数值分析：2025–2026 年度综述

An annual survey of the math.NA arXiv category, covering research published from September 2025 – August 2026. Built on a close-reading set of 144 papers and organised into 40 chapters, it maps the year by the field's research sub-directions.

Book framework (share of the year's output):

• Finite Element and Spectral PDE Solvers (48.9%) — Develops high-order, structure-preserving, and adaptive discretization schemes for linear and nonlinear partial differential equations using finite element and spectral methods.
• Machine Learning and Surrogate PDE Solvers (33.2%) — Integrates neural networks, surrogate modeling, and GPU-accelerated algorithms to efficiently solve and optimize complex differential equations and engineering simulations.
• Sparse Matrix and Quantum Linear Algebra (3.8%) — Focuses on efficient algorithms for sparse matrix operations, eigenvalue problems, preconditioning, and quantum-inspired linear algebra techniques.
• Multiscale and Multiphysics Flow Simulations (3.8%) — Addresses numerical challenges in simulating fluid dynamics, wave propagation, and coupled multiphysics systems across multiple scales and materials.
• Stochastic and Monte Carlo Methods (3.0%) — Applies stochastic differential equation solvers, Monte Carlo sampling, and probabilistic techniques to model random processes and particle transport.
• Structure-Preserving and Inverse Problems (2.6%) — Develops numerical methods that preserve physical structures like conservation laws and entropy, while addressing inverse problems and reconstruction tasks.
• Integral Equations and Parameter Estimation (2.4%) — Utilizes integral transforms, kernel methods, and variational frameworks for solving integral equations and estimating parameters in complex systems.
• Time Integration and Dynamical Systems (1.3%) — Designs explicit, implicit, and hybrid time-stepping schemes for stiff problems, constrained optimization, and long-term dynamical system evolution.
• Geometric and Electromagnetic Numerics (0.4%) — Explores numerical methods on geometric manifolds, electromagnetic field discretizations, and mean-field game perspectives on clustering.
• Adaptive Mesh Refinement and Accuracy (0.3%) — Improves computational efficiency and accuracy through adaptive mesh refinement strategies, error analysis, and optimized discretization schemes.
• Optimal Control and Optimization Theory (0.2%) — Investigates numerical algorithms for optimal control problems, including quantum control, robust control, and optimization on Banach spaces.
• Specialized Numerical Tools and Applications (0.0%) — Covers niche numerical tools and specific applications such as circuit models and communication systems, representing a small cluster of specialized studies.

Each sub-direction is covered across three chapters — how it framed its questions, the strength of its key findings, and its open debates and frontiers — so the reader gets a traceable picture of what this part of the field actually produced over the year.