discretization error analysis

This work addresses the dominant role of discretization error in reverse-time sampling of diffusion models under a fixed inference budget, a factor overlooked by existing non-asymptotic analyses that are often loose and ignore data structure. By deriving first-order asymptotic expansions for both the weak error of the Euler–Maruyama scheme and the Fréchet discretization error, the paper establishes—for the first time—an explicit connection between this error and intrinsic data geometry, such as the covariance spectrum, as well as the diffusion schedule. Under the exact score assumption, the integration of numerical analysis for stochastic differential equations with asymptotic theory yields a computable, geometry-aware optimization objective in the Gaussian setting. This formulation demonstrates strong predictive performance across diverse image generation and posterior sampling tasks, offering both theoretical grounding and practical tools for geometry-informed schedule design.

This work addresses the lack of intuitive, elementary analysis for discretization errors in stochastic differential equations (SDEs) underlying diffusion models, where existing approaches rely heavily on advanced probabilistic tools. We propose a concise, deterministic framework based on Grönwall’s inequality to analyze the convergence of Euler–Maruyama discretization for the variance-preserving SDE (VP-SDE). Our analysis yields, for the first time via elementary techniques, a rigorous $O(1/sqrt{T})$ bound on the sampling error with respect to the number $T$ of discretization steps. We further prove that discrete noise distributions—including Rademacher and uniform—can provably replace Gaussian noise without degrading the convergence rate or sample quality. Experiments validate both the tightness of the theoretical error scaling and the practical efficacy of discrete noise, while also demonstrating that improper noise scaling severely harms performance.

This work addresses the estimation of the mean of discretization errors in numerical solutions of ordinary differential equations by proposing a Bayesian inference framework. The approach models the error as a random variable and constructs a linear Gaussian state-space model by incorporating a Markovian prior informed by classical error analysis and dependent on the solver’s step size. The ensemble Kalman filter is employed to efficiently infer the temporal evolution of the error mean. Theoretical analysis establishes that this prior exhibits a well-defined probabilistic convergence rate as the step size tends to zero. Experimental results on the simple pendulum system and the FitzHugh–Nagumo model demonstrate the method’s effectiveness and practicality in accurately estimating the mean of discretization errors.

This work addresses the reliability of goal-oriented error estimation for nonlinear functional outputs in Galerkin finite element discretizations. Specifically, it investigates whether the classical residual-type estimator (eta = J(z) - B(u_h, z)) bounds the true error (|J(u) - J(u_h)|) via a mesh-independent constant (C), i.e., whether (|J(u) - J(u_h)| leq C|eta|) holds uniformly. The paper provides the first rigorous proof that, even with exact adjoint solution (z), this bound fails for certain combinations of nonlinear functionals (J) and bilinear forms (B): no uniformly bounded reliability constant (C) exists. Through an abstract variational framework and Hilbert space analysis, multiple concrete counterexamples are constructed, explicitly characterizing the (B ext{--}J) coupling mechanisms responsible for estimator failure. These results expose a fundamental limitation of conventional goal-oriented error estimation and deliver critical theoretical guidance for designing reliable error estimators in adaptive algorithms.

This work addresses the convergence guarantees of stochastic line search optimization for over-parameterized models under interpolation conditions. We establish a necessary and sufficient condition on the search direction—applicable to a broad class of methods—that ensures finite termination and bounded backtracking steps, and rigorously prove linear convergence under the Polyak–Łojasiewicz (PL) assumption. The condition unifies major first-order strategies—including momentum, conjugate gradient, and adaptive preconditioning—providing a verifiable theoretical foundation for their principled integration with stochastic line search. Our analysis fills a critical gap in the convergence theory of stochastic line search methods and significantly extends both the applicability and reliability of efficient first-order optimization in interpolation learning regimes.

This work addresses a key limitation of conventional neural PDE surrogates, which model field evolution on fixed grids and thereby overlook the critical role of mesh design in allocating spatial resolution and spectral bandwidth. The study introduces, for the first time, adaptive discretization as a physics-constrained conditional generation task, proposing a two-stage diffusion framework: it first generates an r-adaptive displacement grid conditioned on observed dynamics and then predicts solution evolution on this adaptive mesh. By incorporating physics-aware regularization, geometric validity constraints, and local spectral concentration, the method achieves learnable, interpretable, and numerically stable mesh adaptation. Extensive experiments across five classes of PDE problems demonstrate substantial improvements over traditional adaptive and reduced-order methods, with particularly notable gains in complex domains.

This study addresses the loss of precision caused by rounding errors in finite element computations, which remains difficult to analyze a priori. We propose the first automated a posteriori rounding error estimation framework that leverages running error analysis to track numerical and error propagation in real time. Built upon the FEniCS Form Compiler, this method achieves the first automated error estimation within finite element kernels through a C++ backend, custom arithmetic types, and templated kernel generation techniques. Experimental results demonstrate that the framework successfully detects catastrophic cancellation with only a 2–4× performance overhead. By effectively supporting mixed-precision design and numerical debugging, this work establishes a new paradigm for ensuring reliability in scientific computing.

This work addresses the high memory and computational complexity typically associated with solving three-dimensional partial differential equations on Cartesian grids. By exploiting tensor-product structure, the proposed method decomposes the 3D operator into one-dimensional banded kernels aligned with coordinate axes, thereby avoiding explicit assembly of the global matrix and enabling a matrix-free solution strategy. Within a unified framework that integrates diverse numerical approaches—including Kronecker product algebra, compact finite differences, isogeometric analysis, and direct diagonalization—the study systematically identifies three key techniques: multi-right-hand-side reshaping, sum factorization, and pencil-style MPI decomposition. These innovations collectively enhance hardware affinity and parallel scalability, reducing algorithmic complexity to O(N) and storage requirements to O(Nₓ + Nᵧ + N_z), thus enabling efficient large-scale 3D PDE simulations.

This study addresses the challenge of modeling spatiotemporal convection–diffusion processes defined on two-dimensional manifolds, such as the sphere, by proposing a Gaussian random field framework grounded in convection–diffusion stochastic partial differential equations (SPDEs). The method constructs a covariance structure via Galerkin discretization on smooth, compact Riemannian manifolds and enables scalable spatiotemporal prediction through Bayesian inference. To the best of our knowledge, this work is the first to systematically apply the convection–diffusion SPDE framework to domains with complex geometry, thereby integrating physical interpretability with statistical flexibility. Empirical evaluations demonstrate that the model achieves both high predictive accuracy and computational efficiency on simulated spherical data as well as real-world global aerosol optical thickness observations.

本文提出了一种基于Strang算子分裂近似的框架，用于解决随机波动率模型下的数值期权定价问题，避免了条件积分方差的评估。