Coefficient Inverse Problem

• Coefficient Inverse Problem is a mathematical challenge where unknown spatial or temporal coefficients in PDEs are determined from indirect measurements.
• Key techniques include Carleman estimates, fixed-point methods, and transformation operators that ensure uniqueness, stability, and regularity.
• Numerical methods such as finite element approaches and Tikhonov regularization enable robust reconstruction even with noisy or limited data.

A coefficient inverse problem (CIP) refers to the determination of an unknown coefficient function (typically spatially or temporally varying) in an operator–based model (e.g., PDE, system of PDEs, or integral equation) from partial and indirect measurements of the solution(s) rather than direct observations of the coefficient itself. In mathematical physics and applied mathematics, CIPs are foundational in areas such as medical imaging, geophysics, network inference, quantitative finance, and mean-field games, as they formalize the reconstruction of material, physical, or interaction parameters from accessible observational data.

1. General Structure and Theoretical Setting

The general form of a coefficient inverse problem involves an operator equation parameterized by a function $k(x)$ (the coefficient to be determined): $\mathcal{L}(u; k(x)) = f \quad \text{in} \quad Q$ with boundary and/or initial conditions, where $u$ is the observable state variable and $f$ is a prescribed source. The data consist of indirect measurements—typically traces, Cauchy data, or moments—of $u$ (and possibly derivatives) on a subset of the domain or its boundary, possibly at a sampling of times $t$. The challenge is to reconstruct $k(x)$ given the model and the data.

The coefficient may represent diffusion, reaction, transport, absorption, or interaction effects, and can be scalar, vector, or matrix-valued depending on the application. Identifiability, uniqueness, stability, regularity of data, and the accuracy and quantity of the measurements are central considerations.

2. Model Examples and Formulations

Coefficient inverse problems arise across a range of operator settings:

• Second-order parabolic and elliptic equations: e.g., the determination of the spatially-dependent viscosity coefficient $k(x)$ in a coupled Mean-Field Games System (MFGS) (Klibanov, 2023), or the recovery of the reaction/diffusion coefficients in fractional and classical parabolic models (Ashurov et al., 7 Nov 2025, Cen et al., 18 Mar 2024).
• System settings: The recovery of a matrix coefficient $P(x)$ in an evolution system (parabolic or Schrödinger-type), often with Cauchy-type boundary and initial/final data (Imanuvilov et al., 16 Jul 2024).
• Nonlocal and nonstandard boundary conditions: Problems with integral overdetermination, nonlocal-in-time initial conditions, or data given as internal point measurements (Jing et al., 22 Oct 2024, Ashurov et al., 5 Aug 2025, Durdiev et al., 8 Dec 2025).
• Coupled PDEs and mean-field models: CIPs for mean-field games couple a Hamilton–Jacobi–Bellman (HJB) and a Fokker–Planck (FP) equation, with $k(x)$ multiplying geometric or nonlocal interaction terms, and measurements on both value and density functions (Klibanov, 2023, Klibanov et al., 17 May 2024, Klibanov et al., 2023).
• Extensions to non-standard operators: Coefficients may appear in fractional-time derivatives, higher-order operators (bilaplacian, cascade systems), or as parameters in nonlinear source terms (Durdiev et al., 8 Dec 2025, Murugesan et al., 2023).

The data for these problems typically consist of:

• Lateral Cauchy data: Dirichlet and Neumann traces on boundaries (possibly measured on subsets).
• Interior measurements: Values or derivatives at fixed interior points or times.
• Overdetermination conditions: Integrals of solutions against weight functions (“integral overdetermination” (Ashurov et al., 5 Aug 2025)) or flux measurements.
• Incomplete or single-event data: Minimal boundary time-trace data or a single “polling” experiment (Klibanov, 2023, Klibanov et al., 2023).

3. Analytical Techniques and Main Theoretical Results

Mathematical analysis of CIPs centers on three axes: uniqueness, stability, and regularity.

• Carleman estimates: Weighted integral inequalities provide a sharp tool for proving conditional uniqueness and stability. For second-order parabolic systems (e.g., the MFGS), appropriate Carleman weights in space and time yield observability and three-cylinder inequalities, which underpin Hölder-stability estimates as well as strict convexity of associated cost functionals (Klibanov, 2023, Klibanov et al., 17 May 2024, Klibanov et al., 2023).
  • For the MFGS, a weight of the form $$\phi(x_1, t) = \exp[2\lambda(x_1 - a) - 2\lambda(t - T/2)^2]$$ is used, with large $\lambda \gg 1$. Precise Carleman inequalities are established for forward and backward time operators, controlling $u$, its derivatives, and handling boundary terms appropriately (Klibanov, 2023).
  • Nonlocal or Volterra-type terms arising in coupled mean-field or radiative transport models are handled via either specific kernel structure (e.g., delta or Heaviside kernels) or new Carleman estimates for integral operators (Klibanov et al., 2023, Klibanov et al., 17 May 2024).
• Transformation and operator approaches: In one-dimensional parabolic or Schrödinger systems, transformation operator (Goursat kernel) techniques enable constructive uniqueness proofs, expressing the map from one candidate coefficient to another as the solution of a Volterra-type kernel equation with Grönwall-type contraction arguments (Imanuvilov et al., 16 Jul 2024).
• Spectral and analytic continuation methods: For time-fractional diffusion models, spectral representations via eigenfunction expansions and analytic continuation in time are used to achieve simultaneous uniqueness for multiple coefficients, exploiting the memory effect of fractional operators and the analytic structure of solutions (Jing et al., 22 Oct 2024, Cen et al., 18 Mar 2024).
• Fixed-point and Banach contraction principles: In problems where the coefficient to be recovered enters nonlinearly (e.g., time-dependent reaction/diffusion in fractional models, coefficients in nonlocal terms), the inverse problem reduces to a fixed-point equation for the coefficient, often analyzed in suitable Sobolev or Banach spaces via the Banach contraction mapping theorem (Ashurov et al., 7 Nov 2025, Durdiev et al., 8 Dec 2025, Ashurov et al., 5 Aug 2025).
• Tikhonov regularization, adjoint systems, and variational optimization: For data-perturbed or ill-posed setups, classical or Carleman-weighted Tikhonov functionals can be employed, yielding optimally regularized solutions (Murugesan et al., 2023). Adjoint systems are used to compute gradients, establish first-order conditions, and support conjugate gradient or Levenberg–Marquardt optimization (Murugesan et al., 2023, Klibanov et al., 17 May 2024, Harrach, 2021).
• Stability and uniqueness theorems: Rigorous theorems provide explicit stability constants and rates. For example, in the MFGS, the following estimate holds under positivity and regularity conditions:

  $$\|u_1-u_2\|_{H^{2,1}(Q_{\varepsilon,T-\varepsilon})} + \|m_1-m_2\|_{H^{2,1}(Q_{\varepsilon,T-\varepsilon})} + \|k_1-k_2\|_{L^2(\Omega)} \leq C \varepsilon^{1-p}$$

  for arbitrary $p \in (0,1)$ and $\varepsilon$-close data (Klibanov, 2023).

  Uniqueness for 1D systems is established under boundary and nondegeneracy conditions, and extensions to higher dimensions remain challenging or open (Imanuvilov et al., 16 Jul 2024).

4. Numerical Methods and Computational Aspects

Rigorous theoretical results provide the basis for numerical reconstruction algorithms with provable convergence guarantees.

• Convexification and globally convergent algorithms: For mean-field systems and related high-dimensional CIPs, convexification frameworks use Carleman-weighted (or Carleman-inspired) cost functionals, achieving global strict convexity in suitable function spaces even for nonlinear PDEs (Klibanov et al., 17 May 2024, Klibanov et al., 2023). Minimization via gradient or projected-gradient descent from arbitrary initial guesses in bounded balls yields global minimizers without reliance on a priori proximity.
• Finite element and finite difference implementations: Standard spatial discretization methods are employed, using tailored handling for the forward operator, measurement operator, and their variational (adjoint) derivatives (Harrach, 2021, Murugesan et al., 2023, Klibanov et al., 17 May 2024).
• Stability to noise and regularization: Numerical studies often introduce synthetic or experimental noise and employ Tikhonov regularization, careful discretization, and early-stopping rules to counteract ill-posedness (Murugesan et al., 2023, Klibanov et al., 17 May 2024).
• Practical convergence: Numerical tests for MFGS and related systems demonstrate accurate reconstruction of both shape and contrast of inclusions for a range of synthetic and realistic data, with contrast errors typically below 5% and robustness to noise at similar levels (Klibanov et al., 17 May 2024).
• Operator-specific approaches: For Volterra or integral operator–driven problems, efficient algorithms exploit the reduction to lower-dimensional subproblems or the use of circulant kernels and FFT methods. For transformation-operator settings, matrix inversion and Lyapunov methods are applied (Imanuvilov et al., 16 Jul 2024, Ashurov et al., 5 Aug 2025).

5. Challenges, Open Problems, and Structural Issues

Despite substantial recent advances, several mathematical and computational challenges persist:

• Dimensional constraints: Transformation operator and Volterra kernel methods are essentially one-dimensional; their extension to multidimensional domains remains an open problem (Imanuvilov et al., 16 Jul 2024).
• Boundary and data limitations: Identifiability often depends on sufficient and specific data (Cauchy data at boundaries, internal traces, overdetermination, or full lateral coverage). Single-point or incomplete data generally introduce non-uniqueness or loss of stability; corresponding conditions and optimal measurement configurations are active research topics (Klibanov, 2023, Klibanov et al., 2023).
• Ill-posedness and instability: Many CIPs exhibit severe non-uniqueness, nonlinearity, and instability, especially with minimal data or high noise levels. Regularization theory, especially with Carleman-weights or monotonicity constraints, partially mitigates these issues, but the exponential dependence of stability constants on dimension or noise remains an obstacle (Harrach, 2021).
• Nonlinearity and nonconvexity: The mapping from coefficient to observations is often nonlinear and may exhibit multiple local minima for natural cost functionals. Carleman-weighted strict convexity and appropriate cost design are critical for circumventing local-minima traps (Klibanov et al., 17 May 2024, Klibanov et al., 2023, Beilina et al., 2013).

6. Representative Results and Comparative Table

Setting	Data Configuration	Main Tool/Result	Reference
Mean-Field Games System (MFGS)	Dirichlet/Neumann at $\partial\Omega \times (0,T)$ + $t=T/2$	Carleman estimate, Hölder stability, uniqueness	(Klibanov, 2023)
1D Parabolic System (Matrix Coefficient)	Cauchy data at $x=0$ + initial or final profile	Goursat transformation, Volterra kernel, uniqueness	(Imanuvilov et al., 16 Jul 2024)
Cascade (4th–2nd order) PDE system	Final-time distributed data	Quasi-solution optimization, stability, adjoint equation	(Murugesan et al., 2023)
Fractional diffusion with time-variable coefficients	Cauchy + additional flux	Fixed-point contraction, Banach theorem, unique solvability	(Ashurov et al., 7 Nov 2025)
Parabolic CIPs (generic)	Arbitrary boundary data	Quasi-reversibility, iterative linearization	(Nguyen et al., 2019)
Nonlocal/Integral overdetermined parabolic	Integral constraints in spatial variable	Fourier method, operator reduction	(Ashurov et al., 5 Aug 2025)

These results cover a wide spectrum, demonstrating the diversity of modern techniques and problem formulations for coefficient inverse problems.

7. Impact and Future Directions

Coefficient inverse problems are central to model–based parameter identification and are rapidly evolving as new theoretical insights (e.g., Carleman-based global convexification, advanced transformation operator methods), computational strategies, and application-driven measurement geometries emerge. Open research includes higher-dimensional transformation approaches, rigorous stability under minimal data, quantification of measurement requirements, sharper regularity and error estimates, and the systematic design of robust, efficient algorithms for classes of nonlinear, nonlocal, or fractional models. The potential for cross-fertilization with numerical optimization, learning-based regularization, and hybrid data–model paradigms is substantial as this field advances.

References:

• “A Coefficient Inverse Problem for the Mean Field Games System” (Klibanov, 2023)
• “One-dimensional coefficient inverse problems by transformation operators” (Imanuvilov et al., 16 Jul 2024)
• “Inverse coefficient problem for cascade system of fourth and second order partial differential equations” (Murugesan et al., 2023)
• “Inverse problem of determining a time-dependent coefficient in the time-fractional subdiffusion equation” (Ashurov et al., 7 Nov 2025)
• “Inverse Coefficient Problem for One-Dimensional Subdiffusion with Data on Disjoint Sets in Time” (Cen et al., 18 Mar 2024)
• “Convexification for a Coefficient Inverse Problem for a System of Two Coupled Nonlinear Parabolic Equations” (Klibanov et al., 17 May 2024)
• “Convexification for a Coefficient Inverse Problem of Mean Field Games” (Klibanov et al., 2023)
• “An introduction to finite element methods for inverse coefficient problems in elliptic PDEs” (Harrach, 2021)
• “Economic numerical method of solving coefficient inverse problem for 3D wave equation” (Leonov et al., 2017)
• “Coefficient Identification Problem with Integral Overdetermination Condition for Diffusion Equations” (Ashurov et al., 5 Aug 2025)
• “Globally strongly convex cost functional for a coefficient inverse problem” (Beilina et al., 2013)