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Open research questions in Numerical methods in inverse problems

58 unresolved questions extracted from the limitations and future-work sections of 257 Numerical methods in inverse problems papers in our library. Each links back to the study that raised it.

What the literature leaves open

  • Ill-conditioned linear inverse problems are inherently unstable and exhibit slow iterative convergence. Prior methods, such as Tikhonov regularization, may not be sufficient to address these challenges.

    Stability and Fast Solvers for Ill-Conditioned Linear Inverse Problems via Regularization and Preconditioning · 2026 · DOI

  • The gap in existing methods is the lack of a unified approach to smoothing and regularization. Existing methods have limitations in solving large-scale problems.

    A Unified Approach to Smoothing and Regularization for SOCCPs under Local Error Bound · 2026 · DOI

  • Further research can be conducted to apply the proposed methods to more complex problems, such as nonlinear diffusion equations. The physics-informed neural network approach can be extended to other types of inverse problems.

    Numerical approaches for identifying the time-dependent potential coefficient in the diffusion equation · 2026 · DOI

  • The inverse problem of identifying a time-dependent potential coefficient in a 1D diffusion equation is a challenging problem that requires efficient numerical methods.

    Numerical approaches for identifying the time-dependent potential coefficient in the diffusion equation · 2026 · DOI

  • The problem of finding the extremal eigenvalues is challenging due to the lack of a direct method. The problem of analyzing the behavior of the relaxed problems is challenging due to the complexity of the equations.

    Extremal Steklov–Neumann Eigenvalues · 2026 · DOI

  • The paper identifies the challenge of establishing a rigorous theoretical framework for solving the inverse problem. The paper identifies the challenge of proposing a numerical strategy for recovering the unknown temporal sources. The paper identifies the challenge of demonstrating the accuracy and robustness of the proposed method.

    Inverse t-Source problem and a strict positivity property for coupled subdiffusion systems · 2026 · DOI

  • The gap is that existing methods are not scalable or do not produce stable solutions. The gap is that the literature on this topic is too vast to be surveyed here.

    Scalable Iterative Data-Adaptive RKHS Regularization · 2026 · DOI

  • The optimal filter solution needs to be approximated for most problems. The projection filter has been restricted to univariate dynamics or to Gaussian density families.

    Conjugate continuous-discrete projection filter via sparse-grid quadrature · 2026 · DOI

  • Further work includes method developments for efficiently obtaining tangential interpolations for a system outside the imaginary axis of the complex plane. Theoretical guarantees for the convergence of the proposed parameter recovery procedure need to be established. The advantages of a system TFM derivative in parameter recovery need to be further investigated.

    Parameter recovery from tangential interpolations for systems with an LFT structure · 2026 · DOI

  • The paper identifies a gap in the existing literature regarding parameter recovery for linear time invariant systems from tangential interpolations. The gap is addressed by deriving a necessary and sufficient condition for unique determination of system parameters.

    Parameter recovery from tangential interpolations for systems with an LFT structure · 2026 · DOI

  • The nonlinearity and lack of direct invertibility of the forward operator F. The need for a transport-based distance for comparing signed waveforms. The complexity of the HV metric.

    HV metric for time-domain full-waveform inversion · 2026 · DOI

  • To further analyze the properties of the HV metric. To compare the HV metric with other misfit functions. To apply the HV metric to other applications.

    HV metric for time-domain full-waveform inversion · 2026 · DOI

  • The objective function is nonconvex. The constraints are linear. The Lagrangian multiplier needs to be bounded.

    A Convergent ADMM Algorithm for Grain Boundary Energy Minimization · 2026 · DOI

  • The paper does not provide a comprehensive comparison with other filtering methods. The simulation results are limited to two specific problems.

    Conjugate continuous-discrete projection filter via sparse-grid quadrature · 2026 · DOI

  • Developing a data assimilation scheme for problems with limited regularity. Using non-interpolant observables. Demonstrating the effectiveness of the method.

    Continuous data assimilation for problems with limited regularity using non-interpolant observables · 2026 · DOI

  • To extend the method to other types of problems. To test the method for more complex cases. To compare the method with other data assimilation schemes.

    Continuous data assimilation for problems with limited regularity using non-interpolant observables · 2026 · DOI

  • The nonlinear gray radiative transfer equations are a complex problem. The existing methods have limitations, such as rigorous conservation requirements. The paper needs to demonstrate the effectiveness of the proposed APNN technique through several numerical problems.

    Asymptotic-Preserving Neural Networks Based on Even-odd Decomposition for Multiscale Gray Radiative Transfer Equations · 2026 · DOI

  • The paper mentions that the concept of operator learning still requires further clarification. The results show a deviation in the prediction of the wave front in the Marshak wave problem.

    Asymptotic-Preserving Neural Networks Based on Even-odd Decomposition for Multiscale Gray Radiative Transfer Equations · 2026 · DOI

  • This study investigated stable and efficient solution strategies for ill-conditioned linear inverse problems by integrating regularization techniques with preconditioned iterative solvers. Ill-conditioning, arising from noise, model uncertainty, or near rank-deficiency, poses challenges to numerical stability and computational efficiency. These challenges are addressed by embedding Tikhonov regularization into a preconditioned Krylov subspace method, achieving a balance between stability and convergence speed. Regularization improves the conditioning of the normal equations by suppressing noise amplification, while preconditioning accelerates convergence by improving spectral properties. Numerical results demonstrate that the combined approach is more effective than either technique alone, yielding robust and rapidly convergent solutions. The results confirm that preconditioned conjugate gradient methods applied to regularized systems produce stable and accurate approximations within less iteration. The proposed framework is flexible and applicable to a wide range of inverse problems in scientific and engineering applications. Future work will extend the framework to high-dimensional settings, incorporating adaptive parameter selection, randomized sketch-based preconditioners, and learning-driven spectral transformations to further enhance scalability and robustness in large-scale computational pipelines.

    Stability and Fast Solvers for Ill-Conditioned Linear Inverse Problems via Regularization and Preconditioning · 2026 · DOI

  • Recovering lower order perturbations of a bi-wave operator requires investigation. The study of bi-wave operators with different wave speeds is an open problem.

    Direct and inverse problem for bi-wave equation with time-dependent coefficients from partial data · 2026 · DOI

  • The inverse magnetic problem is a mathematically ill-posed problem that lacks a unique or stable solution. Existing regularization techniques may not be effective in recovering compact structures.

    MAGNETIC INVERSION TO RECOVER COMPACT STRUCTURES BY AN ARCTANGENT STABILIZER · 2026 · DOI

  • To solve the original extremal eigenvalue problems. To analyze the behavior of the relaxed problems for more general domains and boundary conditions.

    Extremal Steklov–Neumann Eigenvalues · 2026 · DOI

  • The paper notes that the approach developed is limited to the treatment of integral operators involving general singular kernels. The paper also notes that the approach requires significant computational resources and may not be suitable for all applications.

    Novel Approaches for the Reliable and Efficient Numerical Evaluation of Landau-Type Operators · 2026 · DOI

  • The paper suggests that future research should focus on improving the efficiency and accuracy of numerical simulations of plasma physics problems. The paper suggests that future research should explore the application of the proposed approach to a wide range of plasma physics problems.

    Novel Approaches for the Reliable and Efficient Numerical Evaluation of Landau-Type Operators · 2026 · DOI

  • The complexity of formulating inverse problems for parabolic equations with variable time direction has grown. There is a need to develop new methods to solve inverse problems with a time direction that varies.

    Inverse problem for a system of mixed-type second-order partial differential equations · 2026 · DOI

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58 open questions have been extracted from the limitations and future-work passages of 257 Numerical methods in inverse problems papers in our library. Each one below links back to the study that raised it, so you can read the original claim in context.

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