Regularization

Sentinel provides 10 regularization methods in three categories: penalty methods (added to the objective), Hessian modifiers (modify the update direction), and post-processing methods (applied after each iteration).

See the Mathematical Reference for complete functional definitions.

Abstract Interface

Sentinel.AbstractRegularization Type

AbstractRegularization

Abstract supertype for all regularization methods.

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Sentinel.WeightSchedule Type

WeightSchedule

Linear interpolation of a regularization weight from start_weight to end_weight over iterations delay+1 through max_iter.

Before iteration delay, the weight is 0. After max_iter, it stays at end_weight.

Ports the weight ramp-up logic from the Fortran regularization modules.

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Sentinel.evaluate_weight Function

evaluate_weight(ws::WeightSchedule, iteration::Int) -> Float64

Evaluate the scheduled weight at the given iteration. Returns 0 for iterations ≤ delay, then linearly interpolates.

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Penalty Methods

Tikhonov

$R(\mathit{\theta })=\frac{1}{2}\alpha \parallel \mathit{\theta }-{\mathit{\theta }}_{\mathrm{ref}}{\parallel }^2$

Sentinel.TikhonovReg Type

TikhonovReg <: AbstractRegularization

Tikhonov (L2) regularization penalizing deviation from reference values.

Fields

• prop_index::Int: which material property (1-based) this applies to

• weight_real::WeightSchedule: weight schedule for real part

• weight_imag::WeightSchedule: weight schedule for imaginary part

• theta_ref_real::Vector{Float64}: (npr,) reference values for real part

• theta_ref_imag::Vector{Float64}: (npi,) reference values for imaginary part

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Total Variation

$R(\mathit{\theta })=\alpha \sum _e\sum _{\mathrm{gp}}{w}_{\mathrm{gp}}|J|\sqrt{|\mathrm{\nabla }\theta {|}^2+{\epsilon }^2}$

Sentinel.TotalVariationReg Type

TotalVariationReg <: AbstractRegularization

Total Variation regularization with smooth (Huber-like) approximation.

Fields

• prop_index::Int: which material property (1-based)

• weight_real::WeightSchedule: weight schedule for real part

• weight_imag::WeightSchedule: weight schedule for imaginary part

• epsilon::Float64: smoothing parameter δ for √(|∇θ|² + ε²)

• active::Bool: false on first iteration, true after update_weights!

• mtrl_mesh::MaterialMesh: hex8 material mesh for integration

• node_to_elements::Vector{Vector{Int}}: precomputed inverse connectivity

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Soft Prior

Region-based Laplacian penalty: $R(\mathit{\theta })=\alpha \sum _j(L\theta {)}_j^2$

Sentinel.SoftPriorReg Type

SoftPriorReg <: AbstractRegularization

Soft prior regularization using region-based Laplacian penalty.

Fields

• prop_index::Int: which material property (1-based)

• weight_real::WeightSchedule: weight schedule for real part

• weight_imag::WeightSchedule: weight schedule for imaginary part

• region_info::RegionInfo: region assignment for material mesh nodes

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Sentinel.RegionInfo Type

RegionInfo

Region assignment information for nodes in a material mesh. Used by soft prior regularization to penalize deviation within regions.

Fields

• node_region::Vector{Int}: (nn,) region ID per node (0 = no region)

• nregions::Int: number of distinct regions

• region_node_count::Vector{Int}: (nregions,) node count per region

• region_node_lists::Vector{Vector{Int}}: (nregions,) node lists per region

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Exterior Constraint

$R(\mathit{\theta })=\alpha \sum _k[max(0,{\theta }_k-{\theta }_{max}{)}^2+max(0,{\theta }_{min}-{\theta }_k{)}^2]$

Sentinel.ExteriorConstraintReg Type

ExteriorConstraintReg <: AbstractRegularization

Exterior constraint penalty that enforces parameter bounds.

Penalty is zero when parameters are within bounds and grows quadratically with violation magnitude. The scalar s = prop.scalar[1] (real) or prop.scalar[2] (imaginary) converts from stored normalized values to physical units before checking bounds.

Fields

• prop_index::Int: which material property (1-based)

• weight::Float64: penalty weight

• min_real::Float64: minimum allowed physical real value

• max_real::Float64: maximum allowed physical real value

• min_imag::Float64: minimum allowed physical imaginary value

• max_imag::Float64: maximum allowed physical imaginary value

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Hessian Modifiers

Joachimowicz

Scales the diagonal Hessian: $H_{kk}\leftarrow \overline{H}\cdot ϵ\cdot w$

Sentinel.JoachimowiczReg Type

JoachimowiczReg <: AbstractRegularization

Joachimowicz Hessian regularization.

Adds avg_diag · total_error · w to each diagonal entry of the Hessian for the specified property parameters.

Fields

• prop_index::Int: which material property (1-based)

• weight::WeightSchedule: weight schedule

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Marquardt

Normalized Levenberg-Marquardt damping: normalize $H$, add $\lambda I$, rescale solution.

Sentinel.MarquardtReg Type

MarquardtReg <: AbstractRegularization

Levenberg-Marquardt Hessian regularization with adaptive weight.

Fields

• prop_index::Int: which material property (1-based)

• current_weight::Float64: current λ value

• start_weight::Float64: initial λ value (for reset)

• min_weight::Float64: minimum λ value (floor)

• delta::Float64: reduction factor (< 1.0) applied on successful steps

• adjust_threshold::Float64: |1−α| threshold for adaptation

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Sentinel.normalize_and_regularize! Function

normalize_and_regularize!(H, grad_vec, reg::MarquardtReg, grad::MaterialGradient,
                           material::Material) -> Vector{Float64}

Perform the Marquardt normalization and regularization in-place:

1. Compute D_k = √|H[k,k]| for relevant parameters

2. Normalize: H[i,j] /= D_i·D_j, grad[k] /= D_k

3. Add λ to diagonal: H[k,k] += reg.current_weight

Returns the scaling vector D for post-solve rescaling.

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Sentinel.rescale_solution! Function

rescale_solution!(delta_theta, D, reg::MarquardtReg, grad::MaterialGradient,
                   material::Material)

Rescale the Marquardt solution: Δθ[k] /= D_k for relevant parameters.

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Post-Processing Methods

Van Houten

Clips power-law exponents to $[{\alpha }_{min},{\alpha }_{max}]$.

Sentinel.VanHoutenReg Type

VanHoutenReg <: AbstractRegularization

Van Houten post-processing regularization.

Clips power-law exponent values prop.rvalue[k, 2] and prop.ivalue[k, 2] to the range [-level, level] for properties with nvpp >= 3.

Fields

• level::Float64: maximum allowed power-law exponent magnitude

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Sentinel.apply_van_houten! Function

apply_van_houten!(material::Material, reg::VanHoutenReg)

Clip power-law exponents in all properties with nvpp >= 3 to [-level, level].

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Spatial Filter

Gaussian smoothing with widths $({\sigma }_x,{\sigma }_y,{\sigma }_z)$.

Sentinel.SpatialFilterReg Type

SpatialFilterReg <: AbstractRegularization

Spatial filter (Gaussian smoothing) post-processing regularization.

Fields

• prop_index::Int: which material property (1-based)

• sigma_real_init::Float64: initial Gaussian width for real part

• sigma_real_final::Float64: final Gaussian width for real part

• sigma_imag_init::Float64: initial Gaussian width for imaginary part

• sigma_imag_final::Float64: final Gaussian width for imaginary part

• max_iter::Int: total global iterations (for weight interpolation)

• const_reg_iters::Int: constant regularization iterations at end

• cutoff_multiplier::Float64: neighbor search radius = max_sigma × cutoff_multiplier

• mtrl_mesh::MaterialMesh: material mesh for coordinate lookup

• neighbor_indices::Vector{Vector{Int}}: precomputed neighbor lists per node

• neighbor_distances::Vector{Vector{Float64}}: precomputed distances per node

• region_info::Union{RegionInfo, Nothing}: optional region constraints

• sensitivity_mask::Union{BitVector, Nothing}: optional mask for GP-covered nodes (Fortran nodsense)

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Sentinel.apply_spatial_filter! Function

apply_spatial_filter!(material::Material, reg::SpatialFilterReg;
                      iteration::Int=1)

Apply Gaussian spatial smoothing to material property values: θ_filtered[i] = Σ_j w_j·θ_j / Σ_j w_j where w_j = exp(−d²/(2σ²)).

Sigma is interpolated between init and final values based on iteration.

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Bounds (McGarry)

Enforces Poisson ratio bounds via clipping.

Sentinel.BoundsReg Type

BoundsReg <: AbstractRegularization

McGarry bounds post-processing regularization.

Fields

• max_poisson::Float64: maximum allowed Poisson's ratio (e.g., 0.485) for Model 3

• property_bounds::Vector{PropertyBounds}: per-property min/max constraints

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Sentinel.apply_mcgarry_bounds! Function

apply_mcgarry_bounds!(material::Material, reg::BoundsReg)

Enforce material property bounds (post-processing, no gradient contribution).

1. General min/max clamping: for each constrained property, clamp scalar * rvalue to [min, max] (matching Fortran mcgarrybounds.f90).

2. Model 3 Poisson ratio: clamp λ so that ν ≤ ν_max.

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Force Balance

Physics-consistency penalty: the equilibrium residual of the measured displacement field under the current property estimate, $R(\mathit{\theta })=\frac{1}{2}w\parallel MK(\mathit{\theta }){\hat{u}}_{\mathrm{meas}}{\parallel }^2$ over non-Dirichlet displacement DOFs. Model 1 (isotropic incompressible) only.

Weights are configured as fractions of the initial data misfit (relative mode, the default for config-driven runs): the fraction is resolved to an absolute weight once at solve start via $w=\mathrm{frac}\cdot D_0/S_0$, which makes the same number portable across meshes and datasets. MGH calibration (benchmark/results/mgh_fb_calibration/): fractions ≤ 0.2 are gentle (data misfit at or slightly better than baseline, per-voxel correlation ≥ 0.99 against the unregularized reconstruction); ≥ 5 over-regularizes. Enabling force balance roughly doubles per-iteration cost (~2.05× on the MGH batched-GPU benchmark, down from 2.26× via exact affine line-search caches for the penalty and gradient — see fb_begin_line_search!); the dominant remaining cost is the per-iteration gradient evaluations, with bounds-clamped line-search trials falling back to full evaluation.

macOS: relative-weight resolution is a one-time iterative solve

Resolving the relative weight (w = frac · D0/S0) needs one global forward solve for D0, once at solve start. On macOS this uses a condensed, factorization-free iterative solve (seconds to a few minutes) instead of a direct factorization, which avoids an upstream Apple-libmalloc crash; the resolved value is accurate and Linux is unaffected (it keeps the faster direct solve). A slow ForceBalanceReg: relative weights resolved, D0/S0 = … step on macOS is the expected one-time cost — see macOS FB / xzone Crash.

Choosing the fraction — noise matters (ground-truth phantom arbitration, benchmark/results/fb_phantom_arbitration/): on equilibrium-consistent (clean / high-SNR / denoised) data, strong force balance genuinely improves accuracy (frac = 1.0: −21 % RMSE vs truth, better contrast recovery). On realistically noisy data it becomes a bias — the equilibrium pull drives the properties to "explain" measurement noise (frac = 1.0 at 2 % noise: +25 % RMSE, flattened contrast). Keep the default frac = 0.1 (neutral) for noisy in-vivo data; reserve larger fractions for high-SNR or denoised acquisitions.

Sentinel.ForceBalanceReg Type

ForceBalanceReg <: AbstractRegularization

Force-balance (equilibrium residual on measured displacements) regularization.

Fields

• weight::WeightSchedule: penalty weight schedule

• relative::Bool: when true, weight holds fractions of the natural scale D0/S0 (initial data misfit over raw equilibrium penalty at the initial material) rather than absolute weights. run_inverse_solve! resolves relative weights to absolute ones once at solve start (one extra global forward solve); a relative instance cannot be bound directly.

• bound context fields (set by bind_context!; untyped because ForwardProblemContext is defined after the regularization includes)

Construct with ForceBalanceReg(weight::WeightSchedule; relative=false) or ForceBalanceReg(w::Float64; relative=false); the context is bound later per zone.

The raw penalty ½‖M·K(θ)·û_meas‖² has force² units and its magnitude is mesh- and dataset-dependent (MGH calibration 2026-07-12: raw penalty 1.3e5 vs data misfit 5.2e-2 — a natural scale of ~4e-7). Relative mode makes the configured number portable: w = 0.2, relative = true means "an equilibrium penalty worth 20% of the initial data misfit", on any mesh and any dataset.

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Sentinel.bind_context! Function

bind_context!(reg::AbstractRegularization, ctx; kwargs...)

Bind a ForwardProblemContext to a regularization that needs FE-level data (mesh, DOFs, boundary conditions, measured displacements) beyond the Material passed to compute_penalty/add_gradient!. Called from solve_zone! after the context is built. No-op for most regularizations.

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bind_context!(regs, ctx; kwargs...)

Bind a context to every regularization in a collection (nothing allowed).

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bind_context!(reg::ForceBalanceReg, ctx;
              gradient_backend=nothing, ka_edata=nothing,
              ka_workspace=nothing)

Bind a zone's ForwardProblemContext to the regularizer so that compute_penalty/add_gradient! can assemble the equilibrium residual. Called from solve_zone!. Throws for unsupported material models.

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Sentinel.resolve_fb_weights! Function

resolve_fb_weights!(regularizations, ctx, material; K=nothing) -> Union{Float64, Nothing}

Resolve every relative-weighted ForceBalanceReg in regularizations to an absolute weight, anchored at the current (initial) material:

w_abs = w_frac · D0 / S0

where S0 = ½‖M·K(θ₀)·û_meas‖² is the raw equilibrium penalty (computed with a temporary weight-1 instance bound to the global ctx) and D0 is the data misfit of the forward solution at θ₀ (one global forward solve — the only nontrivial cost, comparable to a single global iteration's solve; pass a preallocated global stiffness via K to avoid an extra allocation).

On macOS (default), D0 is computed via a condensed, factorization-free iterative solve (_fb_d0_condensed_iterative) to avoid the xzone-malloc crash; the K preallocation is used only by the direct path (Linux, or macOS with SENTINEL_ALLOW_MACOS_FB_RELATIVE=1).

Returns the scale D0/S0, or nothing if no unresolved relative instance is present (in which case nothing is computed). Called by run_inverse_solve! before the first zone solve; safe to call directly for custom drivers.

MGH calibration (2026-07-12, 12-iter batched-GPU sweep): fractions ≤ 0.2 are gentle (data misfit slightly better than baseline, per-voxel corr ≥ 0.99 vs unregularized), 1.0 gave the best misfit with substantive reconstruction changes, and 5.0 over-regularized. See benchmark/results/mgh_fb_calibration/.

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Sentinel.fb_begin_line_search! Function

fb_begin_line_search!(regs, material, delta_theta, grad) -> nothing

Arm the line-search caches on every bound ForceBalanceReg in regs, for a search from material (θ0) along delta_theta (the direction apply_material_update! applies). Both FB terms are exact in the step α because the Model-1 element stiffness is affine in μ (K_uu = μ·G − ω²ρ·M; K_up, K_pp are μ-independent):

• penalty: ½w(c0 + 2α·c1 + α²·c2) from residuals at θ0 and θ0 + 1·d;

• gradient: g(α) = g0 + α·g_d, where g0 reuses the CG-point gradient (add_gradient! remembers its last full evaluation keyed by the μ values, and the CG driver always evaluates at the search base), so arming costs the two residual loops plus a single extra gradient evaluation.

Trial-point compute_penalty/add_gradient! then cost O(nμ)/O(nparams). Only activates when the direction touches property 1 (μ) exclusively — a direction with κ content breaks affinity (K_pp ∝ 1/κ) and leaves the cache off (full evaluation per trial, exact as before). Trials knocked off the cached line (e.g. by bounds clamping) are detected by an α-projection consistency guard and evaluated fully. Call fb_end_line_search! after the search commits.

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Sentinel.fb_end_line_search! Function

fb_end_line_search!(regs) -> nothing

Disarm the line-search quadratic cache on every ForceBalanceReg in regs (the buffers are kept for reuse by the next search).

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CG Update Scaling

Sentinel.cg_update_scale Function

cg_update_scale(alpha, material_orig, material_new, threshold) -> Float64

Scale the CG step size alpha so that the relative material change does not exceed threshold.

Computes eps = material_epsilon(orig, new), then:

• If eps > threshold: return alpha * threshold / eps

• Otherwise: return alpha unchanged

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Specification API

The high-level regularization spec types passed to InversionProblem (regularization = [...]). These are declarative descriptions that the solver lowers to the runtime regularizers documented above; they are distinct from the legacy *Reg classes.

Weights on TotalVariation, SpatialFilter, Tikhonov, Joachimowicz, SoftPrior, Marquardt, ExteriorConstraint, and CGResidualScaling may be per property: pass a vector aligned with props (e.g. weight = [1e-16, 2e-16]). Legacy .dat runfiles that set distinct weights per property now survive load_inversion and the TOML round trip without collapsing to the first property: per-property weights are preserved for every regularizer (no weight collapses). The one remaining scalar collapse is TotalVariation's delta (the Charbonnier smoothing epsilon), which the spec carries as a single value; a legacy runfile with per-property/complex delta collapses to the first value with a warning.

Sentinel.Tikhonov Type

Tikhonov(; props=:all, weight=1e-18)

Tikhonov (L2) penalty toward the initial property values (indicator 3).

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Sentinel.TotalVariation Type

TotalVariation(; props=:all, weight=5e-16, delta=1e-19)

Total-variation penalty (runfile indicator 1).

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Sentinel.SpatialFilter Type

SpatialFilter(; props=:all, sigma=0.003 => 0.0015)

Gaussian spatial filtering of the property update (post-processing smoothing, runfile indicator 2). sigma is the Gaussian width in meters, not a penalty weight.

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Sentinel.Marquardt Type

Marquardt(; props=:all, start_weight=100.0, min_weight=1e-11,
            delta=0.5, adjust_threshold=0.25)

Marquardt adaptive damping (indicator 4).

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Sentinel.Joachimowicz Type

Joachimowicz(; props=:all, weight=5.0)

Joachimowicz parameter-magnitude penalty (indicator 5).

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Sentinel.ExteriorConstraint Type

ExteriorConstraint(; props=:all, weight=1e-14, bounds=nothing)

Soft penalty for property values outside bounds (indicator 6). bounds is one (min_re, max_re, min_im, max_im) tuple per property; nothing uses the standard EMPIRE bounds (3-property layout only).

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Sentinel.CGResidualScaling Type

CGResidualScaling(; weight=0.04 => 0.01)

Conjugate-gradient residual scaling (indicator 7). Applies to all properties.

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Sentinel.VanHouten Type

VanHouten(level=1.2)

Van Houten update-limiting post-processing (indicator 8).

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Sentinel.McGarryBounds Type

McGarryBounds(; bounds=nothing)

Hard clamping of property values to bounds after each iteration (indicator 9). Bound semantics as in ExteriorConstraint; when both are present they must agree (the runfile format shares one bounds table).

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Sentinel.SoftPrior Type

SoftPrior(; props=:all, weight=1e-11, delay=0)

Region-based soft-prior penalty (indicator 10).

Warning

Not yet wired into the solver: nothing reads the region stack into a RegionInfo, so InversionProblem warns and the term is ignored at setup. The type exists so the TOML schema is stable for when the loader lands.

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Sentinel.ForceBalance Type

ForceBalance(; weight=0.1, relative=true)

Force-balance (equilibrium residual on the measured displacement field) penalty — a physics-consistency term for Model-1 (isotropic incompressible) reconstructions. It has no Fortran indicator slot; it is a Julia-side extension carried alongside the runfile regularizers.

In relative mode (the default) weight is a fraction of the initial data misfit D0/S0, resolved to an absolute weight once at solve start — the same number is portable across meshes and datasets. With relative=false it is an absolute weight (mesh- and dataset-dependent). weight may be a scalar or an init => final ramp.

Calibrated guidance (benchmark/results/mgh_fb_calibration/, benchmark/results/fb_phantom_arbitration/): weight ≤ 0.2 is gentle and safe on noisy in-vivo data; larger fractions improve accuracy only on clean/high-SNR/denoised data.

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