12.1 Numerical methods for inverse problems

Inverse problems are like detective work in math. You're trying to figure out what caused something by looking at the results. It's tricky because there might be multiple answers or small changes can lead to big differences.

To solve these puzzles, we use optimization techniques. We try to find the best fit between what we observe and what our model predicts. We also use tricks like regularization to make sure our solutions make sense and aren't too wild.

Inverse Problems as Optimization

Formulating and Solving Inverse Problems

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• Inverse problems determine unknown causes based on observations of their effects, often expressed as finding parameters that produce a desired output
• Formulate inverse problems as optimization problems by minimizing the discrepancy between observed data and model predictions
• Objective functions typically include:
  • Data fidelity term measures how well the model fits the observed data
  • Prior knowledge or constraints incorporate additional information about the solution
• Least squares methods commonly used for linear inverse problems minimize the sum of squared differences between observed and predicted values
• Ill-posedness in inverse problems leads to:
  • Non-uniqueness multiple solutions may fit the observed data equally well
  • Instability small changes in input data can cause large changes in the solution
• Tikhonov regularization converts ill-posed problems into well-posed optimization problems by adding a penalty term to the objective function
• Choice of norm in the objective function impacts the nature of the solution:
  • L2 norm (Euclidean distance) promotes smooth solutions
  • L1 norm promotes sparse solutions with many zero or near-zero values

Optimization Techniques for Inverse Problems

• Gradient-based methods utilize the gradient of the objective function to iteratively improve the solution:
  • Steepest descent moves in the direction of the negative gradient
  • Conjugate gradient uses a more efficient search direction
• Newton's method and quasi-Newton methods incorporate second-order derivative information for faster convergence
• Convex optimization techniques applicable when the objective function is convex:
  • Interior point methods
  • Proximal algorithms
• Global optimization methods for non-convex problems:
  • Simulated annealing
  • Genetic algorithms
• Multi-objective optimization approaches balance multiple competing objectives (data fit, solution smoothness, sparsity)

Regularization for Ill-Posed Problems

Tikhonov Regularization and Variants

• Tikhonov regularization (ridge regression) adds a penalty term to the objective function:
  • Controls magnitude or smoothness of the solution
  • Objective function: $\min_x \|Ax - b\|^2 + \lambda \|Lx\|^2$
    • A forward operator
    • b observed data
    • x solution
    • L regularization operator (often identity or derivative)
    • λ regularization parameter
• L1 regularization (Lasso) promotes sparsity in the solution:
  • Effective for feature selection in inverse problems
  • Objective function: $\min_x \|Ax - b\|^2 + \lambda \|x\|_1$
• Total Variation (TV) regularization preserves edges in image reconstruction:
  • Penalizes total variation of the solution
  • Useful in medical imaging (CT, MRI) and remote sensing

Regularization Parameter Selection

• Regularization parameter λ balances data fidelity and solution stability
• Methods for selecting λ:
  • L-curve plots solution norm vs. residual norm for different λ values
  • Generalized cross-validation minimizes prediction error
  • Discrepancy principle chooses λ to match noise level in the data
• Iterative regularization methods provide implicit regularization:
  • Early stopping in iterative algorithms
  • Landweber iteration with appropriate stopping criteria
• Multi-parameter regularization combines different regularization terms:
  • Incorporates various prior knowledge or desired solution properties
  • Example: $\min_x \|Ax - b\|^2 + \lambda_1 \|x\|_1 + \lambda_2 \|Lx\|^2$

Iterative Methods for Inverse Problems

Gradient-Based and Krylov Subspace Methods

• Landweber iteration simple yet effective for linear inverse problems:
  • Update formula: $x_{k+1} = x_k + \alpha A^T(b - Ax_k)$
  • α step size parameter
• Conjugate gradient method efficient for symmetric positive definite systems:
  • Converges in at most n iterations for n×n matrix
  • Requires less memory than direct methods for large-scale problems
• Krylov subspace methods powerful for large-scale linear inverse problems:
  • GMRES (Generalized Minimal Residual) for non-symmetric systems
  • LSQR for least squares problems
• Non-linear inverse problems require specialized iterative methods:
  • Gauss-Newton algorithm approximates Hessian with Jacobian product
  • Levenberg-Marquardt algorithm adds regularization to Gauss-Newton

Advanced Iterative Techniques

• Alternating direction method of multipliers (ADMM) effective for problems with multiple regularization terms or constraints:
  • Splits problem into smaller, easier-to-solve subproblems
  • Useful for TV regularization and compressed sensing
• Stochastic iterative methods handle large-scale problems with noisy or redundant data:
  • Stochastic gradient descent randomly samples data points
  • Mini-batch methods balance computational efficiency and convergence
• Implementation considerations for iterative methods:
  • Stopping criteria based on relative change, residual, or maximum iterations
  • Step size selection methods (fixed, line search, adaptive)
  • Preconditioning techniques to improve convergence rate

Convergence and Stability of Numerical Methods

Convergence Analysis

• Convergence analysis studies conditions under which an iterative method approaches the true solution
• Contractivity crucial for proving convergence of fixed-point iterative methods:
  • Iteration operator T must be contractive: $\|T(x) - T(y)\| \leq q\|x - y\|$ for some q < 1
• Convergence rates classify how quickly methods approach the solution:
  • Linear convergence: error reduces by constant factor each iteration
  • Superlinear convergence: convergence factor approaches zero
  • Quadratic convergence: error squared each iteration (e.g., Newton's method)
• Spectral analysis of iteration operator reveals convergence properties:
  • Eigenvalues determine convergence rate and potential instabilities
  • Spectral radius ρ(T) < 1 ensures convergence for linear fixed-point methods

Stability and Error Analysis

• Stability analysis examines how small perturbations in input data affect the solution:
  • Particularly important for ill-posed inverse problems
  • Condition number of forward operator A measures problem sensitivity:
    • κ(A) = σ_max(A) / σ_min(A) ratio of largest to smallest singular values
    • Large condition number indicates ill-conditioning
• Error estimates depend on:
  • Smoothness of true solution
  • Properties of regularization method
  • Noise level in the data
• Numerical experiments assess convergence and stability:
  • Parameter studies vary regularization parameters, noise levels
  • Noise sensitivity analysis adds controlled perturbations to input data
  • Cross-validation techniques evaluate generalization performance