Data-Consistent Inversion: A Distributions-based Approach for Characterizing Uncertain Model Parameters from Observed Variability

Core Concepts

The core message of this article is to formulate and analyze a novel constrained quadratic optimization approach for solving data-consistent inverse (DCI) problems, which involve characterizing a probability measure on model parameters such that the push-forward of this measure matches an observed probability measure on specified quantities of interest.

Abstract

The article presents a novel approach to solve a class of stochastic inverse problems, referred to as data-consistent inverse (DCI) problems. DCI problems involve characterizing a probability measure on the parameters of a computational model such that the subsequent push-forward of this measure matches an observed probability measure on specified quantities of interest (QoI) associated with the model outputs.
The key contributions are:

The authors develop and analyze a constrained quadratic optimization approach that estimates push-forward measures using weighted empirical distribution functions (EDFs). This method is more suitable for low-data regimes or high-dimensional problems than previous density-based methods, and can handle cases where the probability measure does not admit a density.

The authors prove theoretical results showing that the optimization-based EDF approximation converges in the L2-norm to the target distribution, and that this L2-convergence implies weak convergence of the associated probability measures.

Numerical examples are provided to demonstrate the performance of the method and compare it to the density-based approach where applicable. The examples include cases where the observed distribution does not admit a density, highlighting the advantages of the proposed EDF-based approach.

The article provides a robust and flexible framework for solving DCI problems, particularly in situations with limited data or when the distributions involved do not have well-defined densities.

Stats

"We require the solution to this inverse problem to be data-consistent, meaning that the push-forward of the probability measure on the parameter space matches a given target distribution on the observed data space."
"The key to building upon the optimization-based approach for push-forward EDFs to solve the DCI problem is through the addition of a critical binning step in the output space."

Quotes

"The method proposed here is more suitable for low-data regimes or high-dimensional problems than the density-based method, as well as for problems where the probability measure does not admit a density."
"We emphasize that the explicit construction of these cells is never actually required in either the theory or in practice, but we do reference the assumed continuity set property of these cells in the theoretical analysis."

Key Insights Distilled From

A Distributions-based Approach for Data-Consistent Inversion

by Kirana Bergs... at arxiv.org 04-19-2024

https://arxiv.org/pdf/2404.11886.pdf

Deeper Inquiries

How can the binning approach be extended to handle cases with highly irregular or complex data spaces

The binning approach can be extended to handle cases with highly irregular or complex data spaces by implementing more sophisticated partitioning techniques. Instead of using a regular grid or K-means clustering, which may not capture the intricate structure of the data space, advanced clustering algorithms like DBSCAN or OPTICS can be employed. These algorithms can identify clusters of varying shapes and densities, allowing for a more accurate representation of the data distribution. Additionally, hierarchical clustering methods can be utilized to create a hierarchy of partitions, enabling a more nuanced approach to capturing the complexity of the data space. By incorporating these advanced clustering techniques, the binning approach can adapt to irregular and complex data spaces more effectively.

What are the potential limitations or drawbacks of the optimization-based EDF approach compared to the density-based method, beyond the examples provided

While the optimization-based EDF approach offers several advantages over the density-based method, there are some potential limitations and drawbacks to consider:

Computational Complexity: The optimization-based approach involves solving a quadratic optimization problem, which can be computationally intensive, especially for large datasets or high-dimensional spaces. This complexity may limit the scalability of the method for extremely large datasets.

Sensitivity to Initialization: The optimization process in the EDF approach may be sensitive to the initial weights or parameters chosen. Convergence to the optimal solution could be influenced by the starting point, leading to potential suboptimal solutions if not initialized correctly.

Assumption of Continuity: The EDF approach assumes continuity in the data space, which may not always hold true for real-world datasets. In cases where the data distribution is highly discontinuous or sparse, the optimization-based method may struggle to accurately capture the underlying distribution.

Interpretability: The weights obtained from the optimization process may not be as easily interpretable as probability densities. Understanding the significance of these weights and their impact on the solution could be more challenging compared to traditional density-based methods.

How might the proposed framework be adapted or combined with other techniques to solve DCI problems in real-world applications with additional constraints or requirements

The proposed framework for solving Data-Consistent Inversion (DCI) problems can be adapted and combined with other techniques to address real-world applications with additional constraints or requirements. Some potential adaptations and combinations include:

Incorporating Bayesian Methods: The framework can be extended to incorporate Bayesian inference techniques to handle prior knowledge or uncertainties in the model parameters. By integrating Bayesian priors with the optimization-based approach, a more robust and probabilistic solution to DCI problems can be achieved.

Feature Engineering: In scenarios where the data space is high-dimensional, feature engineering techniques such as dimensionality reduction or feature selection can be applied to improve the efficiency and effectiveness of the optimization process. By reducing the dimensionality of the data space, the framework can handle complex datasets more efficiently.

Ensemble Methods: Combining the optimization-based EDF approach with ensemble methods like bootstrapping or model averaging can enhance the robustness and stability of the solution. By aggregating multiple solutions obtained from different optimization runs, the framework can provide more reliable and accurate results in the presence of noise or variability in the data.

By adapting and integrating these techniques into the proposed framework, DCI problems in real-world applications can be addressed more effectively, considering additional constraints and requirements inherent in complex datasets.