<p>We consider goal-oriented optimal design of experiments for infinite-dimensional Bayesian linear inverse problems governed by partial differential equations (PDEs). Specifically, we seek sensor placements that minimize the posterior variance of a prediction or goal quantity of interest. The goal quantity is assumed to be a nonlinear functional of the inversion parameter. We propose a goal-oriented optimal experimental design (OED) approach that uses a quadratic approximation of the goal-functional to define a goal-oriented design criterion. The proposed criterion, which we call the <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10915_2025_3073_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="22" /> </InlineMediaObject> <EquationSource Format="TEX">$G_{q}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>G</mi> <mi>q</mi> </msub> </math></EquationSource> </InlineEquation>-optimality criterion, is obtained by integrating the posterior variance of the quadratic approximation over the set of likely data. Under the assumption of Gaussian prior and noise models, we derive a closed-form expression for this criterion. To guide development of discretization invariant computational methods, the derivations are performed in an infinite-dimensional Hilbert space setting. Subsequently, we propose efficient and accurate computational methods for computing the <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10915_2025_3073_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="22" /> </InlineMediaObject> <EquationSource Format="TEX">$G_{q}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>G</mi> <mi>q</mi> </msub> </math></EquationSource> </InlineEquation>-optimality criterion. A greedy approach is used to obtain <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10915_2025_3073_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="22" /> </InlineMediaObject> <EquationSource Format="TEX">$G_{q}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>G</mi> <mi>q</mi> </msub> </math></EquationSource> </InlineEquation>-optimal sensor placements. We illustrate the proposed approach for two model inverse problems governed by PDEs. Our numerical results demonstrate the effectiveness of the proposed strategy. In particular, the proposed approach outperforms non-goal-oriented (A-optimal) and linearization-based (c-optimal) approaches.</p>

Goal Oriented Optimal Design of Infinite-Dimensional Bayesian Inverse Problems using Quadratic Approximations

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2025年10月6日

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J. Nicholas Neuberger,
Alen Alexanderian,
Bart van Bloemen Waanders

摘要

We consider goal-oriented optimal design of experiments for infinite-dimensional Bayesian linear inverse problems governed by partial differential equations (PDEs). Specifically, we seek sensor placements that minimize the posterior variance of a prediction or goal quantity of interest. The goal quantity is assumed to be a nonlinear functional of the inversion parameter. We propose a goal-oriented optimal experimental design (OED) approach that uses a quadratic approximation of the goal-functional to define a goal-oriented design criterion. The proposed criterion, which we call the $G_{q}$ $G_{q}$ -optimality criterion, is obtained by integrating the posterior variance of the quadratic approximation over the set of likely data. Under the assumption of Gaussian prior and noise models, we derive a closed-form expression for this criterion. To guide development of discretization invariant computational methods, the derivations are performed in an infinite-dimensional Hilbert space setting. Subsequently, we propose efficient and accurate computational methods for computing the $G_{q}$ $G_{q}$ -optimality criterion. A greedy approach is used to obtain $G_{q}$ $G_{q}$ -optimal sensor placements. We illustrate the proposed approach for two model inverse problems governed by PDEs. Our numerical results demonstrate the effectiveness of the proposed strategy. In particular, the proposed approach outperforms non-goal-oriented (A-optimal) and linearization-based (c-optimal) approaches.