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Iterative Data-Consistent Inversion with Multiple Push-forward Constraints

Tianyi Jiang, Troy Butler, Timothy Wildey, Tim Kutta, Haonan Wang

TL;DR

This work rigorously establishes the theoretical optimality of the DCI solution to the standard problem, proving that it minimizes the cumulative divergence over the space of all possible pullback measures that satisfy the push-forward constraint, and proposes an iterative solution based on an iterative application of Data-Consistent Inversion.

Abstract

A foundational challenge in uncertainty quantification involves estimating a probability measure on the space of uncertain parameters such that its push-forward through a computational model matches an observed probability measure on the output data associated with quantities of interest (QoI). When multiple, distinct sets of observational data are available, the desired parameter measure should simultaneously satisfy multiple push-forward constraints associated with various subsets of the QoI. In this work, we present a convergent measure-theoretic framework for solving this problem based on an iterative application of Data-Consistent Inversion (DCI). We first rigorously establish the theoretical optimality of the DCI solution to the standard problem, proving that it minimizes the $f$-divergence over the space of all possible pullback measures that satisfy the push-forward constraint. This optimality property provides the foundation for our iterative DCI scheme, which is shown to converge to a solution of the multiple push-forward constraint problem. This iterative solution minimizes the cumulative $f$-divergence across all constraints and, under uniform initializations, represents the maximal entropy solution (the I-projection) onto the intersection of the solution sets. We provide a rigorous convergence analysis for the proposed method and demonstrate its practical utility through numerical examples, including a high-dimensional parameter space governed by partial differential equations, where the iterative approach robustly avoids the complexities associated with approximating high-dimensional joint observed measures.

Iterative Data-Consistent Inversion with Multiple Push-forward Constraints

TL;DR

This work rigorously establishes the theoretical optimality of the DCI solution to the standard problem, proving that it minimizes the cumulative divergence over the space of all possible pullback measures that satisfy the push-forward constraint, and proposes an iterative solution based on an iterative application of Data-Consistent Inversion.

Abstract

A foundational challenge in uncertainty quantification involves estimating a probability measure on the space of uncertain parameters such that its push-forward through a computational model matches an observed probability measure on the output data associated with quantities of interest (QoI). When multiple, distinct sets of observational data are available, the desired parameter measure should simultaneously satisfy multiple push-forward constraints associated with various subsets of the QoI. In this work, we present a convergent measure-theoretic framework for solving this problem based on an iterative application of Data-Consistent Inversion (DCI). We first rigorously establish the theoretical optimality of the DCI solution to the standard problem, proving that it minimizes the -divergence over the space of all possible pullback measures that satisfy the push-forward constraint. This optimality property provides the foundation for our iterative DCI scheme, which is shown to converge to a solution of the multiple push-forward constraint problem. This iterative solution minimizes the cumulative -divergence across all constraints and, under uniform initializations, represents the maximal entropy solution (the I-projection) onto the intersection of the solution sets. We provide a rigorous convergence analysis for the proposed method and demonstrate its practical utility through numerical examples, including a high-dimensional parameter space governed by partial differential equations, where the iterative approach robustly avoids the complexities associated with approximating high-dimensional joint observed measures.
Paper Structure (18 sections, 5 theorems, 57 equations, 17 figures, 1 algorithm)

This paper contains 18 sections, 5 theorems, 57 equations, 17 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Data-Consistent Inversion (DCI): Background, Notation, and Terminology
  3. DCI Solutions Minimize f-Divergences
  4. The Generalized Stochastic Inverse Problem
  5. An Iterative DCI Approach
  6. Numerical Examples
  7. Linear QoI Maps
  8. Numerical Setup
  9. Iterative Results
  10. Joint DCI Results
  11. Increasing the QoI Dimension
  12. A Higher-dimensional PDE-based Example
  13. Numerical Setup
  14. Iterative Results
  15. Iterative Result with Joint Pressure & Velocity Data
  16. ...and 3 more sections

Key Result

Lemma 3.1

Let $P_{\text{obs}}$ denote a probability measure on $(\mathcal{D}, \mathcal{B}_{\mathcal{D}})$ and $P_{\text{init}}$ denote any measure on $(\Lambda,\mathcal{B}_{\Lambda})$ such that $P_{\text{obs}}$ is absolutely continuous with respect to $P_\text{pred}=P_{\text{init}}\circ \phi^{-1}$. Let $\math and, if $P_{\text{init}} \ll P_{\text{up}}$, then where $\tilde{\mathbb{P}}$ is the subspace of $\

Figures (17)

  • Figure 1: Schematic that conceptualizes the iterative DCI process when $k = 3$ and $\mathbb{P}=\mathbb{P}_1\cap\mathbb{P}_2\cap\mathbb{P}_3$.
  • Figure 2: (Left) Uniform initial samples (blue circles) and data-generating samples (orange pluses) on $\Lambda=[0,1]^2\subset\mathbb{R}^2$. (Right) The predicted (blue circles) and observed (orange pluses) samples obtained by evaluating the QoI maps (eqs. \ref{['eq:linear_Q1']} and \ref{['eq:linear_Q2']}) on the initial and data-generating samples.
  • Figure 3: The KDE estimates of densities associated with Figure \ref{['fig:ex1_param_and_QoI_samples']} after the first epoch of iterations. (Top Row): The updated density obtained after using the $Q_1$ map in the first iteration of this epoch (left), after using the $Q_2$ map in the second iteration of this epoch (middle), and the data-generating density estimate (right). (Bottom Row): The marginal QoI densities for $Q_1$ (left) and $Q_2$ (right) associated with the push-forward of the updated density (blue dashed curves) and observed data (orange solid curves).
  • Figure 4: The KDE estimates of densities associated with Figure \ref{['fig:ex1_param_and_QoI_samples']} after the fifth epoch of iterations. (Top Row): The updated density obtained after using the $Q_1$ map in the first iteration of this epoch (left), after using the $Q_2$ map in the second iteration of this epoch (middle), and the data-generating density estimate (right). (Bottom Row): The marginal QoI densities for $Q_1$ (left) and $Q_2$ (right) associated with the push-forward of the updated density (blue dashed curves) and observed data (orange solid curves).
  • Figure 5: The KDE estimates of densities associated with Figure \ref{['fig:ex1_param_and_QoI_samples']} after the final epoch (46th) of iterations. (Top Row): The updated density obtained after using the $Q_1$ map in the first iteration of this epoch (left), after using the $Q_2$ map in the second iteration of this epoch (middle), and the data-generating density estimate (right). (Bottom Row): The marginal QoI densities for $Q_1$ (left) and $Q_2$ (right) associated with the push-forward of the updated density (blue dashed curves) and observed data (orange solid curves).
  • ...and 12 more figures

Theorems & Definitions (7)

  • Lemma 3.1
  • proof
  • Corollary 3.1
  • Lemma 5.1
  • Theorem 5.1
  • proof
  • Corollary 5.1