Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Alen Alexanderian, Ruanui Nicholson, Noemi Petra

TL;DR

The paper tackles optimal experimental design for Bayesian nonlinear inverse problems governed by PDEs with irreducible secondary parameters by integrating model uncertainty via the Bayesian approximation error (BAE) framework. It introduces an uncertainty-aware A-optimal design objective, derived from a Gaussian posterior approximation at the MAP, whose evaluation is designed to be independent of the discretization dimension. Two scalable computational strategies are developed: a Monte Carlo trace estimator and a low-rank eigenmode approach, both compatible with a binary, greedy sensor-placement scheme. The methodology is demonstrated on a three-dimensional elliptic PDE model, revealing that ignoring model uncertainty can yield inferior designs and biased inferences, while the proposed approach produces more informative sensor placements and tighter posterior uncertainty bounds.

Abstract

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.

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

TL;DR

The paper tackles optimal experimental design for Bayesian nonlinear inverse problems governed by PDEs with irreducible secondary parameters by integrating model uncertainty via the Bayesian approximation error (BAE) framework. It introduces an uncertainty-aware A-optimal design objective, derived from a Gaussian posterior approximation at the MAP, whose evaluation is designed to be independent of the discretization dimension. Two scalable computational strategies are developed: a Monte Carlo trace estimator and a low-rank eigenmode approach, both compatible with a binary, greedy sensor-placement scheme. The methodology is demonstrated on a three-dimensional elliptic PDE model, revealing that ignoring model uncertainty can yield inferior designs and biased inferences, while the proposed approach produces more informative sensor placements and tighter posterior uncertainty bounds.

Abstract

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.
Paper Structure (33 sections, 4 theorems, 60 equations, 12 figures, 1 algorithm)

This paper contains 33 sections, 4 theorems, 60 equations, 12 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related work
  3. Our approach
  4. Contributions
  5. Motivation and overview
  6. Preliminaries
  7. Linear models
  8. Incorporating model uncertainty in the inverse problem
  9. Interplay between measurement error and approximation error
  10. Connection to post-marginalization
  11. Nonlinear models
  12. Infinite-dimensional Bayesian inverse problems under uncertainty
  13. A-optimal experimental design under uncertainty
  14. Design of the Bayesian inverse problem
  15. The design criterion
  16. ...and 18 more sections

Key Result

Proposition 3.1

Consider the linearized forward model where we have suppressed the dependence of $m_{\mathup{MAP}}$ to ${\hbox{\boldmath${y}$}}$ for notational convenience. Define the data model where ${\hbox{\boldmath${\nu}$}} \sim \mathcal{N}\!\left( {{\hbox{\boldmath${\varepsilon}$}}_0}, {\mathbf{{\Gamma}}_{\mathup{\nu}}}\right)$. Consider the Bayesian inverse problem of estimating $m$ using equ:linear_data_

Figures (12)

  • Figure 1: Sketch of the physical domain $\Omega$ and location of candidate sensor locations (blue circles).
  • Figure 2: Samples of the primary uncertain parameter $m$.
  • Figure 3: Samples of the secondary uncertain parameter $\xi$.
  • Figure 4: The effect of the secondary uncertain parameter $\xi$ on the state $u$ on top surface for the (fixed) true primary parameter $m$ and four different realizations of $\xi$.
  • Figure 5: Left: the mean of the approximation error at the candidate sensor locations; right: the marginal standard deviation of the approximation error at these locations (i.e., the square root of diagonal entries of $\widehat{\mathbf{{\Gamma}}}_{\mathup{\varepsilon}}$ given in \ref{['equ:BAE_comp']}).
  • ...and 7 more figures

Theorems & Definitions (8)

  • Proposition 3.1
  • proof
  • Proposition 5.1
  • proof
  • Proposition 5.2
  • proof
  • Proposition 5.3
  • proof