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314 recordsLinked to original sources
arXiv2026

Optimal control of fractional diffusion with Dirac measures

We study a PDE-constrained optimization problem for an elliptic equation with the spectral fractional Laplacian and a linear combination of Dirac measures as the forcing term; the controls are the amplitudes of these singular sources. We prove existence and uniqueness of an optimal solution and derive first-order optimality conditions. We then propose a discretization based on finite elements. Since the set of admissible controls is finite dimensional, the control variable itself does not require discretization. We conclude by deriving a priori error bounds

math.OC↗
arXiv2026

Overcoming the spatial order barrier for nonlinear SPDEs with additive space-time white noise

We introduce a fully discrete numerical scheme for semilinear SPDEs with additive space-time white noise that overcomes the previous order barrier for the spatial convergence rate. The scheme achieves a strong convergence rate of $M^{-1+ε}$ in time and $N^{-3/2+ε}$ in space for any $ε>0$, where $M^{-1}$ and $N^{-1}$ are the temporal, respectively the spatial, meshsizes. This substantially improves the standard spatial error bounds of order $N^{-1/2}$ in the literature.

math.NA↗
arXiv2026

Advancements in Spectral Collocation Methods for High-Order Eigenvalue Problems

This paper focuses on computing spectral solutions for high-order eigenvalue problems using an efficient discretization method based on Chebfun spectral discretization algorithms and domain truncation. We solve several numerical eigenvalue problems, demonstrating both the accuracy and computational efficiency of the proposed approach.

math.NA↗
arXiv2026

The Minimum Number of Measurements for Almost-Everywhere Complex Phase Retrieval

Let $d\geq 2$ and let $\bf{f}_1,\ldots,\bf{f}_m\in\mathbb C^d$. We prove that if $m\leq 2d-1$, then the intensity measurement map \[ \bf{x}\longmapsto \bigl( |\langle \bf{x},\bf{f}_1\rangle|^2, \ldots, |\langle \bf{x},\bf{f}_m\rangle|^2 \bigr) \] fails to recover almost every signal in $\mathbb C^d$ uniquely up to a global phase factor. Combined with the known generic sufficiency of $2d$ measurements, our result establishes that the minimum number of measurements required for almost-everywhere phase retrieval in $\mathbb C^d$ is exactly $2d$. This resolves an open problem in phase retrieval by determining the exact measurement threshold for almost-everywhere phase retrieval in ${\mathbb C}^d$.

cs.IT↗
arXiv2026

A General Superconvergence Result for Cubature on Triangulated Polygonal Domains

Cubature rules, which approximate definite integrals as a linear combination of a set of function values, are ubiquitous and necessary for computational methods in the physical sciences. A superconvergence result for cubature rules on polygonal domains is developed, whereby a rule that is exact for all bivariate polynomials of a fixed even degree realize an extra order of convergence under a decrease in the spacing between nodes.

math.NA↗
arXiv2026

The singular Zienkiewicz tetrahedron: Definition and Integration

We extend the two-dimensional singular Zienkiewicz element to three dimensions, leading to a novel $H^2$-conforming finite element on tetrahedral meshes based on rational shape functions. Besides the finite element construction and its conformity analysis, we develop an exact iterative integration procedure for the associated class of rational functions. The resulting formulae allow for the exact integration of the basis functions, their derivatives, and the products occurring in finite element assembly.

math.NA↗
arXiv2026

Spatial symmetry invariance of solution of Kolmogorov flow

We prove a mathematical theorem that solution for all $t > 0$ of the two-dimensional (2D) Kolmogorov flow governed by Navier-Stokes (NS) equations with periodic boundary condition keeps the same spatial symmetry as its smooth initial condition. The proof of a similar theorem for the three-dimensional NS equations is given in the appendix. These mathematical theorems can be used to check the correctness and reliability of numerical simulations of NS turbulence. For example, they support the corresponding CNS (clean numerical simulation) results of the 2D and 3D turbulent Kolmogorov flows [1-3] that remain the same spatial symmetry in the whole time interval of simulation, but do not support the corresponding DNS (direct numerical simulation) results that lose the spatial symmetry quickly. In other words, these DNS results violate these mathematical theorems. Thus, these mathematical theorems rigorously confirm that the spatiotemporal trajectories of NS turbulence given by DNS are indeed quickly polluted by numerical noises badly. All of these indicate that CNS can indeed provide helpful enlightenments to deepen our understanding about turbulence and besides approach some mathematical truths about NS equations.

physics.flu-dyn↗
arXiv2026

A multi-class kinetic traffic flow model: discrete-velocity formulation and diffusively-corrected macroscopic limits

This paper introduces a multi-class extension of a discrete-velocity kinetic traffic flow model based on a non-local Prigogine-Herman framework. We derive a hyperbolically scaled system of equations from a continuous kinetic formulation describing interactions between different vehicle classes through braking and relaxation terms. The model is then discretized with respect to the velocity variable for an arbitrary number of vehicle classes, and the structural properties of the resulting formulation are analyzed. In particular, we prove hyperbolicity and total linear degeneracy. Due to the non-conservative structure of the model, we employ a path-conservative finite volume scheme for the numerical approximation of the system. Finally, we derive the corresponding diffusively-corrected macroscopic multi-class model, investigate its stability and present numerical simulations on a single-lane road to illustrate the theoretical findings.

math.NA↗
arXiv2026

Convergence and acceleration of a nonlinear fixed-point iteration for computing the Fitness Centrality of general graphs

We establish the global convergence of the (non-homogeneous) Fitness Centrality algorithm for general graphs, deriving an explicit convergence bound for the corresponding fixed-point iteration. Furthermore, we show how the convergence can be dramatically improved by Anderson acceleration and by switching to Newton's method once a sufficiently good approximation to the fixed point has been found. The efficacy of this strategy is illustrated by numerical experiments on different types of graphs.

math.NA↗
arXiv2026

A variant of the block preconditioner for indefinite complex symmetric linear systems

In this paper, we propose an efficient preconditioner for solving indefinite complex symmetric linear systems within a block preconditioning framework. We analyze the convergence of the corresponding iterative method and investigate several spectral properties of the preconditioned matrix, including eigenvalue distributions and eigenvector structures. The new preconditioner is used to accelerate the convergence of the flexible version of GMRES. Numerical experiments are presented to illustrate the effectiveness of the proposed preconditioner, and comparisons with existing block preconditioners demonstrate its superior performance.

math.NA↗
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

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

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

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

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

Feasible approximation of matching equilibria for large-scale matching for teams problems

We propose a numerical algorithm for computing feasible and approximately optimal solutions of the matching for teams problem. Specifically, we introduce the notion of approximate matching equilibrium as a feasible approximation of a matching equilibrium with relaxed rationality, and we show that a true equilibrium is recovered in the limit of a sequence of approximate matching equilibria with sub-optimality approaching 0. In our approximation scheme, we parametrize the so-called transfer functions, and we show that tackling the resulting parametric primal and dual optimization problems yields two approximate matching equilibria as well as provable and computable lower and upper bounds for the optimal social welfare. Under a flexible Euclidean setting, we show that the approximation error of our scheme can be controlled to be arbitrarily close to 0, we derive an explicit computational complexity bound, and we develop an algorithm for computing approximate matching equilibria that is efficient for large-scale problems involving a large number of agent populations. We study three problems in our numerical experiments: a retail business problem, the Wasserstein barycenter problem, and a large-scale problem involving up to 1000 agent populations. We show that the proposed algorithm can produce nearly optimal approximate matching equilibria to provide quantitative managerial insights for policymakers, and that the computed sub-optimality estimates are much less conservative than theoretical estimates.

math.OC↗
arXiv2026

An improved estimate of the intermediate internal energy in the energy-consistent HLLD scheme

The robustness of approximate Riemann solutions has been a crucial topic in computational magnetohydrodynamics, from both theoretical and practical perspectives. Recently, the widely used HLLD approximate Riemann solution was revised, becoming significantly more robust under strong magnetic fields. Yet, simplifications were needed as the compressible slow magnetoacoustic mode is not included in the HLLD scheme, and as a result non-physical density distribution has been found when having strong slow shocks. In this note, a simple fix is introduced for the estimated intermediate internal energy, providing robust results in several representative test cases, preserving pressure-positivity when the magnetic field is enhanced by a factor of 1000.

math.NA↗