Optimization

Three optimization algorithms are available for iterative reconstruction. All use Armijo backtracking line search.

See the Mathematical Reference for update rule equations.

Problem Context

Sentinel.ForwardProblemContext Type

ForwardProblemContext

Bundles all immutable problem data needed to evaluate the objective function and gradient. Passed to line search and optimizer routines to avoid long argument lists.

Fields

• grid: Ferrite Grid

• dh: DofHandler

• cv_disp: CellValues for displacement interpolation

• cv_press: CellValues for pressure (or nothing)

• model: AbstractMaterialModel singleton

• gp2mtrs: vector of GP2Mtr per material mesh

• meshes: vector of MaterialMesh

• bcs: BoundaryConditions

• meas: measured Displacement

• ω: angular frequency [rad/s]

• ρ: density [kg/m³]

• regularizations: vector of AbstractRegularization (or nothing)

• solver: linear solver backend

• numdispsets: number of displacement/excitation sets

• fiber_field: per-element fiber direction vectors (or nothing)

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Sentinel.ConvergenceHistory Type

ConvergenceHistory

Records iteration-by-iteration progress of an optimization run.

Fields

• iterations: number of iterations completed

• objective: objective function values per iteration

• disp_error: displacement-only error per iteration

• mat_epsilon: relative material change per iteration

• converged: whether the optimizer declared convergence

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Helper Functions

Sentinel.copy_material Function

copy_material(mat::Material) -> Material

Deep-copy a Material struct, duplicating all SingleProperty arrays.

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Sentinel.apply_material_update! Function

apply_material_update!(material::Material, delta_theta::AbstractVector,
                        alpha::Float64, grad::MaterialGradient)

Update material property values: θ += α·Δθ/scalar, using the gradient's realparam2prop reverse mapping to write each element of delta_theta back to the correct rvalue or ivalue entry.

The division by scalar matches Fortran's incrementmaterial (FEmaterial.f90): realinc = direction%rvalue(direcind) / origmaterial%prop(ii)%scalar(iii) This converts the step from physical units (produced by CG gradient scaling) back to normalized parameter units.

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Sentinel.evaluate_objective! Function

evaluate_objective!(calc::Displacement, K::SparseMatrixCSC{ComplexF64},
                     f_work::Vector{ComplexF64},
                     ctx::ForwardProblemContext, material::Material;
                     iteration::Int=1) -> Float64

Run the forward solve for all displacement sets and compute the objective.

1. Assemble stiffness from current material

2. For each dispset: apply BCs, solve, store in calc

3. Return compute_objective(calc, meas, material, regularizations)

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Sentinel.compute_full_gradient! Function

compute_full_gradient!(grad::MaterialGradient, calc::Displacement,
                        adjnt::Displacement,
                        K::SparseMatrixCSC{ComplexF64},
                        ctx::ForwardProblemContext, material::Material;
                        iteration::Int=1)

Compute the full gradient including adjoint solves and regularization.

1. For each dispset: adjoint_solve!

2. zero!(grad), compute_gradient! (adjoint contribution)

3. add_gradient! for each regularization term

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Sentinel.apply_postprocessing! Function

apply_postprocessing!(material::Material, regularizations)

Apply all post-processing regularizations (Van Houten, Spatial Filter, McGarry bounds) in sequence.

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Conjugate Gradient

Polak-Ribiere+ nonlinear CG:

$${\mathbf{d}}_k=-{\mathbf{g}}_k+{\beta }_k{\mathbf{d}}_{k-1},{\beta }_k=max(0,\frac{{\mathbf{g}}_k^T({\mathbf{g}}_k-{\mathbf{g}}_{k-1})}{{\mathbf{g}}_{k-1}^T{\mathbf{g}}_{k-1}})$$

Sentinel.conjugate_gradient! Function

conjugate_gradient!(material::Material, ctx::ForwardProblemContext;
                     max_iter::Int=50, tol::Float64=1e-6,
                     alpha_init::Float64=1.0,
                     cg_threshold::Float64=0.0,
                     cg_scalar_weights::Union{Matrix{Float64}, Nothing}=nothing,
                     iteration_offset::Int=0) -> (Material, ConvergenceHistory)

Run nonlinear conjugate gradient iterations to minimize the objective.

Arguments

• material: initial material estimate (modified in-place on convergence)

• ctx: forward problem context

• max_iter: maximum number of CG iterations

• tol: convergence tolerance on material_epsilon

• alpha_init: initial line search step size

• cg_threshold: if > 0, cap relative material change per step via cg_update_scale

• cg_scalar_weights: (numprop, 2) matrix of per-property CG scalar weights for gradient direction scaling. If provided, scales the gradient so the average step for each property equals weight * avg_property_value. Ports regind(7) from Fortran.

• iteration_offset: offset for regularization weight schedule evaluation

Returns

• (material, history): updated material and convergence history

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Gauss-Newton

Diagonal Hessian preconditioned descent: $\mathrm{\Delta }{\theta }_k=-g_k/H_{kk}$

Sentinel.gauss_newton! Function

gauss_newton!(material::Material, ctx::ForwardProblemContext;
               max_iter::Int=20, tol::Float64=1e-6,
               alpha_init::Float64=1.0,
               hessian_floor::Float64=1e-30,
               iteration_offset::Int=0) -> (Material, ConvergenceHistory)

Run Gauss-Newton iterations to minimize the objective.

Arguments

• material: initial material estimate (modified in-place)

• ctx: forward problem context (regularizations should include Hessian-contributing regs)

• max_iter: maximum iterations

• tol: convergence tolerance on material_epsilon

• alpha_init: initial line search step size

• hessian_floor: minimum diagonal value to prevent division by zero

• iteration_offset: offset for regularization weight schedules

Returns

• (material, history): updated material and convergence history

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L-BFGS

Limited-memory BFGS with two-loop recursion and initial scaling $\gamma ={\mathbf{s}}^T\mathbf{y}/({\mathbf{y}}^T\mathbf{y})$.

Sentinel.quasi_newton! Function

quasi_newton!(material::Material, ctx::ForwardProblemContext;
               max_iter::Int=50, tol::Float64=1e-6,
               memory::Int=5, alpha_init::Float64=1.0,
               iteration_offset::Int=0) -> (Material, ConvergenceHistory)

Run L-BFGS quasi-Newton iterations to minimize the objective.

Arguments

• material: initial material estimate (modified in-place)

• ctx: forward problem context

• max_iter: maximum iterations

• tol: convergence tolerance on material_epsilon

• memory: number of L-BFGS curvature pairs to store (default 5)

• alpha_init: initial line search step size

• iteration_offset: offset for regularization weight schedules

Returns

• (material, history): updated material and convergence history

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Line Search

Armijo backtracking: reduce $\alpha$ by factor $\rho$ until sufficient decrease $\mathrm{\Phi }(\theta +\alpha d)\le \mathrm{\Phi }(\theta )+c_1\alpha g^{T}d$.

Sentinel.line_search Function

line_search(ctx::ForwardProblemContext, material::Material,
            grad::MaterialGradient, direction::AbstractVector{Float64},
            obj_current::Float64;
            alpha_init::Float64=1.0, c1::Float64=1e-4,
            rho::Float64=0.5, max_iter::Int=20,
            iteration::Int=1) -> (Float64, Float64, Material)

Backtracking line search with Armijo sufficient decrease condition.

Arguments

• ctx: forward problem context

• material: current material state

• grad: current gradient (must be initialized, with rvalue populated)

• direction: search direction vector (same length as grad.rvalue)

• obj_current: current objective value Φ(θ_k)

• alpha_init: initial step size (default 1.0)

• c1: sufficient decrease parameter (default 1e-4)

• rho: backtracking reduction factor (default 0.5)

• max_iter: maximum number of backtracking steps (default 20)

• iteration: current optimizer iteration (for regularization weight schedules)

Returns

• (alpha, obj_new, mat_new): step size, new objective value, updated material If no decrease is found, returns (0.0, obj_current, copy_material(material)).

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