Objectives

AbstractObjective

A reconstruction functional J(σ) with gradient, e.g. the least-squares data misfit through the adjoint state method (AdjointStateObjective) or the Kohn–Vogelius functional (KohnVogeliusObjective). Interface: objective_value, value_and_gradient!.

objective_value(obj, σ)

Value of the objective at the conductivity coefficients σ (forward solves only).

value_and_gradient!(g, obj, σ)

Value of the objective at σ; writes the gradient (in the representation of the objective's gradient Riesz map) into g.

AdjointStateObjective(fm, inputs, data; mode = :neumann, solver = DirectSolver(),
                      misfit = SquaredEuclidean(), gradient = CoefficientGradient())

Least-squares misfit J(σ) = ½ Σₛ ‖U eₛ(σ)‖² between predicted and measured data for the forward model fm, with the gradient from one adjoint solve and the Jacobian on request.

• mode = :neumann: inputs are current patterns (n_inject × s), data the measured voltages (n_measure × s). Voltages are compared after removing their mean (the ground is arbitrary), weighted by fm.measure_weights (boundary lengths for the continuum model).
• mode = :dirichlet: inputs are voltage patterns (n_control × s), data the measured currents (n_inject × s, in the representation of fm.P).

solver is any AbstractLinearSolver; misfit any AbstractMisfit; gradient the AbstractRieszMap applied to the coefficient gradient. All buffers are allocated here; evaluations only allocate inside the linear solvers.

Interface: objective_value, value_and_gradient!, residual!, residual_and_jacobian!, n_residual.

residual!(r, obj, σ)

Whitened residual r = vec(U e) at σ (so that J = ½ ‖r‖²); returns r.

residual(obj, θ)

The residual vector of a least-squares objective (J = ½ ‖r‖²) at θ, allocating; see residual!. For an AdjointStateObjective with zero data and the plain misfit this is the vector of measured voltages, i.e. the forward map. With ChainRulesCore loaded, it has reverse (Jᵀ r̄, adjoint solves) and forward (J δθ, linearized solves) differentiation rules, as does objective_value (reverse: the adjoint-state gradient).

residual_and_jacobian!(r, J, obj, σ)

Whitened residual r and its Jacobian J = ∂r/∂σ (n_residual(obj) × n_σ, coefficient representation) at σ. Costs one block solve with k right-hand sides plus k × s sparse products with the conductivity tensor, k the residual entries per pattern (n_obs, or the rows of a ProjectedMisfit).

n_residual(obj)

Length of the residual vector (n_obs × s, stacked pattern by pattern).

jacobian_operator(obj, σ)

The Jacobian J = ∂r/∂σ of the residual at σ as a matrix-free operator: J * v (one linearized forward solve per pattern) and J' * w (one adjoint solve per pattern), with mul! for both, reusing the factorization of the forward solves. Neumann (current-driven) mode. The operator re-linearizes by itself if the objective has been evaluated at another σ in the meantime. For Krylov methods, Gauss–Newton with linear_solver = :cg, and problems whose Jacobian does not fit into memory.

JacobianOperator

Matrix-free Jacobian of an AdjointStateObjective; see jacobian_operator.

ParametrizedJacobian

Matrix-free Jacobian J_θ = J_σ P of a ParametrizedObjective; see jacobian_operator.

jacobian_column_norms(obj, θ)

Column norms ‖J eⱼ‖ of the Jacobian of the least-squares objective obj at θ (sensitivities), accumulated from row blocks without storing J.

jacobian_gram(obj, θ)

The Gram matrix JᵀJ (n × n) and Jᵀr of the Jacobian and residual of obj at θ, accumulated from row blocks without storing J: the Gauss–Newton Hessian and gradient, also for problems with far more residuals than parameters.

KohnVogeliusObjective(fm, currents, voltages; solver = DirectSolver(), gradient = CoefficientGradient())

Kohn–Vogelius functional J(σ) = ½ Σₛ ‖x_N,ₛ - x_D,ₛ‖²_{A(σ)} for the current patterns currents (n_inject × s) and the measured voltages voltages (n_control × s, at the injection sites: boundary dofs, points or all CEM electrodes). The gradient needs no adjoint solve. After an evaluation, boundary_error returns the voltage misfit of the current-driven solution and pattern_values the contribution of every pattern.

boundary_error(obj::KohnVogeliusObjective)

Measured minus predicted voltages of the current-driven states at the last evaluation (Q x_N - V, mean removed per pattern).

pattern_values(obj::KohnVogeliusObjective)

Contribution ½ ‖x_N,ₛ - x_D,ₛ‖²_A of every pattern at the last evaluation.

AbstractMisfit

Metric in which the data misfit is measured. J = ½ ‖U e‖² for a whitening operator U, e.g. SquaredEuclidean (U = I) or WeightedSquaredEuclidean (UᵀU = W).

SquaredEuclidean()

J = ½ Σₛ ‖eₛ‖² over the patterns s.

WeightedSquaredEuclidean(W)

J = ½ Σₛ eₛᵀ W eₛ for a symmetric positive definite W (n_obs × n_obs), e.g. inverse noise covariances or a boundary mass matrix. Whitened residual r = U e with UᵀU = W.

ProjectedMisfit(U)

J = ½ Σₛ ‖U eₛ‖² for a rectangular U (k × n_obs, k ≤ n_obs): only k combinations of the measurements enter the misfit, e.g. the leading measurement modes of pattern_svd (truncate_patterns(p, K; measurements = M), a two-sided truncation of the data with K M residuals instead of n_obs K). The residual has k entries per pattern.

AbstractEITProblem

A complete reconstruction problem (discretization, forward model, data, objective, current iterate), solved by reconstruct!; see EITProblem.

AbstractSolutionState

State of an iterative reconstruction (iterate, step sizes, histories), e.g. the OptimizationState returned by minimize.