Open research questions in Advanced Numerical Analysis Techniques

What the literature leaves open

• The complexity of the curves. The need for additional flexibility in curve design. The difficulty of determining the equivalence constants for the norm ‖∙‖p of BN,3.

  Chia tách đường cong Bézier bậc ba từng mảnh và các hằng số tương đương cho một vài chuẩn · 2026 · DOI

• The need for a novel recurrence relation for quasi-Bernstein-basis with multiple parameters. The need for an efficient and stable computation algorithm for quasi-Bessel curves. The need for a method to evaluate the approximation performance of quasi-Bessel curves.

  CONSTRUCTION AND GEOMETRIC PROPERTIES OF N-QUASI-BEZIER-CURVES WITH MULTIPLE PARAMETERS · 2026 · DOI

• The need to solve large systems of linear algebraic equations. The approximation of boundaries when using Cartesian grids for cylindrical areas. The high overhead costs and insufficient load of computing cores at coarse grid levels for the W-cycle.

  Multigrid Method for Solving the Poisson Equation in Cylindrical Coordinates on Graphics Processing Unit · 2026 · DOI

• The lack of geometric conditions on the elements makes the error analysis more challenging. The need to use the Babuška-Aziz trick to prove the error estimation.

  A unified approach to the error estimation of Raviart–Thomas interpolation on anisotropic simplices using the Babuška–Aziz trick · 2026 · DOI

• Future work involves proving the existence of minimums. It includes proving the existence and uniqueness of strong solutions. The maximum-bounded principle needs to be proved for the mild solutions.

  Shape Alignment via Allen-Cahn Nonlinear-Convection · 2026 · DOI

• The paper identifies a gap in shape registration methods. It addresses the need for a novel discrepancy measure for shape analysis. The method aims to fill this gap by introducing a generalized convective Allen-Cahn equation.

  Shape Alignment via Allen-Cahn Nonlinear-Convection · 2026 · DOI

• Further study of the properties of the proposed discretization. Application of the discretization to various geometry-processing applications. Comparison with other discretizations and methods.

  Phong‐Rodrigues Extrinsic Vector‐Field Processing · 2026 · DOI

• Existing discretizations typically sacrifice either continuity or the ability to evaluate derivatives pointwise. There is a lack of a discretization that jointly enables extrinsic continuous vector fields, bounded derivatives, and pointwise evaluation of operators.

  Phong‐Rodrigues Extrinsic Vector‐Field Processing · 2026 · DOI

• For cases where these conditions are inconclusive, we introduce a hierarchical subdivision strategy that progressively localizes ambiguous regions and resolves them through refinement.

  Bijectivity analysis of rational T-spline surfaces via Bernstein representations · 2026

• Computational time is substantial (5144 seconds for 100 iterations on a simple square pattern), but no discussion of computational complexity scaling or optimization strategies for larger problems is provided.

  Curvilinear Mask Optimization for Inverse Lithography Based on B-splines and Constrained Delaunay Triangulation · 2026 · DOI

• The residual-based error indicator (Equation 63) for adaptive refinement/coarsening in THB-splines uses Dörfler marking with fixed parameters (θr=0.75, θc=0.02). No analysis exists regarding sensitivity to these marking parameters or comparison with alternative error indicators specifically designed for structure-preserving IGA methods on adaptively refined THB-spline meshes.

  Macro-element refinement schemes for THB-splines: Applications to Bézier projection and structure-preserving discretizations · 2026 · DOI

• The adaptive refinement framework has only been validated numerically on the incompressible Navier-Stokes equations at Re=10³ with polynomial degree p=[2,2]. Extension to higher Reynolds numbers, variable polynomial degrees, and 3D applications of the structure-preserving (p+1)-box refinement strategy remains untested.

  Macro-element refinement schemes for THB-splines: Applications to Bézier projection and structure-preserving discretizations · 2026 · DOI

• The generalized Heron problem with ball constraints is tested only for geometric configurations where either the geodesic segment intersects the constraint set (infinitely many solutions) or does not intersect (unique solution); intermediate cases with multiple disconnected feasible regions on manifolds are not examined.

  Efficient Douglas–Rachford Methods on Hadamard Manifolds with Applications to the Heron Problems · 2026 · DOI

• The parameter selection strategy uses manual tuning (testing 'several reasonable values') for the inertial parameter θ_n and acceleration parameter p; a systematic or adaptive parameter selection framework for Douglas–Rachford methods on Hadamard manifolds is not developed.

  Efficient Douglas–Rachford Methods on Hadamard Manifolds with Applications to the Heron Problems · 2026 · DOI

• Further study on the application of subdivision schemes to other machine learning models. Investigation of the use of subdivision schemes for other signal processing tasks.

  Integrating subdivision schemes into SVM for improved signal classification · 2026 · DOI

• The lack of effective methods for integrating subdivision schemes into machine learning models. The need for a systematic study on the impact of subdivision schemes on SVM performance.

  Integrating subdivision schemes into SVM for improved signal classification · 2026 · DOI

• The paper suggests future research on the extension of the method to other partitionings of the domain. The paper suggests future research on the application of the method in various fields. The paper suggests future research on the development of more efficient algorithms for computing the Lagrange basis.

  Bivariate polynomial histopolation techniques on Padua, Fekete, and Leja triangles · 2026 · DOI

• The paper identifies the gap in the development of a well-defined histopolation scheme. The paper identifies the need for a method to approximate functions using bivariate polynomials. The paper identifies the challenge of solving the histopolation problem using the Padua triangles.

  Bivariate polynomial histopolation techniques on Padua, Fekete, and Leja triangles · 2026 · DOI

• The mean value theorem does not provide concrete insights in terms of well-posedness and stability. The lack of general closed formulae for the Lagrange basis requires a numerical approach.

  Polynomial approximation from diffused data: unisolvence and stability · 2026 · DOI

• The need for a weighting factor to achieve satisfactory enforcement of the Dirichlet boundary condition. The requirement for a large number of training samples to achieve improved accuracy.

  IGANets: Isogeometric analysis networks and their applications to linear structural analysis problems · 2026 · DOI

• The application of IGANets to more complex problems, such as nonlinear structural analysis. The investigation of the use of IGANets for multi-physics problems. The development of more efficient training methods for IGANets.

  IGANets: Isogeometric analysis networks and their applications to linear structural analysis problems · 2026 · DOI

• The limitations of classical B-spline constructions for refined non-uniform partitions. The need for a fully local formulation of the Hermite interpolant.

  Normalized b-spline construction for smooth splines and quasi-interpolation · 2026 · DOI

• Finding conditions for a best Chebyshev approximation when the knots are also variables remains an open problem. The problem of free knot linear spline approximation is equivalent to neural network approximation with piecewise linear activation functions.

  Best Free Knot Linear Spline Approximation and its Application to Neural Networks · 2026 · DOI

• Future research could investigate the application of the method to other types of curves. Future research could also explore the use of the norm ‖•‖p BN,3 in other contexts.

  Chia tách đường cong Bézier bậc ba từng mảnh và các hằng số tương đương cho một vài chuẩn · 2026 · DOI

• The limitation of classical Bézier curves in terms of local shape adjustment. The need for a novel recurrence relation and angle-cutting algorithm for quasi-Bernstein-basis with multiple parameters.

  CONSTRUCTION AND GEOMETRIC PROPERTIES OF N-QUASI-BEZIER-CURVES WITH MULTIPLE PARAMETERS · 2026 · DOI

Most-cited papers in Advanced Numerical Analysis Techniques

• A T-splines-oriented isogeometric topology optimization for plate and shell structures with arbitrary geometries using Bézier extraction · Computer Methods in Applied Mechanics and Engineering · 2024 · 75 citations
• Implementation of Curved Shape Grammars · Environment and Planning B Planning and Design · 2011 · 35 citations
• Pre-shape calculus and its application to mesh quality optimization · Control and Cybernetics · 2021 · 8 citations
• CURVATURE BASED CHARACTERIZATION OF RADIAL BASIS FUNCTIONS: APPLICATION TO INTERPOLATION · Mathematical Modelling and Analysis · 2023 · 7 citations
• Efficient estimation of smoothing spline with exact shape constraints · Statistical Theory and Related Fields · 2020 · 5 citations
• 𝐻²-Conformal Approximation of Miura Surfaces · Computational Methods in Applied Mathematics · 2023 · 4 citations
• Smooth Pycnophylactic Interpolation Produced by Density-Equalising Map Projections · Kartografija i geoinformacije · 2022 · 2 citations
• Enhancement of inverse-distance-weighting 2D interpolation using accelerated decline · Reports on Geodesy and Geoinformatics · 2023 · 2 citations
• Integrating subdivision schemes into SVM for improved signal classification · Journal of Computational and Applied Mathematics · 2026 · 1 citations
• Constrained visualisation using Shepard-Bernoulli interpolation operator · Studia Universitatis Babes-Bolyai Matematica · 2020 · 1 citations

Most recent work

• Integrating subdivision schemes into SVM for improved signal classification · Journal of Computational and Applied Mathematics · 2026
• A class of non-stationary ternary 4-point subdivision schemes based on iterations · Punjab University Journal of Mathematics · 2026
• Curvilinear Mask Optimization for Inverse Lithography Based on B-splines and Constrained Delaunay Triangulation · CSIAM Transactions on Applied Mathematics · 2026
• Macro-element refinement schemes for THB-splines: Applications to Bézier projection and structure-preserving discretizations · Computer Methods in Applied Mechanics and Engineering · 2026
• Linear minimum-variance approximants for noisy data · Applied Numerical Mathematics · 2026
• Efficient Douglas–Rachford Methods on Hadamard Manifolds with Applications to the Heron Problems · Applied Mathematics & Optimization · 2026
• An efficient forward-marching construction of the interpolating splines · Computational Mathematics and Modeling · 2026
• Numerical analysis of multi-patch discontinuous Galerkin isogeometric method for full-potential electronic structure calculations · Journal of Computational and Applied Mathematics · 2026
• Bivariate polynomial histopolation techniques on Padua, Fekete, and Leja triangles · Advances in Computational Mathematics · 2026
• Polynomial approximation from diffused data: unisolvence and stability · BIT Numerical Mathematics · 2026

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