Optimization-Based Inversion Procedure

• Optimization-based inversion is a computational framework that recovers unknown parameters from noisy, indirect measurements by minimizing a composite objective function.
• It combines data-fidelity terms with regularization methods such as Tikhonov and total variation to enhance stability in high-dimensional, ill-posed problems.
• The approach employs diverse algorithms—including first-order, second-order, and stochastic solvers—to efficiently tackle nonlinear and multi-objective inversion challenges.

An optimization-based inversion procedure refers to the computational framework in which unknown parameters or models are recovered from indirect, noisy, or incomplete measurements by solving a suitably posed optimization problem. Modern approaches span convex and nonconvex formulations, single- and multi-objective regimes, stochastic or deterministic algorithms, and integrate regularization, parameterization, or structure-promoting constraints for stability and fidelity in high-dimensional and ill-posed problems.

1. Mathematical Structure and Objective Formulation

Optimization-based inversion is typically formulated as minimizing a composite objective consisting of a data-fidelity term and a regularizer or constraint set. For discrete model parameters $x$, data $b$, and forward operator $A(x)$:

$\min_{x}\; \phi(A(x),b) + \alpha\,R(x)$

where $\phi$ measures the mismatch (e.g., $\ell_2$-norm for Gaussian noise), and $R(x)$ enforces prior information via, e.g., total variation, sparsity, or smoothness (Ye et al., 2017). Constrained forms (e.g., Morozov principle) and Bayesian (MAP) interpretations are employed in regularization-dominated settings.

Multidimensional or multiobjective inversions (e.g., seismic tomography) frequently adopt a Pareto approach, seeking the set of non-dominated solutions with respect to several objective criteria, such as the data misfit and a smoothness or complexity penalty (Silva et al., 2022):

$F(s) = [\Phi(s),\;\mathcal R(s)]$

2. Algorithmic Families and Solvers

Optimization-based inversion encompasses a broad set of algorithms:

• First-order methods: Gradient descent, Nesterov acceleration, and their variants provide scalable but sometimes slow convergence. Step-sizes are set by Lipschitz constants or line-search (Armijo, Barzilai–Borwein) (Ye et al., 2017).
• Second-order/quasi-Newton schemes: Gauss–Newton is preferred in PDE-constrained nonlinear least squares, while L-BFGS and trust-region Newton methods are effective for large-scale or ill-conditioned inverse problems (Becker et al., 2015, Ye et al., 2017).
• Splitting and proximal methods: ADMM and FISTA handle composite objectives with nonsmooth penalties, such as $\ell_1$ or TV (Ye et al., 2017).
• Stochastic solvers: SGD, SVRG, and sub-sampled Newton methods are effective in large-data regimes (Ye et al., 2017).
• Evolutionary strategies: Genetic algorithms, e.g., NSGA II, are leveraged for Pareto optimization in multiobjective geophysical inversion, enabling simultaneous exploration of the misfit-smoothness trade-off with non-dominated sorting and diversity preservation (Silva et al., 2022).

Jacobian-free approaches, such as Broyden-updated Levenberg–Marquardt, circumvent explicit derivatives for inverse problems where analytic or finite-difference derivatives are computationally prohibitive (Piro et al., 2022).

3. Regularization and Parameterization

Regularization is essential for stabilization and physical plausibility:

• Tikhonov regularization: Penalizes solution norm or smoothness via quadratic forms, frequently integrated in the Moore–Penrose inversion framework (Anikin et al., 2024).
• Analytical regularization: For distributional inverse transforms, such as the Radon inversion, additional terms are appended for the control of the analytic distributional residue $f_A$, yielding modified normal equations (e.g. $A^*A + \alpha\,I + \beta\,L^*L$) and improved noise robustness (Anikin et al., 2024).
• Adaptive bases and spectral methods: Methods such as Adaptive Spectral Inversion (ASI) restrict updates to low-dimensional adaptive eigenspaces, enforcing built-in regularization without explicit penalties and improving accuracy for inverse medium problems (Gleichmann et al., 2023).
• Learned parameterization: Bilevel frameworks train regularization weights, norm exponents, and kernel hyperparameters directly from data, optimizing for empirical Bayes risk and leveraging Krylov-projection accelerated inner solvers for scalability (Chung et al., 2021).

4. Extension to Nonlinear and High-Dimensional Inverse Problems

Optimization-based inversion frameworks are extensible to complex forward models including nonlinear PDEs and dynamical systems. High-dimensionality is handled via variable projection (as in FWIME for full waveform inversion), which eliminates auxiliary variables and enables efficient adjoint-gradient computation that combines Born and tomographic components (Barnier et al., 2022).

For nonlinear inverse problems with expensive forward solvers, multilevel optimization schemes utilize hierarchies of discretizations, dynamically allocating computational resources and reducing total inversion cost by a logarithmic factor over single-level schemes (Weissmann et al., 2022).

Inverse graphical methods, relying on certified sublevel-set membership and box classification oracles, further enable global optimization under rigorous guarantees for moderate-dimensional design centering problems (Karpukhin, 2019).

5. Robustness, Convergence, and Practical Enhancements

Robust formulations manage noise and model mismatch:

• Outlier-robust misfits: Use of Huber, Student's $t$, and learned $p$-norm penalties buffer against outliers in the data-fidelity term and, in a bilevel framework, learn heavy-tailed distributions optimal for the uncertainty mixture (Becker et al., 2015, Chung et al., 2021).
• Cycle-skipping mitigation: Model-extension approaches such as FWIME expand the basin of attraction by explaining phase-shifted data with extended parameters and penalizing non-physical corrections (Barnier et al., 2022).
• Convergence: Angle conditions and Armijo-backtracking implement global and local convergence criteria, sometimes proving Zoutendijk-type results for gradient-norm decay (Gleichmann et al., 2023, Piro et al., 2022).

Empirical and cross-validation methodologies inform the selection of regularization weights ($\alpha, \beta$) and parameterization, e.g., via Morozov discrepancy, L-curves, or held-out Bayes risk minimization (Becker et al., 2015, Anikin et al., 2024, Chung et al., 2021).

6. Applications and Extensions

Optimization-based inversion is central across domains:

Domain	Key Features	Example Algorithms/Approaches
Seismic/geophysical imaging	PDE constraints, nonlinear forward, cycle-skipping	Gauss–Newton, FWIME, NSGA II, adjoint-state
Tomography and image reconstruction	Ill-posedness, distributed errors, regularization	Tikhonov, analytical regularization, TN, ADMM
Material and stress evaluation	Acoustoelastic relations, full waveform inversion	Adjoint-state, gradient-descent, regularization
Optimal design/inverse optimization	Parameter recovery, uncertainty quantification	Bi-level IO, conformal prediction, deep unrolled
High-dimensional learning-based inversion	Bilevel learning, Krylov subspaces	MM–GKS, genHyBR, learned RC parameters

Inverse optimization is a natural extension wherein observed decisions are used to recover the underlying objective, cost, or constraints, yielding LP/QP/MILP-based parametric identification as specified by optimality principles, duality, or KKT conditions (Chan et al., 2021, Lin et al., 2024). Deep inverse optimization unrolls interior-point solvers into computation graphs enabling backpropagation-based recovery of embedded weights and coefficients, connecting machine learning with traditional inversion (Tan et al., 2018).

Multilevel and distributed optimization frameworks, unrolled learned iterations, and plug-and-play prior incorporation represent current frontiers for flexibility and scalability in the inversion of physical and abstract systems (Ye et al., 2017, Chung et al., 2021, Gleichmann et al., 2023).

7. Limitations and Frontier Directions

• High-dimensional nonconvexity: Robustness deteriorates or computational cost explodes with severe nonlinearity or insufficient prior information; dimension-reduction or adaptive parametrization is critical (Gleichmann et al., 2023, Karpukhin, 2019).
• Oracle and model structure: Inverse graphical methods, adaptive bases, or learned kernels depend on efficient subproblem or oracle implementations.
• Data and benchmarking: Evaluation is sensitive to perceptual, semantic, and fidelity metrics; overemphasis on semantic alignment can conceal deficiencies in visual realism, as seen in training-free diffusion priors via optimization-based inversion (Dell'Erba et al., 25 Nov 2025).
• Calibration and uncertainty: Conformal prediction and robust IO provide finite-sample guarantees and reduce both actual and perceived optimality gaps (Lin et al., 2024).

As computational and mathematical tools evolve, optimization-based inversion continues to serve as the foundational paradigm for extracting information from noisy, indirect, or incomplete measurements, adapting seamlessly to varying model structures, data regimes, and application domains.