Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Local sensitivity analysis for Bayesian inverse problems

Jürgen Dölz, David Ebert

TL;DR

This work extends local sensitivity analysis to Bayesian inverse problems by developing Taylor-type perturbation expansions for posterior moments with respect to perturbations in the input data and forward map. It derives first- and second-order expansions for posterior means, covariances, and correlations, and shows how these can be iteratively improved by refining the reference point, with connections to Tikhonov regularization. The method is formulated in infinite-dimensional Banach spaces and implemented via affine-parametric representations, enabling efficient computation in PDE and ODE inverse problems. Numerical experiments on Darcy flow and Lotka–Volterra models validate the theoretical rates and demonstrate substantial computational advantages over sampling-based references when small-to-moderate uncertainties prevail.

Abstract

We present an extension of local sensitivity analysis, also referred to as the perturbation approach for uncertainty quantification, to Bayesian inverse problems. More precisely, we show how moments of random variables with respect to the posterior distribution can be approximated efficiently by asymptotic expansions. This is under the assumption that the measurement operators and prediction functions are sufficiently smooth and their corresponding stochastic moments with respect to the prior distribution exist. Numerical experiments are presented to the illustrate the theoretical results.

Local sensitivity analysis for Bayesian inverse problems

TL;DR

This work extends local sensitivity analysis to Bayesian inverse problems by developing Taylor-type perturbation expansions for posterior moments with respect to perturbations in the input data and forward map. It derives first- and second-order expansions for posterior means, covariances, and correlations, and shows how these can be iteratively improved by refining the reference point, with connections to Tikhonov regularization. The method is formulated in infinite-dimensional Banach spaces and implemented via affine-parametric representations, enabling efficient computation in PDE and ODE inverse problems. Numerical experiments on Darcy flow and Lotka–Volterra models validate the theoretical rates and demonstrate substantial computational advantages over sampling-based references when small-to-moderate uncertainties prevail.

Abstract

We present an extension of local sensitivity analysis, also referred to as the perturbation approach for uncertainty quantification, to Bayesian inverse problems. More precisely, we show how moments of random variables with respect to the posterior distribution can be approximated efficiently by asymptotic expansions. This is under the assumption that the measurement operators and prediction functions are sufficiently smooth and their corresponding stochastic moments with respect to the prior distribution exist. Numerical experiments are presented to the illustrate the theoretical results.

Paper Structure

This paper contains 32 sections, 7 theorems, 69 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Motivation
  3. Numerical methods for Bayesian inverse problems
  4. Local sensitivity analysis for forward problems
  5. Contributions
  6. Outline
  7. Preliminaries
  8. Bayesian inverse problems
  9. Stochastic moments
  10. Taylor expansions in Banach spaces
  11. Local sensitivity analysis for Bayesian inverse problems
  12. Perturbation model
  13. Expansions for first and second moments
  14. Comparison of the Perturbation Approximation to the Laplace Approximation
  15. Iterative improvement of the reference value
  16. ...and 17 more sections

Key Result

Theorem 2.1

Assume that the potential eq:potential is $\pi_{\xi}$-measur-able. Then the posterior distribution of $\xi$ conditioned to $\eta^\delta$ exists and is denoted by ${\pi_{\xi}^\delta} = \pi_{\xi|\eta=\eta^\delta}$. It is absolutely continuous with respect to $\pi_{\xi}$ and given through the Radon--Ni

Figures (10)

  • Figure 1: Illustration of the mapping properties considered for computing stochastic moments with respect to the posterior distribution.
  • Figure 2: Darcy model: Top left: reference solution. Top right: realization of random field. Bottom left: perturbed solution according to random field top right. Bottom right: Derivative in direction of random field top right. Evaluation points highlighted.
  • Figure 3: Convergence for Darcy model and uncentered prior distributions.
  • Figure 4: Convergence for Darcy model and centered prior distributions.
  • Figure 5: $\|\cdot\|_{L^2(\mathcal{D})}$-norm of updates $d^{(n)}$ for each $\alpha$ for $100$ iterations for Darcy model.
  • ...and 5 more figures

Theorems & Definitions (14)

  • Theorem 2.1: Stu2010
  • Remark 2.2
  • Lemma 2.3
  • Remark 3.1
  • Theorem 3.2
  • proof
  • Corollary 3.3
  • proof
  • Corollary 4.1
  • proof
  • ...and 4 more