Complex Geometric Optics for Inverse Problems

• Complex Geometric Optics is a method that constructs complex exponential solutions to probe and uniquely determine unknown interior coefficients from boundary measurements.
• It leverages advanced techniques such as Carleman estimates and integral identities to achieve stability and uniqueness in inverse boundary value problems.
• The method extends across various PDE classes—including elliptic, parabolic, hyperbolic, and nonlinear systems—offering critical insights for applications like medical imaging and geophysical exploration.

The Complex Geometric Optics (CGO) solution method is a central analytic technique in the theory of inverse boundary value problems (IBVPs) for partial differential equations (PDEs) of elliptic, parabolic, and hyperbolic type. It facilitates unique identifiability and stability results for unknown coefficients (potentials, conductivities, vector fields, or higher-order parameters) from boundary or partial data by constructing special classes of solutions with prescribed exponential and oscillatory behavior. These solutions are designed to probe the interior of the domain and extract Fourier or microlocal information about the sought-after coefficients. The method originated in the study of Calderón’s problem and Sylvester–Uhlmann’s work for the Schrödinger equation, and has since been extended and adapted to a wide array of settings, including higher-order PDEs, systems (Maxwell, elasticity), time-dependent problems, and nonlinear/quasilinear regimes.

1. Core Structure of the CGO Method

CGO solutions are special complex-valued solutions to PDEs of the form $L_q u = 0$ where $L_q$ is a linear or quasilinear operator depending on an unknown coefficient $q$ (or a tuple of coefficients). A canonical CGO ansatz adopts the form

$$u(x) = e^{\zeta \cdot x}(1 + r(x)),\qquad \zeta \in \mathbb{C}^n,\,\, \zeta\cdot \zeta=0,$$

where the phase $\zeta$ is chosen so that the exponential solves the principal part of the operator (e.g., for $-\Delta$, this means harmonicity in the complexified sense), and $r$ is a corrector term whose norm can be controlled for $|\Im\,\zeta|$ large. Analogous constructions appear in parabolic/heat problems (with weights $e^{\phi(x,t)/h}$), in time-dependent hyperbolic settings (with phases designed to focus on bicharacteristics), and for variable-coefficient or nonlinear operators via suitable microlocal weights and Carleman estimates.

The method comprises the following essential steps:

1. CGO Construction: Building $u$ so that $L_q u=0$ approximately, with errors controlled.
2. Dual Solution or Adjoint: Constructing a similar solution for $L_q^* v=0$ (or the appropriate adjoint).
3. Integral Identity: Deriving a bilinear (or multilinear) identity, typically of the form

$$\int_\Omega (q_1 - q_2)u_1 u_2\, dx = \text{boundary terms or zero},$$

by equating boundary or Cauchy data for two operators.

1. Asymptotic/Fourier Limit: Choosing the phases so that the product $u_1 u_2$ recovers a plane wave or oscillates at a prescribed frequency, extracting the Fourier transform of the coefficient difference.
2. Uniqueness/Stability Extraction: Using limiting arguments (Riemann–Lebesgue lemma, stationary phase) and unique continuation/gauge arguments to deduce injectivity, stability, or (in systems) constraint relations.

2. CGO in Classical Elliptic Inverse Problems

The CGO method was first exploited via the Sylvester–Uhlmann construction in the multidimensional Calderón problem:

• Schrödinger case: For $(\Delta + q)u = 0$ in $\Omega\subset\mathbb{R}^n$, CGO solutions are $u(x)=e^{\zeta\cdot x}(1+o(1))$ for complex $\zeta$ with $\zeta\cdot\zeta=0$ and $|\Im\,\zeta|\gg1$.
• Given full Dirichlet–to–Neumann data, inserting pairwise CGOs into the volume integral identity yields

$$\widehat{q_1-q_2}(\xi) = \lim_{|\zeta|\to \infty} \int (q_1 - q_2)u_1u_2\, dx$$

for $\xi = \zeta_1 + \zeta_2 \in \mathbb{R}^n$, facilitating pointwise recovery of the Fourier transform of $q_1 - q_2$ (Isakov et al., 2013, Liu et al., 20 Dec 2025).

In two dimensions, the method is adapted to complex-analytic Carleman weights (e.g., holomorphic phases), with further regularity reduction possible: uniqueness is achieved down to $q\in L^p$, $p>2$, via $\bar\partial$-equations and Cauchy-integral corrections (Imanuvilov et al., 2012, Blåsten et al., 2017).

When applied to the magnetic Schrödinger operator, an additional gauge freedom appears, with recovery of $\nabla\times A$ and $q$ up to the natural invariance, using partial data and boundary-reflection arguments (Pohjola, 2012).

3. Extensions to Higher-Order and Time-Dependent Operators

For higher-order equations like the biharmonic or fourth-order Schrödinger operator, CGO solutions of the form $u(x)=e^{i\theta\cdot x}(1+p(x))$ are constructed, with $\theta\in\mathbb{C}^n$, $\theta\cdot\theta=0$. The corrector $p$ is solved via iterative or Neumann-type approaches, with precise Sobolev bounds. Volume or Cauchy data comparisons remain the backbone for extracting stability and uniqueness, with the limiting process yielding sharp (logarithmic or Hölder) stability exponents (Liu et al., 20 Dec 2025).

In parabolic (e.g., heat) and time-dependent convection–diffusion equations, the approach is adapted by using weightings $e^{\phi(x,t)/h}$, with phases $\phi$ (limiting Carleman weights) solving eikonal equations appropriate for the operator (e.g., $|\nabla\phi|^2 = \partial_t \phi$ for the heat equation). Carleman estimates provide the core analytic foundation for existence and approximation of such solutions, and the associated integral identities allow identification of time-dependent and even anisotropic coefficients (Feizmohammadi, 2022, Kumar et al., 25 Jul 2024).

4. CGO for Systems and Nonlinear PDEs

For hyperbolic systems (e.g., Maxwell, elasticity), the method is generalized using vectorial or tensorial CGOs, Gaussian beams, and, in nonlinear settings, high-order linearizations:

• For quasilinear wave and elastic equations, one builds iterative corrections, encoding higher-order interactions (trilinear, quartic, etc.) via multilinear integral identities involving combinations of CGO or Gaussian beam solutions. This enables the unique determination of nonlinear material parameters, such as the full Taylor expansion of analytic nonlinearities or all quadratic and cubic elastic constants (Nakamura et al., 2017, Uhlmann et al., 2019, Uhlmann et al., 2022).
• Microlocal analysis, interaction geometry (e.g., focusing at interior points), and the precise calculation of principal symbols are crucial in handling these higher-order inverse problems.

5. Role of Carleman Estimates and Integral Identities

Carleman estimates are foundational to the CGO method, providing:

• The large-parameter (semiclassical) weighted inequalities necessary for constructing CGO solutions with controlled norms and error terms, even in the presence of lower regularity or partial boundary data.
• Unique continuation and injectivity properties crucial for reduction of the inverse problem to identification via Fourier or geodesic ray transforms.
• Extension to partial data scenarios via reflection techniques and limiting weights.

The boundary or volume integral identities—typically exploited via stationarity (stationary phase, Riemann–Lebesgue lemma)—are then used to isolate the impact of the difference of coefficients within the class of available CGO solutions.

6. Gauge Invariance and Rigidity

In many settings (e.g., convection–diffusion, magnetic Schrödinger), the CGO-based identification is unique only up to a natural gauge:

• For convection-diffusion with inaccessible boundary, $(v,\rho)$ is only recovered up to the gauge $$(v+\nabla\psi, \rho+\partial_t\psi+v\cdot\nabla\psi)$$ with $\psi|_{\partial\Omega}=0$ (Kumar et al., 25 Jul 2024).
• For the magnetic Schrödinger operator, only $\nabla\times A$ and the electric potential $q$ are determined, modulo gauge transformations vanishing on the accessible boundary (Pohjola, 2012).
• The method makes these ambiguities explicit and characterizes the full set of obstructions to uniqueness.

7. Advances, Stability, and Extensions

• Refinements and quantitative variants of CGO arguments yield sharp stability estimates, delineating the boundary between logarithmic and Hölder stability as a function of a priori regularity, frequency, or data regime (Isakov et al., 2013, Liu et al., 20 Dec 2025, Kow et al., 2021).
• The method underpins numerical algorithms through linearization and one-step reconstructions, especially in the context of Fréchet differentiability for nonlinear PDEs (e.g., $p$-Laplace, nonlinear elasticity) (Hannukainen et al., 2018, Chapko et al., 2019).
• Multi-level and compressed sensing approaches leverage the CGO-based stability framework for effective piecewise constant or hierarchical coefficient reconstructions in computational settings (Beretta et al., 2014).

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Overall, the Complex Geometric Optics solution method is an indispensable analytic and microlocal tool for addressing uniqueness, stability, and reconstruction in inverse boundary value problems for a broad spectrum of linear and nonlinear PDEs, with its flexibility enabling adaptation to diverse operator classes, dimensions, data regimes, and geometric constraints (Isakov et al., 2013, Kumar et al., 25 Jul 2024, Liu et al., 20 Dec 2025, Feizmohammadi, 2022, Nakamura et al., 2017, Uhlmann et al., 2019, Pohjola, 2012, Blåsten et al., 2017, Imanuvilov et al., 2012, Uhlmann et al., 2022).