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arXiv2026

Mathematical and numerical analysis of quantum signal processing

Quantum signal processing (QSP) provides a representation of scalar polynomials of degree $d$ as products of matrices in $\mathrm{SU}(2)$, parameterized by $(d+1)$ real numbers known as phase factors. QSP is the mathematical foundation of quantum singular value transformation (QSVT), which is often regarded as one of the most important quantum algorithms of the past decade, with a wide range of applications in scientific computing, from Hamiltonian simulation to solving linear systems of equations and eigenvalue problems. In this article we survey recent advances in the mathematical and numerical analysis of QSP. In particular, we focus on its generalization beyond polynomials, the computational complexity of algorithms for phase factor evaluation, and the numerical stability of such algorithms. The resolution to some of these problems relies on an unexpected interplay between QSP, nonlinear Fourier analysis on $\mathrm{SU}(2)$, fast polynomial multiplications, and Gaussian elimination for matrices with displacement structure.

quant-ph↗

arXiv2026

GraHTP: A Provable Newton-like Algorithm for Sparse Phase Retrieval

This paper investigates the sparse phase retrieval problem, which aims to recover a sparse signal from a system of quadratic measurements. In this work, we propose a novel non-convex algorithm, termed Gradient Hard Thresholding Pursuit (GraHTP), for sparse phase retrieval with complex sensing vectors. GraHTP is theoretically provable and exhibits high efficiency, achieving a quadratic convergence rate after a finite number of iterations, while maintaining low computational complexity per iteration. Numerical experiments further demonstrate GraHTP's superior performance compared to state-of-the-art algorithms.

math.NA↗

arXiv2026

Numerical Ergodicity and Optimal Strong Error Estimates for a Class of Novel Tamed Schemes to Superlinear SPDEs

We construct a class of novel tamed schemes for superlinear stochastic partial differential equations (SPDEs), including the stochastic Allen--Cahn equation driven by either multiplicative or additive noise. The schemes preserve the same Lyapunov structure as the original system, and we rigorously establish their longtime unconditional stability. Furthermore, we prove that the corresponding Galerkin-based fully discrete tamed schemes inherit the unique ergodicity of the underlying SPDEs and achieve optimal strong convergence rates in both the multiplicative and additive noise cases.

math.NA↗

arXiv2026

Dynamical Low-Rank Smoothing

Computational costs often make smoothing procedures prohibitive for high-dimensional data assimilation problems. To address this challenge, we propose a dynamical low-rank approximation (DLRA) methodology for smoothing concerning frameworks based on stochastic differential equations. We extend the previously developed joint mean-and-covariance optimization (JMCO) filtering setting to derive a reduced-order smoother via the Rauch--Tung--Striebel recursion and establish the corresponding Kalman--Bucy smoothing for affine drift dynamics. The resulting algorithms retain the adaptive nature of DLRA while significantly reducing the computational time and storage of the whole smoothing procedure.

math.NA↗

arXiv2026

Geometric Ergodicity of Affine Invariant Ensemble Langevin and its Discrete Time Variants

Affine-invariant ensemble samplers are widely used in Bayesian applications. However, their quantitative convergence theory, in particular geometric ergodicity, remains a basic open question. We study the affine invariant ensemble Langevin dynamics, an interacting particle system that uses the empirical covariance of the whole ensemble as a preconditioner. While effective in practice, theoretical understanding of this method is not available beyond plain qualitative convergence in total variation; a central difficulty is that the empirical covariance can approach singularity. This paper addresses this challenge. For potentials with bounded Hessian that are strongly convex outside a ball, we prove geometric ergodicity using a novel Lyapunov function that combines an inverse-covariance barrier with a coercive exponential energy. We then show that directly applying the Euler--Maruyama scheme can diverge with positive probability, even for a one-dimensional Gaussian target. This motivates a covariance-trace time regularization. We prove geometric ergodicity of the regularized diffusion and, for sufficiently small step size, of its unadjusted Euler--Maruyama discretization. We also show that the invariant distributions of the discretization converge weakly to the product target distribution as the step size tends to zero.

math.ST↗

arXiv2026

Tangential-Normal Decompositions of the Second Family of Finite Element Differential Forms

This paper introduces a novel tangential-normal ($t$-$n$) decomposition for the second family of finite element differential forms, presenting a new framework for constructing bases in finite element exterior calculus. The main contribution is the development of a $t$-$n$ basis in which degrees of freedom and shape functions are explicitly dual, a property that streamlines stiffness matrix assembly and enhances the efficiency of interpolation and numerical integration. Additionally, the integration of the well-documented Lagrange element basis supports practical implementation of finite element differential forms in applications.

math.NA↗

arXiv2026

An Immersed Interface Method for Parabolic Interface Problems with Nonlinear Jump Conditions

We develop an immersed interface finite difference method for a one-dimensional nonlinear parabolic interface problem with jump condition \[ [u]_α=λu^+u^-. \] The method combines a Crank--Nicolson immersed interface discretization with an $s$-parameter reduction of the nonlinear interface condition, thereby reducing the nonlinear coupling to a scalar quadratic equation. We also discuss two different viewpoints for combining Newton iteration with immersed interface discretization, namely the IIM--Newton and Newton--IIM formulations. Numerical experiments are presented to illustrate the behavior and accuracy of the method.

math.NA↗

arXiv2026

Data-efficient Kernel Methods for Learning Hamiltonian Systems

Hamiltonian dynamics describe a wide range of physical systems. As such, data-driven simulations of Hamiltonian systems are important for many scientific and engineering problems. In this work, we propose kernel-based methods for identifying and forecasting Hamiltonian systems directly from trajectory data. We present two approaches: a 2-step method that reconstructs trajectories before learning the Hamiltonian, and a 1-step method that jointly infers both. Across several benchmark systems, including mass-spring dynamics, a nonlinear pendulum, and the Henon-Heiles system, we demonstrate that our framework achieves accurate, data-efficient predictions and outperforms 2-step kernel-based baselines, particularly in scarce-data regimes, while preserving the Hamiltonian structure. Moreover, we prove a priori error estimates, ensuring reliability of the learned models. We also provide a more general, problem-agnostic numerical framework that goes beyond Hamiltonian systems and can be used for data-driven learning of arbitrary dynamical systems.

math.NA↗

arXiv2026

Residual neural networks overcome the curse of dimensionality for semilinear heat equations

Rigorous results show that feedforward neural networks can overcome the curse of dimensionality in the numerical approximation of high-dimensional partial differential equations (PDEs), but comparatively little is known about residual neural networks (ResNets) in the nonlinear PDE setting. We prove that ResNets overcome the curse of dimensionality in the numerical approximation of solutions of semilinear heat equations with globally Lipschitz continuous, gradient-independent nonlinearities: under polynomial growth and network approximability hypotheses on the PDE data, there exist $η\in(0,\infty)$ and ResNets $Ψ_{d,\varepsilon}$, $d\in\mathbb{N}$, $\varepsilon\in(0,1]$, with at most $ηd^η\varepsilon^{-η}$ parameters whose realizations approximate the solution in dimension $d$ with an $L^2$-error of at most $\varepsilon$. The proof represents one deterministic realization of a multilevel Picard estimator by a ResNet whose shortcut connections transmit the spatial variable and a scalar accumulator, while the residual branches successively add the summands of the estimator. For ridge-sum initial conditions, admissible sigmoidal activations, and globally Lipschitz truncations of the nonlinearity, we obtain, for every $ξ>0$, the explicit bound $C_ξd^{4+ξ}\varepsilon^{-(3+ξ)}$ on the number of parameters.

math.NA↗

arXiv2026

Optimal error estimates for the half-way bounce-back lattice Boltzmann method for the Stokes equations

We give a mathematical proof of the optimal convergence rates for the D2Q9 BGK lattice Boltzmann method with the half-way bounce-back rule for the incompressible Stokes equations in a flat channel. The convergence rates are second-order for the velocity and first-order for the pressure as the lattice spacing $h$ tends to zero, in agreement with formal analyses and numerical experiments, whereas the available rigorous convergence theorems only yield an $O(h^{1/2})$ bound for the velocity error. A key step in the proof is a decomposition of the leading boundary consistency error into macroscopic and kinetic components. These components are absorbed by suitably constructed Stokes and discrete Knudsen layer correctors, respectively. Incorporating these correctors into the prediction function used in previous rigorous analyses, we obtain a refined prediction function with consistency errors of sufficiently high order. Combined with the known weighted $L^2$-stability estimate, this gives the optimal convergence rates.

math.NA↗

arXiv2026

Transversality Conditions for Boundary Constraints Defined by Differential Equations

What are the transversality conditions for an optimal control problem when the boundary conditions are defined by differential equations? This seemingly bizarre question is motivated by trajectory optimization problems in the $N$-body system. The question, however, is more fundamental and goes beyond problems in astrodynamics to nonintegrable dynamical systems in general. The main contribution of this paper is the development of generic initial- and final-time transversality conditions for optimal control problems whose boundary conditions are defined in terms of differential equations with side conditions. The mathematical definition of differential boundary conditions are part of the foundations developed in this paper. To support the new fundamentals, the concept of coordinated/uncoordinated clock times and weak adjoint covectors are introduced. In the case of uncoordinated clock times, the new transversality conditions reveal that there exists a special situation where a weak adjoint covector is orthogonal to the vector field of the boundary differential equation. This condition is sharply different from the classical statement of orthogonality with respect to the endpoint manifold. The theorems developed in this paper are generic. An application of the theorems to several cases in the three-body problem are described in separate papers.

math.OC↗

arXiv2026

Spectral convergence of random feature method in one dimension

We first prove the spectral convergence of the random feature method (RFM) when used to solve second-order elliptic equations and eigenvalue problems in one dimension, provided that the solutions belong to Gevrey classes or Sobolev spaces. Second, we derive the convergence rate of RFM when integrated with the Partition of Unity Method (PUM) in terms of the patch size. Finally, we show that the singular values of the resulting random feature matrix decay exponentially, leading to exponential growth of the condition number. We also prove that PUM can mitigate this excessive singular-value decay.

math.NA↗

arXiv2026

Geometric Optics Approximation Sampling: A Reflector-Induced Transport Map Framework

In this paper, we propose Geometric Optics Approximation Sampling (GOAS), a reflector-induced transport-map framework for sampling from target measures. Once a reflecting surface is constructed, the associated transport map is explicitly determined by the physical law of reflection. As a concrete realization, we develop a supporting-hyperellipsoid construction that requires only a discrete approximation of the target measure and does not require gradient information of the target density. The formulation accommodates both density-based and sample-based target representations. A softmin smoothing technique is introduced to obtain a smooth approximate transport map from this piecewise hyperellipsoidal construction. We establish well-posedness and stability of the reflector-induced push-forward measure and derive quantitative error estimates in the maximum mean discrepancy metric, and convergence of continuous statistical observables, including fixed-order moments. Numerical experiments on an analytically tractable example, strongly non-Gaussian targets, sample-based target approximations, and Bayesian inverse problems demonstrate the accuracy and flexibility of GOAS.

math.NA↗

arXiv2026

Conformal Uncertainty Quantification Guarantees for Neural Operators

Neural operators provide fast surrogate models for approximating operators between function spaces, but their predictions often lack uncertainty quantification. We develop a split conformal framework to guarantee that a calibrated pointwise band around the neural operator output contains the true solution on at least a $1-γ$ fraction of the evaluation domain, with probability at least $1-α$ over test and calibration inputs, where $α,γ\in(0,1)$. Our method reduces a normalized residual field to its spatial $(1-γ)$-quantile and computes a scaling factor using a held-out calibration dataset. We prove marginal coverage guarantees for measurable residual fields defined on arbitrary probability spaces, covering both continuum domains and fixed discretizations. Under mild assumptions on the data distribution, we show that the coverage conditional on the calibration set follows a Beta distribution, which we verify with numerical experiments on Darcy flow and Navier--Stokes equations, where our calibration yields bands consistently tighter than existing corrections while retaining the target coverage.

math.NA↗

arXiv2026

Randomized inexact block triangular preconditioners for double saddle-point systems in PDE-constrained optimization

We develop a new class of inexact block triangular preconditioners for double saddle-point systems arising from PDE-constrained optimization. The proposed preconditioners are constructed through matrix factorization techniques while preserving the inherent block structure of the original systems. A comprehensive spectral analysis of the preconditioned matrices is provided, yielding explicit bounds for both real and nonreal eigenvalues. To enable efficient construction of the inexact preconditioners, randomized strategies are introduced to select the required subblocks. We establish high-probability bounds for the expected approximation error, with the error estimates explicitly characterized in terms of the eigenvalues of the associated matrices. Numerical experiments demonstrate the effectiveness, robustness, and scalability of the proposed preconditioners, and validate the efficiency of the randomized construction strategies.

math.NA↗

arXiv2026

Accelerated Primal-Dual Proximal Gradient Splitting Methods for Convex-Concave Saddle-Point Problems

In this paper, based a novel primal-dual dynamical model with adaptive scaling parameters and Bregman divergences, we propose new accelerated primal-dual proximal gradient splitting methods for solving bilinear saddle-point problems with optimal nonergodic convergence rates. For the first, using the spectral analysis, we show that a naive extension of acceleration to a quadratic game is unstable. Motivated by this, we present an accelerated primal-dual gradient flow which combines acceleration with careful velocity correction. To work with non-Euclidean distances, we also equip our continuous model with general Bregman divergences and prove the exponential decay of a Lyapunov function. Then, new primal-dual splitting methods are developed based on proper semi-implicit Euler schemes of the continuous model, and the theoretical convergence rates are nonergodic and optimal with respect to the matrix norms, Lipschitz constants and convexity parameters. Moreover, we introduce efficient restart variants to further improve the proposed methods and provide some numerical results to validate the practical performance.

math.OC↗

arXiv2025

Machine learning of continuous and discrete variational ODEs with convergence guarantee and uncertainty quantification

The article introduces a method to learn dynamical systems that are governed by Euler--Lagrange equations from data. The method is based on Gaussian process regression and identifies continuous or discrete Lagrangians and is, therefore, structure preserving by design. A rigorous proof of convergence as the distance between observation data points converges to zero and lower bounds for convergence rates are provided. Next to convergence guarantees, the method allows for quantification of model uncertainty, which can provide a basis of adaptive sampling techniques. We provide efficient uncertainty quantification of any observable that is linear in the Lagrangian, including of Hamiltonian functions (energy) and symplectic structures, which is of interest in the context of system identification. The article overcomes major practical and theoretical difficulties related to the ill-posedness of the identification task of (discrete) Lagrangians through a careful design of geometric regularisation strategies and through an exploit of a relation to convex minimisation problems in reproducing kernel Hilbert spaces.

math.NA↗

arXiv2026

Numerical Analysis of the Virtual Element Approximation for the Smagorinsky Turbulence Model

In this paper, we consider the Smagorinsky model for the Navier-Stokes equations within a virtual element framework. Under the standard assumption of small data, we prove the existence and uniqueness of a solution. Assuming more regularity to the solutions, we derive the known convergence rates $h$ for the a priori error estimates of the Smagorinsky model in two dimensional domains. We additionally prove that divergence-free virtual discretizations provide improved convergence orders, with weaker regularity assumptions than in the finite element literature. We conclude the paper with numerical results that corroborate the theory.

math.NA↗