🔍Inverse Problems Unit 15 – Computational Methods in Inverse Problems

Key Concepts and Definitions

• Inverse problems aim to determine unknown causes based on observed effects or measurements
• Well-posed problems satisfy existence, uniqueness, and stability of the solution
• Ill-posed problems violate at least one of the well-posedness conditions
  • Often lack stability, where small changes in input can lead to large changes in the solution
• Forward problems involve predicting the effects given the causes
• Regularization introduces additional information to stabilize ill-posed problems
• Noise in measurements can significantly impact the solution of inverse problems
• Ill-conditioning occurs when the solution is highly sensitive to perturbations in the input data

Mathematical Foundations

• Linear algebra concepts such as vector spaces, matrices, and linear transformations are essential
• Functional analysis deals with infinite-dimensional vector spaces and operators
  • Hilbert spaces, Banach spaces, and linear functionals are key concepts
• Optimization theory provides techniques for finding the best solution among feasible options
• Probability theory and statistics help quantify uncertainty and noise in inverse problems
  • Bayesian inference incorporates prior knowledge and updates it with observed data
• Partial differential equations (PDEs) often describe the forward problem
  • Elliptic, parabolic, and hyperbolic PDEs are common in inverse problems
• Fourier analysis and wavelet theory are used for signal and image processing applications

Inverse Problem Formulation

• The forward problem is represented by an operator $F$ mapping the model parameters $m$ to the data $d$: $F(m) = d$
• The inverse problem aims to find the model parameters $m$ given the observed data $d$
• Least squares formulation minimizes the discrepancy between the predicted and observed data: $\min_m \|F(m) - d\|^2$
• Tikhonov regularization adds a penalty term to the least squares objective: $\min_m \|F(m) - d\|^2 + \alpha \|Lm\|^2$
  • $L$ is a regularization operator, and $\alpha$ is the regularization parameter
• Bayesian formulation treats the model parameters and data as random variables
  • Prior probability distribution represents the initial knowledge about the model parameters
  • Likelihood function describes the probability of observing the data given the model parameters
  • Posterior probability distribution combines the prior and likelihood using Bayes' theorem

Regularization Techniques

• Regularization addresses ill-posedness by introducing additional constraints or information
• Tikhonov regularization is a common approach that penalizes the norm of the solution or its derivatives
  • The regularization parameter $\alpha$ controls the balance between data fitting and regularization
• Truncated singular value decomposition (TSVD) regularizes by truncating small singular values
  • Reduces the influence of noise and improves stability
• Iterative regularization methods, such as Landweber iteration and conjugate gradient, solve the problem iteratively
  • The number of iterations acts as a regularization parameter
• Total variation (TV) regularization preserves sharp edges and discontinuities in the solution
  • Suitable for image reconstruction and denoising
• Sparsity-promoting regularization, such as $\ell_1$-norm regularization, encourages sparse solutions
  • Useful when the desired solution is known to be sparse in some domain

Numerical Methods for Solving Inverse Problems

• Discretization techniques convert the continuous problem into a discrete form
  • Finite difference, finite element, and finite volume methods are commonly used
• Matrix formulation represents the discretized problem as a system of linear equations: $Am = d$
• Direct methods, such as Gaussian elimination and LU decomposition, solve the system exactly
  • Suitable for small to medium-sized problems
• Iterative methods, such as Jacobi, Gauss-Seidel, and Krylov subspace methods, solve the system approximately
  • Efficient for large-scale problems and sparse matrices
• Regularization is often incorporated into the numerical solution process
  • Tikhonov regularization modifies the system to $(A^TA + \alpha L^TL)m = A^Td$
• Preconditioning techniques improve the convergence of iterative methods
  • Jacobi, Gauss-Seidel, and incomplete LU preconditioners are commonly used

Optimization Algorithms

• Optimization algorithms are used to solve the regularized inverse problem
• Gradient-based methods, such as steepest descent and conjugate gradient, use the gradient information
  • Efficient for smooth and convex problems
• Newton's method and its variants (Gauss-Newton, Levenberg-Marquardt) use second-order information
  • Converge faster but require computing the Hessian matrix or its approximation
• Quasi-Newton methods, such as BFGS and L-BFGS, approximate the Hessian using gradient information
  • Provide a balance between convergence speed and computational cost
• Proximal algorithms, such as proximal gradient and alternating direction method of multipliers (ADMM), handle non-smooth regularizers
  • Useful for problems with sparsity-promoting or total variation regularization
• Stochastic optimization methods, such as stochastic gradient descent (SGD), use random subsets of data
  • Suitable for large-scale problems and online learning

Error Analysis and Uncertainty Quantification

• Error analysis quantifies the discrepancy between the true and estimated solutions
• Sensitivity analysis studies how changes in the input data or model parameters affect the solution
  • Singular value decomposition (SVD) provides insights into the sensitivity of the problem
• Uncertainty quantification assesses the reliability and variability of the solution
  • Bayesian inference provides a probabilistic framework for uncertainty quantification
  • Markov chain Monte Carlo (MCMC) methods sample from the posterior distribution
  • Gaussian processes and polynomial chaos expansions are used for uncertainty propagation
• Residual analysis examines the difference between the observed and predicted data
  • Helps identify model inadequacies and outliers
• Cross-validation techniques, such as k-fold and leave-one-out, assess the model's predictive performance
  • Used for model selection and parameter tuning

Applications and Case Studies

• Geophysical imaging: Seismic inversion, ground-penetrating radar, and gravity surveys
  • Estimate subsurface properties from surface measurements
• Medical imaging: Computed tomography (CT), magnetic resonance imaging (MRI), and positron emission tomography (PET)
  • Reconstruct images from projection data or Fourier measurements
• Image processing: Deblurring, denoising, and super-resolution
  • Recover high-quality images from degraded or low-resolution observations
• Machine learning: Parameter estimation, model selection, and feature extraction
  • Learn models from data and make predictions or decisions
• Atmospheric science: Data assimilation and remote sensing
  • Combine observations with numerical models to estimate the state of the atmosphere
• Inverse scattering: Acoustic, electromagnetic, and elastic wave scattering
  • Determine the properties of scatterers from the scattered field measurements