Small-noise approximation for Bayesian optimal experimental design with nuisance uncertainty

Bartuska, Arved; Espath, Luis; Tempone, Raúl

doi:10.1016/j.cma.2022.115320

Cited by 7 publications

(6 citation statements)

References 28 publications

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“…[6,14,24,27,33,38] for a small sample of the literature addressing inverse problems under uncertainty. Methods for OED in such inverse problems have been studied in [5,11,18,28]. The works [5,28] concern optimal design of infinite-dimensional Bayesian linear inverse problems governed by PDEs.…”

Section: Related Workmentioning

confidence: 99%

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Optimal design of large-scale nonlinear Bayesian inverse problems under model uncertainty

Alexanderian,

Nicholson,

Petra

2024

Inverse Problems

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We consider optimal experimental design (OED) for Bayesian nonlinear inverse problems governed by partial diﬀerential equations (PDEs) under model uncertainty. Speciﬁcally, we consider inverse problems in which, in addition to the inversion parameters, the governing PDEs include secondary uncertain parameters. We focus on problems with inﬁnite-dimensional inversion and secondary parameters and present a scalable computational framework for optimal design of such problems. The proposed approach enables Bayesian inversion and OED under uncertainty within a uniﬁed framework. We build on the Bayesian approximation error (BAE) approach, to incorporate modeling uncertainties in the Bayesian inverse problem, and methods for A-optimal design of inﬁnite-dimensional Bayesian nonlinear inverse problems. Speciﬁcally, a Gaussian approximation to the posterior at the maximum a posteriori probability point is used to deﬁne an uncertainty aware OED objective that is tractable to evaluate and optimize. In particular, the OED objective can be computed at a cost, in the number of PDE solves, that does not grow with the dimension of the discretized inversion and secondary parameters. The OED problem is formulated as a binary bilevel PDE constrained optimization problem and a greedy algorithm, which provides a pragmatic approach, is used to ﬁnd optimal designs. We demonstrate the eﬀectiveness of the proposed approach for a model inverse problem governed by an elliptic PDE on a three-dimensional domain. Our computational results also highlight the pitfalls of ignoring modeling uncertainties in the OED and/or inference stages.

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Section: Related Workmentioning

confidence: 99%

“…On the other hand, [5] targets OED for linear inverse problems with reducible sources of uncertainty. The efforts [11,18] focus on inverse problems with finite-and low-dimensional inversion and secondary parameters. These articles devise sampling approaches for estimating the expected information gain in such problems.…”

Section: Related Workmentioning

confidence: 99%

Optimal design of large-scale nonlinear Bayesian inverse problems under model uncertainty

Alexanderian,

Nicholson,

Petra

2024

Inverse Problems

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“…There has also been an increased interest in parameter inversion and design of experiments in systems governed by uncertain forward models; see e.g., [6,14,24,26,32,34] for a small sample of the literature addressing inverse problems under uncertainty. Methods for OED in such inverse problems have been studied in [5,11,18,27]. The works [5,27] concern optimal design of infinite-dimensional Bayesian linear inverse problems governed by PDEs.…”

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confidence: 99%

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Optimal Design of Large-scale Bayesian Linear Inverse Problems Under Reducible Model Uncertainty: Good to Know What You Don't Know

Alexanderian¹,

Petra²,

Stadler³

et al. 2021

SIAM/ASA J. Uncertainty Quantification

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We consider optimal experimental design (OED) for Bayesian nonlinear inverse problems governed by partial differential equations (PDEs) under model uncertainty. Specifically, we consider inverse problems in which, in addition to the inversion parameters, the governing PDEs include secondary uncertain parameters. We focus on problems with infinite-dimensional inversion and secondary parameters and present a scalable computational framework for optimal design of such problems. The proposed approach enables Bayesian inversion and OED under uncertainty within a unfied framework. We build on the Bayesian approximation error (BAE) framework, to incorporate modeling uncertainties in the Bayesian inverse problem, and methods for A-optimal design of infinite-dimensional Bayesian nonlinear inverse problems. Specifically, a Gaussian approximation to the posterior at the maximum a posteriori probability point is used to define an uncertainty aware OED objective that is tractable to evaluate and optimize. In particular, the OED objective can be computed at a cost, in the number of PDE solves, that does not grow with the dimension of the discretized inversion and secondary parameters. The OED problem is formulated as a binary bilevel PDE constrained optimization problem and a greedy algorithm, which provides a pragmatic approach, is used to find optimal designs. We demonstrate the effectiveness of the proposed approach for a model inverse problem governed by an elliptic PDE on a three-dimensional domain. Our computational results also highlight the pitfalls of ignoring modeling uncertainties in the OED and/or inference stages.

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Laplace-based strategies for Bayesian optimal experimental design with nuisance uncertainty

Bartuska,

Espath,

Tempone

2024

Stat Comput

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Finding the optimal design of experiments in the Bayesian setting typically requires estimation and optimization of the expected information gain functional. This functional consists of one outer and one inner integral, separated by the logarithm function applied to the inner integral. When the mathematical model of the experiment contains uncertainty about the parameters of interest and nuisance uncertainty, (i.e., uncertainty about parameters that affect the model but are not themselves of interest to the experimenter), two inner integrals must be estimated. Thus, the already considerable computational effort required to determine good approximations of the expected information gain is increased further. The Laplace approximation has been applied successfully in the context of experimental design in various ways, and we propose two novel estimators featuring the Laplace approximation to alleviate the computational burden of both inner integrals considerably. The first estimator applies Laplace’s method followed by a Laplace approximation, introducing a bias. The second estimator uses two Laplace approximations as importance sampling measures for Monte Carlo approximations of the inner integrals. Both estimators use Monte Carlo approximation for the remaining outer integral estimation. We provide four numerical examples demonstrating the applicability and effectiveness of our proposed estimators.

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Small-noise approximation for Bayesian optimal experimental design with nuisance uncertainty

Cited by 7 publications

References 28 publications

Optimal design of large-scale nonlinear Bayesian inverse problems under model uncertainty

Optimal design of large-scale nonlinear Bayesian inverse problems under model uncertainty

Optimal Design of Large-scale Bayesian Linear Inverse Problems Under Reducible Model Uncertainty: Good to Know What You Don't Know

Laplace-based strategies for Bayesian optimal experimental design with nuisance uncertainty

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