Inverse Boundary Value Problem

• Inverse boundary value problems are mathematical frameworks that recover unknown coefficients in PDEs using boundary maps such as Dirichlet-to-Neumann operators.
• Analytical tools like CGO solutions, Alessandrini's identity, and Carleman estimates play a central role in proving uniqueness, stability, and regularity results.
• Numerical reconstruction strategies, including iterative Newton methods and regularization techniques, enable practical recovery of parameters in applications ranging from medical imaging to geophysical exploration.

An inverse boundary value problem (IBVP) is a class of mathematical problem in which the objective is to recover unknown coefficients, parameters, or structures inside a domain from data measured on the boundary. Formally, given access to input-to-output mappings—such as Dirichlet-to-Neumann or Neumann-to-Dirichlet maps—at the boundary of a region, the goal is to reconstruct one or more coefficients appearing in the governing partial differential equations (PDEs). This framework is fundamental in areas such as medical imaging, geophysics, nondestructive evaluation, and mathematical physics.

1. Canonical Formulations and Dirichlet-to-Neumann Maps

In a prototypical setup, let Ω be a bounded domain with boundary ∂Ω, and consider an elliptic PDE such as the stationary Schrödinger equation: $(-\Delta + q(x))\,u(x) = 0,\quad x \in \Omega,$ with Dirichlet data $u|_{\partial\Omega} = f$ and normal derivative (Neumann) measurements $\partial_\nu u|_{\partial\Omega}$. The Dirichlet-to-Neumann (DN) map $\Lambda_q$ is defined by

$$\Lambda_q : H^{1/2}(\partial\Omega) \rightarrow H^{-1/2}(\partial\Omega),\quad f \mapsto \partial_\nu u|_{\partial\Omega},$$

where $u$ solves the above PDE. The central inverse problem is: given $\Lambda_q$, determine $q(x)$ inside $\Omega$ (Blåsten et al., 2017, Imanuvilov et al., 2012).

A wide range of PDE operators (Schrödinger, Helmholtz, Maxwell, parabolic and hyperbolic types, as well as fully nonlinear and variable-order dynamics) admit analogous IBVPs with suitable forward/boundary maps (Beretta et al., 2014, Feizmohammadi, 2022, Nakamura et al., 2017, Uhlmann et al., 2022, Bellassoued et al., 2012).

2. Uniqueness Theorems and Regularity Thresholds

An IBVP is well-posed if one can prove that the boundary measurements uniquely determine the unknown coefficient(s), up to natural obstructions such as gauge invariance. Uniqueness results can depend heavily on dimension, PDE type, boundary data type, and the regularity of coefficients.

For the 2D Schrödinger equation with $q \in L^p(\Omega)$, $p > 4/3$, global uniqueness holds: if $\Lambda_{q_1} = \Lambda_{q_2}$ then $q_1 = q_2$ almost everywhere (Blåsten et al., 2017). In fact, for smoother potentials $q \in L^p(\Omega)$, $p > 2$, the uniqueness persists with even weaker assumptions (Imanuvilov et al., 2012, Imanuvilov et al., 2011). The precise regularity threshold is dictated by the capacity to control the mapping properties of Cauchy-type operators and stationary phase errors in the 2D CGO construction.

For higher dimensions ($n \geq 3$), the sharp threshold is $q \in L^{n/2}(\Omega)$ for the Schrödinger problem, and analogous regularity for general elliptic operators. For models with gauge freedom (such as the magnetic Schrödinger operator), uniqueness is up to gauge transformation, but both the curl of the magnetic vector potential and the electric potential are uniquely determined from boundary data (Pohjola, 2012, Kumar et al., 25 Jul 2024).

In the setting of time-dependent or nonlinear equations, uniqueness often requires additional analytic or geometric hypotheses (e.g., simplicity of the underlying metric, non-trapping or convexity conditions) (Feizmohammadi, 2022, Nakamura et al., 2017, Uhlmann et al., 2022, Uhlmann et al., 2019).

3. Key Analytical Tools

Analytical resolution of IBVPs leverages a suite of techniques:

• Complex Geometrical Optics (CGO) solutions: These are special solutions of the PDE with exponential or Gaussian-type oscillatory behavior, parameterized by large and/or complex spectral parameters. Their construction involves Carleman estimates and explicit conjugation of the operator. In the Schrödinger/Helmholtz case, CGO solutions provide the means to extract Fourier data of the unknown parameter via boundary measurements (Blåsten et al., 2017, Imanuvilov et al., 2012, Beretta et al., 2014, Pohjola, 2012).
• Alessandrini's Identity: Integral identities relating differences of DN maps to integrals of the differences of coefficients, weighted by products of solutions to the PDE. These are central for both uniqueness and stability (Blåsten et al., 2017, Beretta et al., 2014).
• Stationary phase and Carleman estimates: These facilitate the passage from oscillatory integrals to local equality of coefficients, and underpin the unique continuation in both elliptic and hyperbolic inverse problems (Blåsten et al., 2017, Nakamura et al., 2017).
• Higher-order linearization and microlocal analysis: In nonlinear or quasilinear problems, the response to small boundary excitations is expanded in series, with higher-order terms providing access to nonlinear coefficients via multilinear Fourier synthesis or Gaussian beam interactions (Nakamura et al., 2017, Uhlmann et al., 2019, Uhlmann et al., 2022).

4. Stability and Increasing Stability Phenomena

Stability quantifies the sensitivity of the parameter recovery to errors or perturbations in the boundary map. For the classical Calderón problem (stationary Schrödinger), stability is almost always logarithmic or double-logarithmic in the natural topology, indicating severe ill-posedness. However, at higher frequencies (as in the Helmholtz equation), the situation improves: stability becomes Hölder-type or even Lipschitz in certain regimes, a phenomenon dubbed "increasing stability" (Isakov et al., 2013, Kow et al., 2021, Liu et al., 20 Dec 2025, Beretta et al., 2014).

For example, in the high-frequency regime for the Schrödinger (or Helmholtz) IBVP, one obtains

$$\|q_1 - q_2\|_{L^2(\Omega)} \leq C k^4\, \operatorname{dist}(C_{q_1},C_{q_2}) + o(1) \text{ as } k \to \infty,$$

where $k$ is the frequency parameter and $C_{q}$ is the Cauchy data set (Isakov et al., 2013). For nonlinear elliptic or parabolic inverse problems, well-designed Carleman estimates yield explicit modulus of continuity for the parameter recovery; in the spectrum of PDEs, the exact rate is controlled by the regularity and support of the coefficient and the measurement operator (Liu et al., 20 Dec 2025, Beretta et al., 2014).

5. Extensions: Nonlinear, Hyperbolic and Hybrid Problems

Recent research generalizes IBVPs to:

• Nonlinear elliptic and hyperbolic equations: For nonlinear wave and elasticity equations of divergence form, higher-order linearization coupled with multi-wave interactions can uniquely determine both linear and nonlinear coefficients from boundary data (Nakamura et al., 2017, Uhlmann et al., 2019, Hintz et al., 2020, Uhlmann et al., 2022).
• Parabolic and heat-conduction equations: For the heat equation with time-dependent conductivity and capacity, uniqueness is achievable under geometric simplicity assumptions, using Laplace transform and Carleman-weighted exponential solutions (Feizmohammadi, 2022, Chapko et al., 2019). Anisotropic (tensor) coefficients can also be recovered up to natural diffeomorphism or gauge obstructions in two dimensions.
• Inverse shape and obstacle problems: Recovery of an embedded boundary (e.g., the interface of an inclusion or cavity) from scattering or time-resolved data is treated using non-linear boundary integral equations, Fréchet derivatives, and regularized Newton schemes (Chapko et al., 2022, Chapko et al., 2019).
• Partially accessible or locally measured data: Unique recovery may still hold under local data, provided sufficient a priori geometric control, and possibly up to a residual gauge; this has been established for convection-diffusion and magnetic Schrödinger operators with only partial boundary access (Kumar et al., 25 Jul 2024, Pohjola, 2012).

6. Numerical Reconstruction and Algorithmic Strategies

Algorithmic approaches to IBVPs rely on analytically grounded iterative schemes:

• Iterative Newton-type and projected gradient methods: These use the (possibly regularized) linearization of the forward map, e.g., Fréchet derivative-based Newton updates, with Tikhonov or discrepancy-principle regularization for ill-conditioned linear systems (Hannukainen et al., 2018, Chapko et al., 2022, Chapko et al., 2019, Beretta et al., 2014).
• Discretization via spectral, finite element, or boundary element methods: For geometric or shape-based IBVPs, high-order discretizations with specialized quadrature handle singularities in integral kernels, and expansion into trigonometric bases enables dimension reduction in correction steps (Chapko et al., 2022, Chapko et al., 2019).
• Optimization and parametrization effects: The fidelity of coefficient recovery in nonlinear and anisotropic media depends intricately on the parametrization (e.g., log-conductivity, resistivity, or “natural power” for $p$-Laplace models) (Hannukainen et al., 2018).
• Noise-robust schemes: Tikhonov regularization enforces stability in the presence of measurement noise, with regularization parameter schedules tuned to the convergence of the iteration and prior information on the coefficient (Chapko et al., 2022, Chapko et al., 2019).
• Numerical validation: Benchmarks on canonical domains (e.g., disk, rectangle) confirm accuracy and robustness under moderate noise, with precise control over reconstruction error as a function of discretization, regularization, and boundary data (Chapko et al., 2022, Chapko et al., 2019).

7. Outlook and Connections

IBVPs are central to inverse problems in applied mathematics and have driven major advances in microlocal analysis, PDE theory, geometric PDE, and numerical analysis. Open directions include extension to minimal regularity, partial/incomplete boundary data, nonlinear and time-dependent settings, and hybrid models coupling multiple physics. Connections with boundary rigidity, geodesic ray transforms, and advanced Carleman techniques are especially prominent in emerging theory (Feizmohammadi, 2022, Uhlmann et al., 2019, Nakamura et al., 2017, Liu et al., 20 Dec 2025, Hintz et al., 2020).

A comparative table of representative IBVP types and their key features:

PDE Type & Coefficient(s)	Boundary Map/ Data	Stability Type	Uniqueness/ Gauge
Schrödinger (elliptic, 2D/3D)	$q(x)$	DN map	log/ Hölder/ Lipschitz
Maxwell system (time-dependent)	$\mu,\epsilon,\sigma$	tangential traces	Hölder
Heat/ parabolic	$k(x,t),\,c(x,t)$	time DN map	log (parabolic Carleman)
Nonlinear hyperbolic/ elasticity	$\gamma(x),\,C_{kl}$	nonlin. DN map	N/A
Helmholtz (frequency-dependent)	$c(x)$	DN map at $\omega$	increasing (Hölder/Lip)
Convection-diffusion/ mag. Schrod.	$A,q$	partial DN map	unique/gauge, local

For each class, the analytic and algorithmic challenges are tightly intertwined with the geometry, PDE properties, data type (global/local), and physical applications, dictating the landscape of contemporary research and future developments.