Inverse Solver

The inverse solver computes gradients of the objective function with respect to material parameters using the adjoint method. See the Mathematical Reference for the full derivation.

Objective Function

The data-misfit objective is $\mathrm{\Phi }(\mathit{\theta })=\frac{1}{2}\parallel {\mathbf{u}}_{\mathrm{calc}}-{\mathbf{u}}_{\mathrm{meas}}{\parallel }^2$, optionally augmented with regularization penalties.

Sentinel.compute_objective Function

compute_objective(calc::Displacement, meas::Displacement) -> Float64

Compute the displacement-only objective function value (L2 misfit): Φ = ½‖u_calc − u_meas‖²

This calls displacement_error! internally to compute the misfit and returns calc.abserror.

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compute_objective(calc::Displacement, meas::Displacement,
                  material::Material, regularizations;
                  iteration::Int=1) -> Float64

Compute the total objective function value including regularization penalties.

Stores displacement error in material.disperror, per-regularization errors in material.regerror, and the total in material.totalerror.

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Gradient Initialization

The gradient structure maps material parameters to a flat index vector for optimization. Each entry corresponds to a specific (property, real/imaginary, value index) triple.

Sentinel.init_gradient! Function

init_gradient!(grad::MaterialGradient, material::Material;
               realparamind::Union{AbstractMatrix{Bool}, Nothing}=nothing)

Initialize the MaterialGradient struct from a Material, building the index mapping from (property, real/imag, value_index) to a flat gradient vector offset.

Ports initmtrlgrad + propparamindices from FEmaterial.f90.

Arguments

• grad: MaterialGradient to initialize in-place

• material: Material struct with property definitions

• realparamind: optional (2*numprop, max_nvpp) Bool matrix selecting which parameter groups are active. nothing → all active.

Index layout

For each property iprop (1-based) and value index jprop (1-based):

• Row 2*(iprop-1)+1 → real part

• Row 2*(iprop-1)+2 → imaginary part

realprop2param[row, jprop] gives the starting 1-based index in grad.rvalue for the flattened gradient entries of that parameter group. realparam2prop[k, :] = [iprop, jprop, iii] is the reverse mapping (iii=1 for real, 2 for imaginary).

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Gradient Computation

The adjoint gradient is:

$$\frac{\partial \mathrm{\Phi }}{\partial {\theta }_m}=-\mathrm{Re}\left[\sum _e\sum _{\mathrm{gp}}{w}_{\mathrm{gp}}{\bar{\mathit{\lambda }}}_e^T\frac{\partial {\mathbf{K}}_e}{\partial {\theta }_m}{\mathbf{u}}_e\right]$$

Dispatched per material model type for model-specific $\partial \mathbf{K}/\partial \theta$ derivatives.

Sentinel.compute_gradient! Function

compute_gradient!(grad::MaterialGradient,
                  dh::DofHandler, cv_disp::CellValues,
                  calc::Displacement, adjnt::Displacement,
                  model::IsotropicIncompressible,
                  material::Material,
                  gp2mtrs::Vector{GP2Mtr},
                  meshes::Vector{MaterialMesh},
                  ω::Float64; numdispsets::Int=1)

Compute the adjoint gradient for the IsotropicIncompressible model (Model 1).

Ports isoadjointgrad_multimesh from FEgradient.f90 (lines 126–416).

The gradient formula for parameter θ_m at material mesh node m is:

∂Φ/∂θ_m = -Σ_dispsets Re[ Σ_elem Σ_gp  wght · (∂K/∂θ)_ij · λᵢ · uⱼ ]

where wght = basis_m (hex8 shape function at material mesh node m).

Three property derivatives for Model 1:

• ∂K/∂μ: deviatoric stiffness (9 terms per node pair)

• ∂K/∂ρ: mass matrix diagonal (-ω²·Nᵢ·Nⱼ)

• ∂K/∂κ: pressure-pressure block ((1/κ²)·ψ_p·ψ_q)

Arguments

• grad: initialized MaterialGradient (from init_gradient!)

• dh: DofHandler with vector field Lagrange{RefHexahedron, 2}()^3

• cv_disp: CellValues for scalar Lagrange{RefHexahedron, 2}()

• calc: calculated displacement (forward solution)

• adjnt: adjoint displacement

• model: IsotropicIncompressible() singleton

• material: Material with properties μ, ρ, κ

• gp2mtrs: vector of GP2Mtr (one per material mesh)

• meshes: vector of MaterialMesh

• ω: angular frequency [rad/s]

• numdispsets: number of displacement sets (default 1)

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Gradient Type

Sentinel.MaterialGradient Type

MaterialGradient

Stores the adjoint gradient of the objective function with respect to all real material parameter values. Corresponds to the Fortran mtrlgrad derived type.

Fields

• numprop: number of material properties

• npr: per-property count of real parameter points

• npi: per-property count of imaginary parameter points

• nvpp: per-property values per point

• numrealparam: total number of distinct real parameter groups (including real+imag parts)

• totnumrealparam: total flattened gradient vector length

• realprop2param: maps (2*iprop-1/2, ival) → starting index in rvalue

• realparam2prop: maps gradient index → (iprop, ival, 1=real/2=imag)

• nreg: number of regularization terms

• toterrinit: initial total error

• totalerror: complete error including regularization

• disperror: displacement misfit only

• regerror: per-regularization-term error, length nreg

• rvalue: gradient values, length totnumrealparam

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Full Inverse Problem

The full inverse solver orchestrates multi-zone inversion using Zone Decomposition to partition the domain and solve subproblems independently. See the Optimization API for ForwardProblemContext, evaluate_objective!, compute_full_gradient!, and material operation utilities.