8.1 Formulation of linear inverse problems

Linear inverse problems aim to find unknown inputs from known outputs in linear systems. They're represented by the equation Ax = b, where A is the forward operator, x is the unknown input, and b is the known output.

These problems often involve noise and can be ill-posed, violating conditions of existence, uniqueness, or stability. Understanding the mathematical structure and challenges helps in developing effective solution strategies, like regularization techniques.

Mathematical structure of linear inverse problems

Linear systems and their components

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• Linear inverse problems find unknown input (cause) from known output (effect) in linear system
• General form $Ax = b$
  • A represents forward operator
  • x represents unknown input
  • b represents known output
• Dimensionality affects problem properties and solution approaches
  • Overdetermined systems (more equations than unknowns)
  • Underdetermined systems (fewer equations than unknowns)
  • Square systems (equal number of equations and unknowns)
• Real-world inverse problems often include noise and measurement errors
  • Represented as e in equation $Ax = b + e$
• Well-posedness concept (defined by Hadamard) crucial for understanding solution stability and uniqueness
  • Three conditions: existence, uniqueness, and stability of solutions

Mathematical foundations and challenges

• Noise and measurement errors complicate solution process
  • Require robust estimation techniques (least squares, regularization)
• Well-posedness conditions often violated in practice
  • Lead to ill-posed problems
• Dimensionality impacts solution methods
  • Overdetermined systems solved using least squares
  • Underdetermined systems require additional constraints (regularization)
• Condition number of A influences problem sensitivity
  • High condition number indicates ill-conditioning
  • Leads to solution instability

Forward vs inverse operators

Characteristics and roles of operators

• Forward operator A maps model space to data space
  • Represents physical process or system under study
  • Examples: CT scan projection, seismic wave propagation
• Inverse operator A^(-1) (when exists) maps data space to model space
  • Allows input reconstruction from output
  • Examples: image reconstruction from CT projections, subsurface structure estimation from seismic data
• Forward problem typically well-posed
  • Involves applying forward operator to known input
  • Examples: simulating radar reflections, predicting temperature distribution
• Inverse problem often ill-posed
  • Involves applying inverse operator (or approximation) to known output
  • Examples: radar target identification, heat source localization

Operator properties and practical considerations

• Forward operator properties influence inverse problem behavior
  • Rank determines solution uniqueness
  • Condition number affects solution stability
• Forward operator often known or modelable
  • Based on physical laws or empirical relationships
  • Examples: Maxwell's equations for electromagnetic problems, Navier-Stokes equations for fluid dynamics
• Inverse operator may not exist or difficult to compute directly
  • Leads to use of approximate inversion techniques
  • Examples: iterative methods, regularization approaches
• Operator discretization impacts problem formulation
  • Continuous operators approximated by matrices
  • Discretization scheme affects solution accuracy and computational complexity

Linear inverse problems in matrix notation

Matrix equation and its components

• Matrix equation $Ax = b$ represents linear inverse problem
  • A denotes m × n matrix
  • x denotes n × 1 vector
  • b denotes m × 1 vector
• Matrix A columns represent model space basis functions
  • Each column corresponds to a model parameter
  • Examples: basis images in tomography, frequency components in spectral analysis
• Matrix A rows correspond to individual measurements or constraints
  • Each row represents a data point or equation
  • Examples: detector readings in CT scan, seismometer recordings in seismology

Mathematical tools and solution methods

• Singular Value Decomposition (SVD) of A provides valuable problem insights
  • Reveals rank, condition number, and singular vectors
  • Used in truncated SVD regularization
• Null space and range of A crucial for solution understanding
  • Null space relates to solution non-uniqueness
  • Range determines existence of exact solutions
• Moore-Penrose pseudoinverse A^+ finds minimum norm solution
  • Used when A not invertible or rectangular
  • Computed using SVD: $A^+ = V\Sigma^+U^T$
• Regularization techniques stabilize ill-posed problems
  • Tikhonov regularization adds penalty term to objective function
  • L1 regularization promotes sparsity in solutions

Ill-posed linear inverse problems

Characteristics and manifestations of ill-posedness

• Ill-posed problems violate Hadamard's conditions
  • Existence, uniqueness, or stability of solutions compromised
• Non-existence of solutions occurs when b outside range of A
  • Data inconsistent with forward model
  • Examples: noisy measurements in tomography, conflicting constraints in optimization
• Non-uniqueness arises from non-trivial null space of A
  • Infinitely many solutions satisfy the equation
  • Examples: limited-angle tomography, underdetermined systems in geophysics
• Solution instability characterized by high sensitivity to data perturbations
  • Often due to ill-conditioning of forward operator
  • Examples: high-frequency instabilities in deconvolution, small eigenvalue problems in inverse heat conduction

Analysis tools and mitigation strategies

• Discrete Picard condition determines ill-posedness
  • Analyzes decay rates of singular values and data coefficients
  • Helps identify appropriate regularization level
• Regularization methods address ill-posedness
  • Truncated SVD discards small singular values
  • Tikhonov regularization adds smoothness constraints
• L-curve technique selects regularization parameters
  • Balances solution norm and residual norm
  • Graphical method for finding optimal trade-off
• Generalized Cross-Validation (GCV) alternative for parameter selection
  • Minimizes prediction error in leave-one-out cross-validation
  • Applicable to various regularization methods