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Estimating and using information in inverse problems

Wolfgang Bangerth, Chris R. Johnson, Dennis K. Njeru, Bart van Bloemen Waanders

TL;DR

This work introduces information density as a spatially varying measure of how well coefficients in inverse problems can be recovered from indirect measurements, grounded in Bayesian Fisher information and the Cramér–Rao bound. It develops finite- and infinite-dimensional formulations, defines per-cell information j_k and density j(x), and shows how these quantities can inform regularization, discretization, and experimental design. Through a two-dimensional advection–diffusion test problem, the authors demonstrate that meshes refined by information content yield better-posed discretizations and more informative reconstructions than conventional refinement criteria. The approach provides a pre-measurement, physics-based tool to allocate computational and data-collection resources efficiently, with extensions to nonlinear problems and scalable computation discussed for future work.

Abstract

In inverse problems, one attempts to infer spatially variable functions from indirect measurements of a system. To practitioners of inverse problems, the concept of "information" is familiar when discussing key questions such as which parts of the function can be inferred accurately and which cannot. For example, it is generally understood that we can identify system parameters accurately only close to detectors, or along ray paths between sources and detectors, because we have "the most information" for these places. Although referenced in many publications, the "information" that is invoked in such contexts is not a well understood and clearly defined quantity. Herein, we present a definition of information density that is based on the variance of coefficients as derived from a Bayesian reformulation of the inverse problem. We then discuss three areas in which this information density can be useful in practical algorithms for the solution of inverse problems, and illustrate the usefulness in one of these areas -- how to choose the discretization mesh for the function to be reconstructed -- using numerical experiments.

Estimating and using information in inverse problems

TL;DR

This work introduces information density as a spatially varying measure of how well coefficients in inverse problems can be recovered from indirect measurements, grounded in Bayesian Fisher information and the Cramér–Rao bound. It develops finite- and infinite-dimensional formulations, defines per-cell information j_k and density j(x), and shows how these quantities can inform regularization, discretization, and experimental design. Through a two-dimensional advection–diffusion test problem, the authors demonstrate that meshes refined by information content yield better-posed discretizations and more informative reconstructions than conventional refinement criteria. The approach provides a pre-measurement, physics-based tool to allocate computational and data-collection resources efficiently, with extensions to nonlinear problems and scalable computation discussed for future work.

Abstract

In inverse problems, one attempts to infer spatially variable functions from indirect measurements of a system. To practitioners of inverse problems, the concept of "information" is familiar when discussing key questions such as which parts of the function can be inferred accurately and which cannot. For example, it is generally understood that we can identify system parameters accurately only close to detectors, or along ray paths between sources and detectors, because we have "the most information" for these places. Although referenced in many publications, the "information" that is invoked in such contexts is not a well understood and clearly defined quantity. Herein, we present a definition of information density that is based on the variance of coefficients as derived from a Bayesian reformulation of the inverse problem. We then discuss three areas in which this information density can be useful in practical algorithms for the solution of inverse problems, and illustrate the usefulness in one of these areas -- how to choose the discretization mesh for the function to be reconstructed -- using numerical experiments.
Paper Structure (23 sections, 2 theorems, 74 equations, 6 figures)

This paper contains 23 sections, 2 theorems, 74 equations, 6 figures.

Table of Contents

  1. Introduction
  2. References to information in inverse problems in the research literature
  3. Perspectives on inverse problems
  4. A finite-dimensional, linear model problem
  5. Defining an "information content" for components of the solution vector
  6. Estimating the information content for each component of the solution vector
  7. Extension to infinite-dimensional inverse problems
  8. Definition of the inverse problem
  9. Defining the information content
  10. Defining the information density
  11. Using information densities
  12. Using information densities for regularization
  13. Using information densities to guide the discretization of an inverse problem
  14. Using information densities for experimental design
  15. Numerical examples of using information densities
  16. ...and 8 more sections

Key Result

Proposition 1

For the model problem defined above, the Fisher information matrix $I_p$ defined in eq:fisher has the following form: and where $h_k$ satisfies the equation again augmented by appropriate boundary conditions for $h_k$.

Figures (6)

  • Figure 1: Notional positioning of parameter estimation methods. Our approach -- enriching deterministic parameter estimation problems solved as a PDE-constrained problem with information obtained from the Fisher information matrix -- is shown in blue.
  • Figure 2: Left: The solution $u^\ast(\mathbf x)$ of the forward problem from which we generate "synthetic" measurements $z_\ell$ via \ref{['eq:synthetic-measurements']}. Right: The source term $q^\ast(\mathbf x)$ from which we compute synthetic measurements is constant and nonzero only in the solid red circle offset from the center; the detector locations $\xi_\ell, \ell=1,\ldots,L=100$ are marked by dots.
  • Figure 3: The solution of problem \ref{['eq:inverse-problem-system']}, computed on a very fine finite element mesh. Top left: The primal variable $u(\mathbf x)$, shown with the same scale for color and isocontours as in Fig. \ref{['fig:synthetic']}. Top right: The recovered sources $q(\mathbf x)$, i.e., the solution of the inverse problem. The red circle indicates the location of the source term used in generating the synthetic data. Bottom left: The adjoint variable $\lambda(\mathbf x)$. Bottom right: The information density $j(\mathbf x)$ associated with this problem, as defined in \ref{['eq:information-density-definition']}.
  • Figure 4: Reconstructions (top) on a sequence of meshes (bottom) refined based on the information content of each cell of the mesh.
  • Figure 5: Sequences of meshes generated by different mesh refinement criteria. Top: Mesh refinement is driven by an a posteriori error indicator. Bottom: Mesh refinement is driven by a a smoothness indicator.
  • ...and 1 more figures

Theorems & Definitions (8)

  • Remark 1
  • Remark 2
  • Remark 3
  • Proposition 1
  • Proposition 2
  • Remark 4
  • Remark 5
  • Remark 6