Open research questions in Numerical methods in inverse problems

What the literature leaves open

• The experiments are conducted mainly under the exchangeability assumption of Theorem 4. The coverage guarantees established in Theorem 4 hold under the exchangeability assumption on the sequence of cost functions. The paper does not provide a comprehensive comparison with other methods. The numerical experiments are limited to a heterogeneous benchmark and an adaptive cruise control problem.

  Regime-Adaptive Conformal Calibration of Entropic Soft-Min Relaxations for Heterogeneous Optimization Problems · 2026 · DOI

• The inverse problem of recovering sparse initial data in spatial fractional parabolic equations is severely ill-posed. There is a need for a numerical algorithm that can effectively and stably recover sparse initial conditions in spatial fractional parabolic equations.

  Recovering Sparse Initial Conditions in Nonlocal Diffusion · 2026 · DOI

• Future research could explore the application of the paper's methods to other types of manifolds. Future research could investigate the use of other types of local coordinates for Riemannian interpolation.

  Canonically centered coordinates for Grassmann interpolation: Lagrange, Hermite, and errors · 2026 · DOI

• The gap is that Riemannian normal coordinates have limitations, especially when derivative data are to be treated. The gap is that manifolds lack the vector space structure that classical Euclidean interpolation algorithms rely on.

  Canonically centered coordinates for Grassmann interpolation: Lagrange, Hermite, and errors · 2026 · DOI

• Further research can be conducted to apply the results to various real-world materials. The paper provides a framework for fractional-order models, which can be further developed and refined.

  L1 discretization error analysis for regularized k-Prabhakar derivatives in Hölder spaces · 2026 · DOI

• Future research could focus on extending the results to more complex models and scenarios. The development of more efficient and accurate reconstruction algorithms is also a potential direction for future research.

  On uniqueness of coefficient identification in the Bloch-Torrey equation for magnetic resonance imaging · 2026 · DOI

• The lack of uniqueness results for coefficient identification in the Bloch-Torrey equation. The need for a model-based reconstruction approach that takes into account the spatially varying spin density and relaxation times.

  On uniqueness of coefficient identification in the Bloch-Torrey equation for magnetic resonance imaging · 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

• These challenges are addressed by embedding Tikhonov regularization into a preconditioned Krylov subspace method, achieving a balance between stability and convergence speed.

  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

• The lack of stochastic convergence analysis for large-scale linear discrete ill-posed problems with random noise. The need for an adaptive strategy suitable for large-scale computation.

  Stochastic Convergence Analysis for Large-Scale Linear Discrete Ill-posed Problems · 2026

• 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

• 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 extremal eigenvalue problems for the Steklov-Neumann eigenproblem remain unsolved. There is a lack of understanding of the behavior of the relaxed problems.

  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

• 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

• The inverse problem remains ill-posed despite uniqueness. Small perturbations in the over-specified data can cause large deviations in the recovered coefficients.

  Uniqueness and CN–Bell Spectral Reconstruction of Three Time-Dependent Coefficients in a Parabolic Inverse Problem with Quadratic Spatial Diffusivity · 2026 · DOI

• Selecting the inverse temperature parameter in entropic soft-min relaxations to ensure a prescribed tolerance with finite-sample distribution-free validity. Dealing with the unknown geometry of the cost function. Ensuring the distribution-free finite-sample validity of the calibration procedure.

  Regime-Adaptive Conformal Calibration of Entropic Soft-Min Relaxations for Heterogeneous Optimization Problems · 2026 · DOI

• To further improve the efficiency and stability of the proposed algorithm. To apply the proposed algorithm to various applications, including biological systems, quantum mechanics, porous media flow, viscoelasticity, contaminant transport, polymer unzipping, probability theory, and mathematical finance.

  Recovering Sparse Initial Conditions in Nonlocal Diffusion · 2026 · DOI

• Further research could investigate more complex house models, - Further research could explore different types of insulating materials, - Further research could examine the effects of varying boundary…

  Duality-Based Algorithm and Numerical Analysis for Optimal Insulation Problems on Nonsmooth Domains · 2026 · DOI

• The gap between the full lifted state and the reduced trajectory. The gap between the estimated pencil and the exact pencil. The gap between the classical baseline and the McLachlan-projected reduced dynamics.

  McLachlan-projected reduced dynamics for ill-posed Schrödingerized backward diffusion · 2026

• The inverse problem is ill-posed, - Small perturbations in the data can lead to large errors in the reconstructed coefficients, - A trial-and-error approach was used to choose the regularization parameters, - A systematic study of optimal parameter selection is left for future…

  Uniqueness and CN–Bell Spectral Reconstruction of Three Time-Dependent Coefficients in a Parabolic Inverse Problem with Quadratic Spatial Diffusivity · 2026 · DOI

Most-cited papers in Numerical methods in inverse problems

• Elasticities and the Inverse Hyperbolic Sine Transformation · Oxford Bulletin of Economics and Statistics · 2019 · 976 citations
• Partial least squares path modeling: Quo vadis? · Quality & Quantity · 2018 · 475 citations
• A Statistical Perspective on Ill-Posed Inverse Problems · Statistical Science · 1986 · 379 citations
• Regularization methods for near-field acoustical holography · The Journal of the Acoustical Society of America · 2001 · 260 citations
• Inverse Problems · American Mathematical Monthly · 1976 · 212 citations
• The iterative time reversal process: Analysis of the convergence · The Journal of the Acoustical Society of America · 1995 · 175 citations
• A computational trick for delta-method standard errors · Economics Letters · 2004 · 101 citations
• Space Mapping and Defect Correction · Computational Methods in Applied Mathematics · 2005 · 101 citations
• A regularization approach to the many instruments problem · Journal of Econometrics · 2012 · 92 citations
• What to Do When Your Hessian is Not Invertible · Sociological Methods & Research · 2004 · 49 citations

Most recent work

• Physics-informed neural networks for 2D-transient inverse heat conduction problem involving static and moving heat source · International Journal of Heat and Mass Transfer · 2026
• Rigorous Foundations of the Spectral RG Functional: Kernel Uniqueness, Boundary Invariance, and Variational Structure · Zenodo (CERN European Organization for Nuclear Research) · 2026
• New Double Integral Reinforcing Recurrent Neural Network for Solving Matrix Pseudoinverse Problem · IEEE Transactions on Cybernetics · 2026
• Bayesian inference for PDE-based inverse problems using the optimization of a discrete loss · Computer Methods in Applied Mechanics and Engineering · 2026
• An inverse problem for recovering a time-dependent source in time fractional diffusion–wave models governed by the integral fractional Laplacian · Chaos Solitons & Fractals · 2026
• An Efficient Spectral Method for Fourth-Order PDEs in Complex Geometries · Journal of Scientific Computing · 2026
• A general probability density framework for local histopolation and weighted function reconstruction from mesh line integrals · Numerical Algorithms · 2026
• Double inertial contraction method for monotone inclusion problems for medical image recovery and data classifications · Journal of Computational and Applied Mathematics · 2026
• Regularization by denoising diffusion models for solving inverse PDE problems with application to full waveform inversion · Communications Physics · 2026
• Inverse initial data reconstruction for Maxwell’s equations via time splitting method · Journal of Computational Physics · 2026

Related topics in Mathematics