Quantifying Spatial Variability of Information in Inverse Problems to Improve Reconstruction Accuracy

Core Concepts

The information density, based on the variance of reconstructed parameters, can be used to quantify the spatial variability of information available for accurately recovering parameters in inverse problems. This information density can guide practical algorithms for solving inverse problems, such as choosing the discretization mesh.

Abstract

The content discusses the concept of "information" in the context of inverse problems, where one attempts to infer spatially variable functions from indirect measurements. The authors present a definition of information density based on the variance of coefficients derived from a Bayesian reformulation of the inverse problem. The key highlights are: Inverse problems are often ill-posed, meaning small measurement errors can lead to significantly different reconstructed parameters. This is due to a lack of information. The authors propose defining an "information density" that quantifies how much information is available at different spatial locations for accurately recovering the parameters. For a finite-dimensional, linear model problem, the information density is defined as the square root of the diagonal elements of the Fisher information matrix, which approximates the inverse of the parameter covariance matrix. This definition satisfies intuitive properties, such as information being additive with more measurements and inversely proportional to measurement uncertainties. The approach is extended to infinite-dimensional inverse problems, where the information density is defined as a spatially variable function. The authors outline three areas where the information density can be useful in practical algorithms for solving inverse problems, and provide a detailed numerical example on how to use it for choosing the discretization mesh.

Stats

The information density jk for parameter qk is defined as: jk = sqrt(Qkk) where Qkk = sum_l (1/sigma_l^2) (mℓ, hk)^2 + beta |omega_k| and hk satisfies the equation Lhk = sk.

Quotes

"The information density, based on the variance of reconstructed parameters, can be used to quantify the spatial variability of information available for accurately recovering parameters in inverse problems." "This information density can guide practical algorithms for solving inverse problems, such as choosing the discretization mesh."

Key Insights Distilled From

Estimating and using information in inverse problems

by Wolfgang Ban... at arxiv.org 04-24-2024

https://arxiv.org/pdf/2208.09095.pdf

Deeper Inquiries

How can the information density be used to guide the design of measurement systems (e.g. sensor placement) to maximize the information content for inverse problem reconstruction

The information density can play a crucial role in guiding the design of measurement systems, particularly in optimizing sensor placement to maximize the information content for inverse problem reconstruction. By analyzing the spatial distribution of the information density, one can identify regions where the uncertainty in the parameter estimation is high or low. Maximizing Information Content: High Information Density Areas: Sensors should be strategically placed in regions with high information density to capture valuable data that contributes significantly to parameter estimation. These areas provide more accurate and reliable information about the parameters being reconstructed. Low Information Density Areas: In regions with low information density, additional sensors or alternative measurement techniques may be required to improve the overall information content. Optimizing Sensor Placement: Dense Sensor Placement: Areas with high information density should have denser sensor placement to capture detailed information and reduce uncertainty in parameter estimation. Sparse Sensor Placement: Regions with low information density may benefit from targeted sensor placement to gather specific data that can enhance the overall reconstruction process. Iterative Optimization: Adaptive Sensor Placement: Utilizing the information density iteratively, sensor placement can be adjusted during the reconstruction process based on the evolving understanding of the parameter distribution. This adaptive approach ensures that new measurements are strategically acquired to maximize information content. By leveraging the information density to guide sensor placement, the measurement system can be optimized to provide the most informative data for accurate inverse problem reconstruction.

What are the limitations of the information density approach, and how can it be extended to handle highly nonlinear inverse problems or cases with strong prior information

The information density approach, while powerful, does have limitations that need to be addressed for handling highly nonlinear inverse problems or cases with strong prior information. Limitations: Nonlinearity: In highly nonlinear problems, the linear approximation used to compute the information density may not accurately capture the system's behavior, leading to inaccuracies in the information content estimation. Strong Prior Information: When strong prior information is available, the information density approach may need to be modified to incorporate this prior knowledge effectively. Failing to do so could result in biased estimations of the information content. Extensions: Nonlinear Problems: For highly nonlinear inverse problems, the information density approach can be extended by incorporating higher-order terms or utilizing advanced techniques such as Bayesian optimization to handle the nonlinearities more effectively. Incorporating Priors: To address cases with strong prior information, the information density formulation can be adapted to integrate prior knowledge into the estimation of information content. This can help in refining the parameter estimation process and reducing uncertainties. By addressing these limitations and extending the information density approach, it can be applied more effectively to handle complex nonlinear problems and cases with significant prior information.

Can the information density be used to adaptively refine the discretization mesh during the iterative solution of an inverse problem, rather than just for the initial mesh design

The information density can indeed be utilized to adaptively refine the discretization mesh during the iterative solution of an inverse problem, providing a dynamic approach to mesh refinement based on the spatial distribution of information content. Adaptive Mesh Refinement: Information-Guided Refinement: The information density can guide the adaptive refinement of the mesh by identifying regions with high information content that require finer discretization for accurate parameter estimation. Dynamic Mesh Adjustment: As the solution progresses iteratively, the information density can be used to identify areas where the mesh resolution needs to be increased or decreased based on the changing information content. Iterative Process: Real-Time Adjustment: By continuously evaluating the information density during the iterative solution process, the mesh refinement can be dynamically adjusted to focus computational resources on areas with the highest information content. Enhanced Convergence: Adaptive mesh refinement based on information density can lead to faster convergence and more accurate solutions by concentrating computational efforts where they are most beneficial for parameter estimation. By incorporating information density-guided adaptive mesh refinement strategies, the iterative solution of inverse problems can be optimized for efficient and accurate reconstruction.