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Goal oriented optimal design of infinite-dimensional Bayesian inverse problems using quadratic approximations

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

TL;DR

A goal-oriented optimal experimental design (OED) approach that uses a quadratic approximation of the goal-functional to define a goal-oriented design criterion that outperforms non-goal-oriented (A-optimal) and linearization-based (c-optimal) approaches.

Abstract

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 Gq-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 Gq-optimality criterion. A greedy approach is used to obtain Gq-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.

Goal oriented optimal design of infinite-dimensional Bayesian inverse problems using quadratic approximations

TL;DR

A goal-oriented optimal experimental design (OED) approach that uses a quadratic approximation of the goal-functional to define a goal-oriented design criterion that outperforms non-goal-oriented (A-optimal) and linearization-based (c-optimal) approaches.

Abstract

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 Gq-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 Gq-optimality criterion. A greedy approach is used to obtain Gq-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.

Paper Structure

This paper contains 25 sections, 7 theorems, 109 equations, 15 figures, 1 table, 3 algorithms.

Table of Contents

  1. Introduction
  2. Background
  3. Bayesian linear inverse problems
  4. Classical optimal experimental design
  5. Goal-oriented OED
  6. Computational Methods
  7. A randomized algorithm
  8. Algorithm based on low-rank spectral decomposition of $\tilde{\mathbf{H}}_{\text{mis}}$
  9. An approach based on low-rank SVD of $\mathbf{F}$
  10. Computational cost
  11. Computational experiments
  12. Model problem with a quadratic goal functional
  13. Model and goal
  14. The inverse problem
  15. Optimal design and uncertainty
  16. ...and 10 more sections

Key Result

theorem 1

Let $\mathcal{A}$ be a bounded selfadjoint linear operator on a Hilbert space $\mathscr{M}$ and let $b \in \mathscr{M}$. Consider the quadratic functional $\mathcal{Z}:\mathscr{M}\to\mathbb{R}$ given by Let $\mu = \mathsf{N}(m_{0}, \mathcal{C})$ be a Gaussian measure on $\mathscr{M}$. Then, we have

Figures (15)

  • Figure 1: The true inversion parameter $m_{\text{true}}$ (left) and corresponding state solution $p(m_{\text{true}})$ (right). The subdomain $\Omega^{*}$ (black rectangles) is depicted in the left figure.
  • Figure 2: MAP point (leftmost) and three posterior samples. The posterior is obtained using data collected from the entire set of $N_{s}$ candidate sensor locations.
  • Figure 3: Classical (A-optimal) and goal-oriented ($G_{q}$-optimal) designs of size $k\in\{5, 10, 15, 20\}$ plotted over the true state solution. The subdomain $\Omega^{*}$ (black rectangles) are plotted over the goal-oriented plots.
  • Figure 4: Bottom row: classical (A-optimal) and goal-oriented ($G_{q}$-optimal) designs of size $k=5$ plotted over the respective posterior standard deviation fields. Top row: posterior goal-densities constructed by propagating posterior samples through $Z$. The dashed line is the true goal value.
  • Figure 5: Goal-densities for classical and goal-oriented approaches and $k \in\{ 3, 4,\cdots, 20\}$. The dashed line is $q(\boldsymbol{m_{\text{true}}})$.
  • ...and 10 more figures

Theorems & Definitions (17)

  • theorem 1: Variance of a quadratic functional
  • proof
  • theorem 2: Goal-oriented criterion
  • proof
  • proposition 1
  • proof
  • proposition 2
  • proof
  • lemma 1
  • lemma 2
  • ...and 7 more