the-art methods in projection-based parametric model reduction, describing the different approaches within each class of methods for handling parametric variation and providing a comparative discussion that lends insights to potential advantages and disadvantages in applying each of the methods. We highlight the important role played by parametric model reduction in design, control, optimization, and uncertainty quantification-settings that require repeated model evaluations over different parameter values.

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Section: Parametric Model Reduction and Surrogate Modelingmentioning

confidence: 99%

A Survey of Projection-Based Model Reduction Methods for Parametric Dynamical Systems

2015

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“…Thus, combining (31), (32), along with (30), we get E µpr E y|θ ρ y post − ρ 2 = tr PΓ post (H misfit + Γ −1 pr )Γ post P * = tr(PΓ post P * ).…”

Section: Appendix: Proofs and Derivationsmentioning

confidence: 92%

Goal-oriented optimal design of experiments for large-scale Bayesian linear inverse problems

2018

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We develop a framework for goal-oriented optimal design of experiments (GOODE) for largescale Bayesian linear inverse problems governed by PDEs. This framework differs from classical Bayesian optimal design of experiments (ODE) in the following sense: we seek experimental designs that minimize the posterior uncertainty in the experiment end-goal, e.g., a quantity of interest (QoI), rather than the estimated parameter itself. This is suitable for scenarios in which the solution of an inverse problem is an intermediate step and the estimated parameter is then used to compute a QoI. In such problems, a GOODE approach has two benefits: the designs can avoid wastage of experimental resources by a targeted collection of data, and the resulting design criteria are computationally easier to evaluate due to the often low-dimensionality of the QoIs. We present two modified design criteria, A-GOODE and D-GOODE, which are natural analogues of classical Bayesian A-and D-optimal criteria. We analyze the connections to other ODE criteria, and provide interpretations for the GOODE criteria by using tools from information theory. Then, we develop an efficient gradient-based optimization framework for solving the GOODE optimization problems. Additionally, we present comprehensive numerical experiments testing the various aspects of the presented approach. The driving application is the optimal placement of sensors to identify the source of contaminants in a diffusion and transport problem. We enforce sparsity of the sensor placements using an 1 -norm penalty approach, and propose a practical strategy for specifying the associated penalty parameter.

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“…Finding this direction involves identifying parameter modes that are both informed by the observational data and also required for estimating the quantity of interest. We adopt the inference-forprediction (IFP) algorithm presented more generally for multiple quantities of interest in [52]. Denoting the Hessian square root by := −1/2 (β MAP ) and using the linearized parameter-to-prediction map (β MAP ), we compute the eigendecomposition ΨΣ 2 Ψ * of * * (β MAP ) (β MAP ) .…”

Section: Prediction With Quantified Uncertainty: Forward Propagation mentioning

confidence: 99%

“…Since this operator has rank 1, the eigenvector Ψ corresponding to the only nonzero eigenvalue is given by * * (β MAP ), and the corresponding eigenvalue is Σ 2 = * * (β MAP ) 2 . Following [52], the influential direction for prediction (based on linearizations of the parameter-to-observable map and the parameter-to-prediction map) is given by = ΨΣ −1/2 .…”

Section: Prediction With Quantified Uncertainty: Forward Propagation mentioning

confidence: 99%

“…On the other hand, in the center and right figures in the bottom row, focusing on East Antarctica, where the posterior samples suggest lower uncertainties, the modes show mixed uncertainty and sensitivity influence. Figure 10: The (logarithm of the) gradient of the prediction quantity Q with respect to the uncertain basal sliding parameter β (top row) and the directions that jointly maximize uncertainty and sensitivity for the prediction (bottom row) following [52]. For visualization purposes, in the top row we plot ln(| (β MAP )| + 10 −10 ).…”

Section: Prediction With Quantified Uncertainty: Forward Propagation mentioning

confidence: 99%

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Scalable and efficient algorithms for the propagation of uncertainty from data through inference to prediction for large-scale problems, with application to flow of the Antarctic ice sheet

Isaac

Petra

Stadler

et al. 2015

Journal of Computational Physics

161

216

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The majority of research on efficient and scalable algorithms in computational science and engineering has focused on the forward problem: given parameter inputs, solve the governing equations to determine output quantities of interest. In contrast, here we consider the broader question: given a (large-scale) model containing uncertain parameters, (possibly) noisy observational data, and a prediction quantity of interest, how do we construct efficient and scalable algorithms to (1) infer the model parameters from the data (the deterministic inverse problem), (2) quantify the uncertainty in the inferred parameters (the Bayesian inference problem), and (3) propagate the resulting uncertain parameters through the model to issue predictions with quantified uncertainties (the forward uncertainty propagation problem)?We present efficient and scalable algorithms for this end-to-end, data-to-prediction process under the Gaussian approximation and in the context of modeling the flow of the Antarctic ice sheet and its effect on loss of grounded ice to the ocean. The ice is modeled as a viscous, incompressible, creeping, shear-thinning fluid. The observational data come from satellite measurements of surface ice flow velocity, and the uncertain parameter field to be inferred is the basal sliding parameter, represented by a heterogeneous coefficient in a Robin boundary condition at the base of the ice sheet. The prediction quantity of interest is the present-day ice mass flux from the Antarctic continent to the ocean.We show that the work required for executing this data-to-prediction process-measured in number of forward (and adjoint) ice sheet model solves-is independent of the state dimension, parameter dimension, data dimension, and the number of processor cores. The key to achieving this dimension independence is to exploit the fact that, despite their large size, the observational data typically provide only sparse information on model parameters. This property can be exploited to construct a low rank approximation of the linearized parameter-toobservable map via randomized SVD methods and adjoint-based actions of Hessians of the data misfit functional.

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Goal-Oriented Inference: Approach, Linear Theory, and Application to Advection Diffusion

Cited by 15 publications

References 20 publications

A Survey of Projection-Based Model Reduction Methods for Parametric Dynamical Systems

A Survey of Projection-Based Model Reduction Methods for Parametric Dynamical Systems

Goal-oriented optimal design of experiments for large-scale Bayesian linear inverse problems

Scalable and efficient algorithms for the propagation of uncertainty from data through inference to prediction for large-scale problems, with application to flow of the Antarctic ice sheet

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