Coupled Complex Boundary Method (CCBM)

• CCBM is a reformulation strategy that encodes overdetermined Dirichlet and Neumann data into a single complex Robin-type boundary condition for solving inverse and free boundary problems.
• It employs optimization, adjoint-based sensitivity analysis, and numerical regularization to recover shapes, coefficients, and source supports across various PDE settings.
• The method has demonstrated improved stability and computational efficiency in applications such as Bernoulli problems, Stokes flows, and bioluminescence tomography.

The Coupled Complex Boundary Method (CCBM) is a reformulation strategy for inverse Cauchy problems, free boundary problems, and geometric inverse problems in which overdetermined Dirichlet and Neumann data are encoded into a single complex Robin-type boundary condition for a complex-valued state. First put forward by Cheng et al. (2014) to deal with inverse source problems and subsequently developed in shape optimization and coefficient-identification settings, CCBM replaces boundary overdeterminedness by the requirement that the imaginary part of the complex solution vanish in the domain, and then solves the resulting problem by optimization, adjoint-based sensitivity analysis, and numerical regularization (Rabago, 2022, Rabago et al., 2023, Rabago et al., 2024, Cherrat et al., 23 Aug 2025, Gong et al., 2022, Nainggolan et al., 2 Feb 2026, Wu et al., 11 Sep 2025, Essahraoui et al., 12 May 2026).

1. Conceptual principle and historical development

The central idea of CCBM is to transform an overdetermined boundary value problem into a single complex boundary value problem with a complex Robin boundary condition coupling Dirichlet and Neumann data. In the exterior Bernoulli setting, this was introduced as a shape optimization framework in which the free boundary is identified by optimizing a cost function constructed from the imaginary part of the solution in the whole domain (Rabago, 2022). In inverse Cauchy problems for elliptic equations, the same principle was used to combine measured data on the accessible boundary and missing data on the inaccessible boundary into a single complex formulation, leading to an operator equation suitable for iterative regularization (Gong et al., 2022).

In this formulation, one writes the state as $u=u_1+i u_2$ or $w=w_r+i\,\Im\{w\}$, depending on the application. The real part carries one component of the physical boundary information and the imaginary part carries the other. If the geometry, source, or coefficient is correct, the imaginary component vanishes in the domain, so reconstruction reduces to minimizing a volumetric misfit built from that imaginary part. In the 2022 Bernoulli paper, this was described as shifting the least-squares criterion from the boundary to the interior (Rabago, 2022).

Subsequent work broadened the method across PDE classes. The same complex-coupling mechanism was used for Stokes free boundary problems and Stokes obstacle identification, for advection–diffusion obstacle recovery from a single pair of Cauchy data, for diffusion coefficient reconstruction, for bioluminescence tomography, and for cavity reconstruction under a homogeneous Robin condition (Rabago et al., 2023, Rabago et al., 2024, Cherrat et al., 23 Aug 2025, Nainggolan et al., 2 Feb 2026, Wu et al., 11 Sep 2025, Essahraoui et al., 12 May 2026). This suggests a unifying CCBM template: encode the available Cauchy data into one complex boundary condition, solve a well-posed complex PDE, and identify the unknown by enforcing vanishing imaginary part.

2. Canonical mathematical formulations

Across the cited literature, CCBM appears in several PDE settings, but the complex coupling pattern remains stable.

Setting	Complex boundary formulation	Primary unknown
Exterior Bernoulli problem (Rabago, 2022)	$\partial_n u + i u = \lambda$ on the free boundary $\Sigma$	Free boundary
Stokes free boundary problem (Rabago et al., 2023)	$-pn + \alpha u + i (u\cdot n)n = 0$ on $\Sigma$	Free boundary
Stokes obstacle identification (Rabago et al., 2024)	$\sigma(u,p)n + i u = g + i f$ on the accessible boundary $\Sigma$	Obstacle shape
Advection–diffusion obstacle recovery (Cherrat et al., 23 Aug 2025)	$\sigma\,\partial_n u + i u = g + i f$ on $\Sigma$, $w=w_r+i\,\Im\{w\}$0 on $w=w_r+i\,\Im\{w\}$1	Inclusion shape
Diffusion coefficient recovery (Nainggolan et al., 2 Feb 2026)	$w=w_r+i\,\Im\{w\}$2 on $w=w_r+i\,\Im\{w\}$3	Diffusion coefficient
Bioluminescence tomography (Wu et al., 11 Sep 2025)	$w=w_r+i\,\Im\{w\}$4 on $w=w_r+i\,\Im\{w\}$5	Source support and intensity
Cavity reconstruction with homogeneous Robin condition (Essahraoui et al., 12 May 2026)	$w=w_r+i\,\Im\{w\}$6 on $w=w_r+i\,\Im\{w\}$7, $w=w_r+i\,\Im\{w\}$8 on $w=w_r+i\,\Im\{w\}$9	Unknown interior boundary

For the advection–diffusion problem, the forward model is

$\partial_n u + i u = \lambda$0

with $\partial_n u + i u = \lambda$1 on the unknown boundary $\partial_n u + i u = \lambda$2 and $\partial_n u + i u = \lambda$3 on the accessible boundary $\partial_n u + i u = \lambda$4. Decomposition $\partial_n u + i u = \lambda$5 yields two coupled real PDEs, and the inverse problem is equivalent to enforcing $\partial_n u + i u = \lambda$6 in $\partial_n u + i u = \lambda$7 (Cherrat et al., 23 Aug 2025).

For the Stokes obstacle problem, CCBM uses a complex-valued velocity and pressure, with the complex Robin-like condition $\partial_n u + i u = \lambda$8 on the accessible boundary and $\partial_n u + i u = \lambda$9 on the obstacle boundary. The obstacle is recovered by minimizing a cost functional based on the imaginary parts of the velocity and pressure over the whole domain (Rabago et al., 2024). For free boundary Stokes flow, the overdetermined stress-free and no-penetration conditions are merged into $\Sigma$0, again with recovery achieved by forcing the imaginary components to vanish (Rabago et al., 2023).

In BLT, the complex coupling is parameter-dependent: $\Sigma$1 and the support $\Sigma$2 of the source is sought together with its intensity $\Sigma$3 (Wu et al., 11 Sep 2025). In diffusion coefficient recovery, the unknown is no longer a shape but the spatially varying coefficient $\Sigma$4, yet the same principle is retained: solve a complexified PDE and reconstruct $\Sigma$5 by driving $\Sigma$6 toward zero (Nainggolan et al., 2 Feb 2026).

3. Cost functionals, adjoints, and shape calculus

The defining CCBM objective is a domain integral built from the imaginary part of the complex state. In the exterior Bernoulli problem and in advection–diffusion obstacle recovery, the standard choice is

$\Sigma$7

and minimizing $\Sigma$8 over admissible shapes yields the sought domain (Rabago, 2022, Cherrat et al., 23 Aug 2025). In the Stokes free boundary problem, the analogue is

$\Sigma$9

where $-pn + \alpha u + i (u\cdot n)n = 0$0 and $-pn + \alpha u + i (u\cdot n)n = 0$1 are the imaginary parts of velocity and pressure (Rabago et al., 2023). In obstacle identification for Stokes flow, the cost is

$-pn + \alpha u + i (u\cdot n)n = 0$2

and $-pn + \alpha u + i (u\cdot n)n = 0$3 precisely when the current shape matches the true obstacle (Rabago et al., 2024).

These functionals are paired with adjoint equations and boundary representations of the shape derivative. For advection–diffusion, the first-order shape derivative in direction $-pn + \alpha u + i (u\cdot n)n = 0$4 is

$-pn + \alpha u + i (u\cdot n)n = 0$5

where the adjoint $-pn + \alpha u + i (u\cdot n)n = 0$6 solves

$-pn + \alpha u + i (u\cdot n)n = 0$7

with $-pn + \alpha u + i (u\cdot n)n = 0$8 on $-pn + \alpha u + i (u\cdot n)n = 0$9 and $\Sigma$0 on $\Sigma$1 (Cherrat et al., 23 Aug 2025).

For the Stokes obstacle problem, the rearrangement method gives the boundary form

$\Sigma$2

with $\Sigma$3 the adjoint state (Rabago et al., 2024). For cavity reconstruction with homogeneous Robin condition, the derivative is written as

$\Sigma$4

where

$\Sigma$5

and $\Sigma$6 is the mean curvature (Essahraoui et al., 12 May 2026).

The Bernoulli analysis goes further: it proves the existence of the shape derivative of the complex state, computes the shape gradient of the cost functional, characterizes its shape Hessian at the optimal domain under a strong and then a mild regularity assumption, and proves compactness of the latter expression (Rabago, 2022). The associated Riesz operator is compact in $\Sigma$7, not coercive in $\Sigma$8, which the paper interprets as instability typical of free boundary inverse problems (Rabago, 2022).

4. Numerical algorithms and regularization mechanisms

A recurrent numerical component in CCBM is Sobolev-gradient descent. Direct use of the boundary shape gradient can damage mesh quality, so several papers replace the raw boundary density by an $\Sigma$9-type Riesz representative. In advection–diffusion, the regularized deformation field $\sigma(u,p)n + i u = g + i f$0 is obtained from

$\sigma(u,p)n + i u = g + i f$1

and the domain is updated by $\sigma(u,p)n + i u = g + i f$2 (Cherrat et al., 23 Aug 2025). Closely related $\sigma(u,p)n + i u = g + i f$3-type systems appear in the Bernoulli, Stokes, and Robin-cavity papers, where finite elements are used for the state, adjoint, and gradient equations, and the stepsize is chosen by Armijo or adaptive line search (Rabago, 2022, Rabago et al., 2023, Rabago et al., 2024, Essahraoui et al., 12 May 2026).

Two later developments introduce explicit constraint handling through ADMM. In advection–diffusion shape recovery, an auxiliary variable $\sigma(u,p)n + i u = g + i f$4 enforces bounds $\sigma(u,p)n + i u = g + i f$5 on the real part of the state, and the augmented Lagrangian

$\sigma(u,p)n + i u = g + i f$6

leads to alternating updates of the shape, the projected auxiliary variable, and the multiplier (Cherrat et al., 23 Aug 2025). In cavity reconstruction with a homogeneous Robin condition, the same strategy is applied to $\sigma(u,p)n + i u = g + i f$7 through the constraint $\sigma(u,p)n + i u = g + i f$8, with a projection step

$\sigma(u,p)n + i u = g + i f$9

followed by the multiplier update $\Sigma$0 (Essahraoui et al., 12 May 2026). The advection–diffusion paper also compares exact and partial shape gradients and reports that partial gradient selection can improve efficiency (Cherrat et al., 23 Aug 2025).

CCBM has also been used outside shape descent. In inverse Cauchy problems, the complex reformulation yields an operator equation $\Sigma$1, with $\Sigma$2, and this is solved by Golub–Kahan bidiagonalization with Givens rotation under a discrepancy-principle stopping rule. The resulting CCBM-GKB method is analyzed as an iterative regularization scheme (Gong et al., 2022).

For coefficient recovery, a modified CCBM introduces a gradient-weighted $\Sigma$3-type term in the misfit,

$\Sigma$4

together with Tikhonov regularization and Sobolev-gradient descent. A projection-based extension and a pick-a-point strategy are then used for piecewise-constant coefficients (Nainggolan et al., 2 Feb 2026). In BLT, the method is embedded in a perimeter- and volume-regularized level-set framework, and first-order optimality makes the source intensity explicit through $\Sigma$5, reducing the algorithm to a shape-only optimization problem (Wu et al., 11 Sep 2025).

5. Applications and reported empirical behavior

The Bernoulli study reports successful reconstructions in two and three spatial dimensions. In 2D, concentric circles, L-shape, and ribbon domains were examined; in 3D, the tests included a perturbed sphere, torus, and four disjoint spheres. The paper states that CCBM converges robustly to exact solutions, sometimes outperforming the Kohn–Vogelius method for coarse meshes and in computational time per iteration, and that the moving-mesh strategy avoided remeshing (Rabago, 2022).

For Stokes free boundary problems, the 2D experiments showed that CCBM and classical tracking Dirichlet data converge to essentially the same optimal shape and display similar cost and gradient norm decay. In 3D, CCBM yielded smoother and more stable optimal surfaces, better preserved topology, and showed greater robustness and faster convergence for the same computational cost (Rabago et al., 2023). For immersed-obstacle detection in Stokes flow, the method reconstructed convex, nonconvex, sharply concave, and cornered obstacles in both two and three dimensions, remained robust under Gaussian noise up to 30%, and did so without perimeter or volume functional penalization (Rabago et al., 2024).

In advection–diffusion shape recovery, standard CCBM was reported to faithfully recover convex and mildly non-convex shapes in noise-free scenarios, while struggling with concavities under noisy data and under the combined effects of advection and diffusion. The ADMM-modified CCBM substantially improved reconstruction for non-convex and concave shapes, worked without explicit perimeter regularization, and was described as robust to moderate noise up to 5–7% Gaussian noise in the reported tests. Cost functional histories showed monotonic decrease and stabilization, with gradient norms confirming convergence (Cherrat et al., 23 Aug 2025).

For diffusion coefficient recovery, the modified CCBM with the weighted $\Sigma$6-type misfit was observed to yield more stable reconstructions and to reduce certain high-frequency artifacts when the $\Sigma$7-weight was sufficiently large but not excessive. Across the reported numerical scenarios, it often showed favorable stability and robustness relative to several classical boundary-based formulations, and the projection-based extension supported stable recovery of piecewise-constant diffusion coefficients when all subdomains shared a portion of the boundary (Nainggolan et al., 2 Feb 2026).

In BLT, the parameter-dependent CCBM was combined with perimeter and volume regularizations and a level-set representation. The reported 2D experiments included smooth, non-convex, polygonal, closely situated, and nested sources. The method reconstructed both support and intensity, handled multiple and nested sources, and achieved lower area and intensity errors than the comparison method labeled “G.Z.”, while remaining robust under noisy boundary data (Wu et al., 11 Sep 2025). In cavity reconstruction with a homogeneous Robin condition, unconstrained CCBM gave good recoveries for smooth or convex cavities and large $\Sigma$8, whereas the ADMM-constrained version reduced sensitivity to initialization and noise and improved reconstructions for challenging concave geometries (Essahraoui et al., 12 May 2026).

For inverse Cauchy completion, the CCBM-GKB algorithm was proved to be a regularization method and was described numerically as much faster than the classic Landweber method. The numerical discussion states that the required number of iterations for a desired error is an order of magnitude smaller for CCBM-GKB than for CCBM-Landweber (Gong et al., 2022).

6. Limitations, interpretive issues, and related terminology

The published CCBM literature does not claim that the complex reformulation removes ill-posedness. In the Bernoulli analysis, compactness of the shape Hessian is proved and linked directly to instability typical of free boundary inverse problems (Rabago, 2022). In cavity reconstruction with a homogeneous Robin condition, it is stated explicitly that a single measurement may correspond to infinitely many admissible domains, even though shape optimization can still yield reasonable reconstructions (Essahraoui et al., 12 May 2026). A plausible implication is that CCBM regularizes the computational treatment of such problems without converting them into uniquely solvable inverse maps.

Several limitations recur across applications. In the Stokes obstacle paper, reconstructions degrade for very small obstacles far from the boundary or for multiple distant inclusions, and the Lagrangian boundary-deformation approach cannot change topology unless one introduces tools such as topological gradients (Rabago et al., 2024). By contrast, the BLT framework uses a level-set representation and therefore naturally accommodates topological changes such as splitting, merging, and nesting (Wu et al., 11 Sep 2025). In diffusion coefficient recovery, performance is explicitly problem- and parameter-dependent, and excessive $\Sigma$9-weights may oversmooth or slow convergence (Nainggolan et al., 2 Feb 2026). In the Robin-cavity and BLT settings, the efficacy of the method depends on algorithmic parameter tuning and on the richness of the boundary data; BLT further links the coupling parameter and Tikhonov parameter through the condition $\sigma\,\partial_n u + i u = g + i f$0 to obtain uniform boundedness of the reconstructed intensity (Essahraoui et al., 12 May 2026, Wu et al., 11 Sep 2025).

The Stokes free boundary paper notes additional practical issues: CCBM requires a PDE solver that can handle complex-valued systems efficiently, and the relation between the imaginary part and the physical quantities is indirect, which can make diagnostics less intuitive (Rabago et al., 2023). These are method-intrinsic consequences of replacing an overdetermined real problem by a single complex one.

A nomenclature issue also arises. In the inverse-problem literature discussed here, CCBM denotes the Coupled Complex Boundary Method. This should not be conflated with the distinct “complex-boundary treatment method” for cut-Cartesian finite-volume schemes (Qin, 2023), nor with the “fully coupled-channel complex scaling method” used for the $\sigma\,\partial_n u + i u = g + i f$1 system in few-body quantum physics (Doté et al., 2017). The shared use of “complex boundary” or “coupled complex” language does not indicate methodological identity.