Glossary

1.4 Mathematical formulation of inverse problems

5 min read•july 30, 2024

Inverse problems are mathematical puzzles that work backward, finding causes from effects. They're tricky because solutions might not exist, be unique, or stay stable with small changes. This chapter dives into the math behind these challenges.

We'll explore the key components: forward models, inverse models, and data. We'll also look at norms, metrics, and problem formulation. Understanding these basics is crucial for tackling real-world inverse problems effectively.

Mathematical Framework for Inverse Problems

Inverse Problem Fundamentals

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  • Inverse problems determine unknown causes based on observed effects, contrasting with forward problems that predict effects from known causes
  • Mathematical framework for inverse problems comprises three main components
    • Forward model
    • Inverse model
    • Data
  • Forward model represented by operator F mapping model parameters m to observed data d: F(m)=dF(m) = d
  • aims to find estimate of m given d, expressed as m=F−1(d)m = F^{-1}(d), where F−1F^{-1} denotes inverse operator
  • Ill-posedness characterizes many inverse problems
    • Solutions may not exist
    • Solutions may not be unique
    • Solutions may not depend continuously on data
  • techniques address ill-posedness and obtain stable solutions ()

Norms, Metrics, and Problem Formulation

  • Choice of norms and metrics in model and data spaces crucial for proper formulation and solution of inverse problems
  • Common norms used in inverse problems
    • L2 norm (Euclidean norm)
    • L1 norm (Manhattan norm)
    • Lp norms for p > 0
  • Metrics define distances between elements in model and data spaces
    • Euclidean distance
    • Mahalanobis distance
    • Kullback-Leibler divergence (for probability distributions)
  • Problem formulation involves selecting appropriate norms and metrics based on
    • Nature of the physical problem
    • Desired properties of the solution
    • Computational considerations

Model Parameters vs Observed Data

Characteristics of Model Parameters

  • Model parameters (m) represent unknown quantities or properties of the system to be estimated
  • Types of model parameters
    • Discrete (finite number of parameters)
    • Continuous (infinite-dimensional function spaces)
  • Physical constraints on model parameters
    • Non-negativity (concentrations, densities)
    • Boundedness (probabilities between 0 and 1)
  • Prior information about model parameters
    • Statistical distributions (Gaussian, uniform)
    • Smoothness assumptions
  • Dimensionality of model parameter space affects problem complexity
    • Low-dimensional (few parameters)
    • High-dimensional (many parameters or functions)

Properties of Observed Data

  • Observed data (d) are measurable quantities resulting from underlying model parameters and system behavior
  • Relationship between model parameters and observed data: d=F(m)+εd = F(m) + ε, where ε represents or noise
  • Types of observed data
    • Direct measurements (temperature readings)
    • Indirect measurements (seismic wave travel times)
    • Different types of sensors (optical, acoustic, electromagnetic)
    • Various experimental techniques (tomography, spectroscopy)
  • Dimensionality of data space
    • Under-determined problems (fewer data points than model parameters)
    • Over-determined problems (more data points than model parameters)
  • Uncertainty quantification essential for observed data
    • Measurement errors
    • Systematic biases
    • (Gaussian, Poisson)

Objective Functions and Constraints

Formulation of Objective Functions

  • quantifies discrepancy between observed data and predicted data from forward model
  • Common forms of objective functions
    • Least squares: J(m)=∣∣F(m)−d∣∣2J(m) = ||F(m) - d||^2
    • Weighted least squares: J(m)=(F(m)−d)TW(F(m)−d)J(m) = (F(m) - d)^T W (F(m) - d)
    • Maximum likelihood estimators: J(m)=−log⁡p(d∣m)J(m) = -\log p(d|m)
  • General form of objective function with regularization: J(m)=∣∣F(m)−d∣∣2+αR(m)J(m) = ||F(m) - d||^2 + αR(m)
    • R(m) denotes regularization term
    • α represents regularization parameter
  • Choice of objective function depends on
    • Noise characteristics of data
    • Expected properties of solution
    • Computational efficiency considerations

Constraints and Regularization

  • Constraints in inverse problems
    • Equality constraints: h(m)=0h(m) = 0
    • Inequality constraints: g(m)≤0g(m) ≤ 0
  • Constraints reflect physical limitations or prior knowledge about model parameters
  • Regularization techniques address ill-posedness and promote desired solution properties
  • Common regularization methods
    • Tikhonov regularization: R(m)=∣∣Lm∣∣2R(m) = ||Lm||^2, where L is a differential operator
    • Total variation regularization: R(m)=∫∣∇m∣dxR(m) = \int |\nabla m| dx
    • Sparsity-promoting regularization: R(m)=∣∣m∣∣1R(m) = ||m||_1
  • Choice of regularization term depends on desired solution properties
    • Smoothness
    • Sparsity
    • Adherence to prior information
  • Regularization parameter α balances data fit and regularization
    • Small α: focus on data fit
    • Large α: emphasis on regularization

Uniqueness, Stability, and Existence of Solutions

Uniqueness and Identifiability

  • refers to existence of only one solution satisfying inverse problem formulation
  • Non-uniqueness can occur due to
    • Insufficient data
    • Inherent ambiguity in problem
  • Identifiability analysis determines whether available data and forward model sufficiently determine model parameters
  • Methods for assessing uniqueness
    • Null space analysis for linear problems
    • Local and global optimization techniques for nonlinear problems
  • Examples of non-unique inverse problems
    • Gravity inversion (mass distributions with same surface gravity)
    • Electrical impedance tomography (different conductivity distributions with same boundary measurements)

Stability and Sensitivity Analysis

  • addresses how small changes in observed data affect estimated model parameters
  • Ill-posed problems may exhibit high sensitivity to data perturbations
  • analysis assesses stability of linear inverse problems
    • Condition number κ(A) = ||A|| ||A^(-1)||
    • Large condition number indicates ill-conditioning
  • techniques for nonlinear inverse problems
    • Adjoint methods
    • Monte Carlo simulations
    • Finite difference approximations
  • Regularization improves stability by adding prior information or constraints
  • Examples of unstable inverse problems
    • Numerical differentiation
    • Backward heat equation

Existence of Solutions

  • Existence of solutions concerns whether any model parameters can exactly reproduce observed data within given problem formulation
  • Hadamard conditions for require
    • Existence of solution
    • Uniqueness of solution
    • Continuous dependence of solution on data
  • Inverse problems often violate one or more Hadamard conditions
  • Approaches to address non-existence of exact solutions
    • Least squares formulation
    • Relaxation of constraints
    • Introduction of data uncertainties
  • Examples of inverse problems with existence issues
    • Over-determined systems with inconsistent data
    • Inverse heat conduction with noisy boundary measurements

Key Terms to Review (24)

Bayesian Inversion: Bayesian inversion is a statistical approach used to solve inverse problems by incorporating prior knowledge and observational data to update beliefs about unknown parameters. This method applies Bayes' theorem, which combines prior distributions with likelihoods from observed data to produce a posterior distribution that reflects the updated knowledge about the parameters of interest. The effectiveness of Bayesian inversion lies in its ability to quantify uncertainty and incorporate different sources of information, making it a powerful tool in understanding and solving inverse problems.
Condition Number: The condition number is a measure of how sensitive the solution of a mathematical problem is to changes in the input data. In the context of inverse problems, it indicates how errors in data can affect the accuracy of the reconstructed solution. A high condition number suggests that small perturbations in the input can lead to large variations in the output, which is particularly important in stability analysis, numerical methods, and when using techniques like singular value decomposition.
Data Acquisition Methods: Data acquisition methods refer to the various techniques and processes used to collect and record information from physical systems or processes for analysis. These methods are crucial in inverse problems, as they help gather the necessary data needed to infer unknown parameters or structures from observed results. Understanding how to effectively acquire data is essential for accurately solving inverse problems, which often involve interpreting complex signals or measurements.
Discretization: Discretization is the process of transforming continuous mathematical models and equations into discrete counterparts that can be solved numerically. This step is essential in various fields such as numerical analysis and computational science, as it allows for the approximation of solutions to problems that may not have closed-form solutions. By breaking down continuous domains into finite elements or grid points, discretization plays a crucial role in understanding and solving inverse problems, ensuring well-posedness, and implementing numerical methods like finite difference and finite element techniques.
Error Analysis: Error analysis refers to the systematic study of errors that occur in the process of solving mathematical problems, especially in the context of estimating unknown parameters from given data. This analysis helps identify how inaccuracies in data or computational methods can affect the accuracy of solutions. Understanding error analysis is crucial when dealing with inverse problems, as it allows for better assessment and minimization of uncertainties arising from ill-posedness and numerical methods used to find solutions.
Forward problem: A forward problem involves predicting the outcome or observations of a system based on known inputs and parameters. This type of problem is fundamental to understanding the relationship between inputs and outputs in a system, and it forms the basis for defining and solving inverse problems, where one aims to deduce the inputs from observed outputs.
Fredholm Integral Equation: A Fredholm integral equation is a type of integral equation that plays a significant role in inverse problems, characterized by an unknown function being under an integral sign and dependent on both a variable and an integral kernel. These equations can be categorized into two main types: the first kind, which involves determining the unknown function directly from the integral, and the second kind, which includes an additional term that usually represents known data. Understanding Fredholm integral equations is essential for solving many inverse problems, where one seeks to recover information about a system from indirect measurements.
Geophysical Imaging: Geophysical imaging is a technique used to visualize the subsurface characteristics of the Earth through the analysis of geophysical data. This method often involves inverse problems, where data collected from various sources are used to estimate properties such as density, velocity, and composition of subsurface materials. By combining mathematical formulations, numerical methods, and statistical frameworks, geophysical imaging plays a crucial role in understanding geological structures, resource exploration, and environmental assessments.
Ill-posed problem: An ill-posed problem is a situation in mathematical modeling or inverse problems where at least one of the conditions for well-posedness, such as existence, uniqueness, or stability of solutions, is not satisfied. This means that the problem may not have a solution, may have multiple solutions, or small changes in input can lead to large variations in the output, making it difficult to find reliable answers.
Image Reconstruction: Image reconstruction is the process of creating a visual representation of an object or scene from acquired data, often in the context of inverse problems. It aims to reverse the effects of data acquisition processes, making sense of incomplete or noisy information to recreate an accurate depiction of the original object.
Inverse Problem: An inverse problem involves determining the causal factors that produce a set of observed data, essentially working backwards from effects to causes. This type of problem is characterized by its complexity and the often ill-posed nature, where solutions may not exist, may not be unique, or may not depend continuously on the data.
Iterative methods: Iterative methods are computational algorithms used to solve mathematical problems by refining approximate solutions through repeated iterations. These techniques are particularly useful in inverse problems, where direct solutions may be unstable or difficult to compute. By progressively improving the solution based on prior results, iterative methods help tackle issues related to ill-conditioning and provide more accurate approximations in various modeling scenarios.
Kalman Filter: The Kalman Filter is a mathematical algorithm used to estimate the state of a dynamic system from a series of incomplete and noisy measurements. It combines prior knowledge of the system dynamics with observed data, providing optimal estimates by minimizing the mean of the squared errors. This filter is crucial for applications in control systems, navigation, and signal processing, especially in contexts involving inverse problems where accurate parameter estimation is essential.
Linear Inverse Problem: A linear inverse problem involves reconstructing an unknown quantity from observed data using linear equations. This type of problem arises in various fields where the relationship between the observed data and the unknowns can be expressed as a linear equation, making it possible to apply techniques for solving such equations to find the unknowns. The key challenge is that the observed data may contain noise or be incomplete, complicating the reconstruction process.
Measurement Errors: Measurement errors refer to the discrepancies between the actual value of a quantity and the value obtained from a measurement process. These errors can arise from various sources, including inaccuracies in instruments, environmental factors, and human mistakes. Understanding measurement errors is crucial in inverse problems as they can significantly affect the reliability and accuracy of the solutions derived from the data collected.
Noise Characteristics: Noise characteristics refer to the properties and behavior of noise present in data or measurements, which can impact the accuracy and reliability of results in inverse problems. Understanding these characteristics is crucial for developing effective algorithms that can separate useful signals from unwanted disturbances, allowing for better reconstruction of the underlying model or solution.
Objective Function: An objective function is a mathematical expression that quantifies the goal of an optimization problem, typically aiming to minimize or maximize some value. It plays a crucial role in evaluating how well a model fits the data, guiding the search for the best solution among all possible options while considering constraints and trade-offs.
Parameter Estimation: Parameter estimation is the process of using observed data to infer the values of parameters in mathematical models. This technique is essential for understanding and predicting system behavior in various fields by quantifying the uncertainty and variability in model parameters.
Regularization: Regularization is a mathematical technique used to prevent overfitting in inverse problems by introducing additional information or constraints into the model. It helps stabilize the solution, especially in cases where the problem is ill-posed or when there is noise in the data, allowing for more reliable and interpretable results.
Sensitivity analysis: Sensitivity analysis is a technique used to determine how the variation in the output of a model can be attributed to changes in its input parameters. This concept is crucial for understanding the robustness of solutions to inverse problems, as it helps identify which parameters significantly influence outcomes and highlights areas that are sensitive to perturbations.
Stability: Stability refers to the sensitivity of the solution of an inverse problem to small changes in the input data or parameters. In the context of inverse problems, stability is crucial as it determines whether small errors in data will lead to significant deviations in the reconstructed solution, thus affecting the reliability and applicability of the results.
Tikhonov Regularization: Tikhonov regularization is a mathematical method used to stabilize the solution of ill-posed inverse problems by adding a regularization term to the loss function. This approach helps mitigate issues such as noise and instability in the data, making it easier to obtain a solution that is both stable and unique. It’s commonly applied in various fields like image processing, geophysics, and medical imaging.
Uniqueness: Uniqueness refers to the property of an inverse problem where a single solution corresponds to a given set of observations or data. This concept is crucial because it ensures that the solution is not just one of many possible answers, which would complicate interpretations and applications in real-world scenarios.
Well-posedness: Well-posedness refers to a property of mathematical problems, especially in the context of inverse problems, where a problem is considered well-posed if it satisfies three criteria: it has a solution, the solution is unique, and the solution's behavior changes continuously with initial conditions. This concept is crucial for ensuring that solutions to inverse problems are reliable and meaningful, impacting how these problems are formulated and addressed, particularly when dealing with non-linear scenarios that require careful handling to avoid ill-posedness.
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