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arXiv2026

Well-posedness and numerical reconstruction of a source term for linear parabolic problems with an integral constraint

This work investigates a time-dependent source identification problem for linear parabolic equations subject to an integral constraint and Neumann boundary conditions in a domain of $\mathbb{R}^d$, $d\ge 1$. We establish well-posedness and higher regularity of the solution pair in parabolic Hölder spaces. A numerical algorithm based on a finite element discretization in space and an implicit time-stepping scheme is then developed for the reconstruction of the unknown source. The resulting discrete inverse problem involves a well-conditioned operator. We show that identity Tikhonov regularization provides only uniform, nonselective shrinkage in this setting, whereas Tikhonov regularization with derivative-based penalties, analyzed through the generalized singular value decomposition, provides an effective denoising strategy. The regularization parameter is selected using the Morozov discrepancy principle. Numerical errors are evaluated using full parabolic Hölder norms, which provide a more comprehensive assessment of the reconstruction by incorporating errors in the solution, its derivatives, and the associated Hölder seminorms. Numerical experiments for smooth and piecewise constant sources demonstrate accurate and robust reconstructions under increasing levels of noise.

math.AP↗

arXiv2026

Families of relative periodic orbits in the planar three-body problem via consecutive alignments

Relative periodic orbits (RPOs) are solutions of the three-body problem that are periodic in a uniformly rotating reference frame and, in general, quasi-periodic in inertial coordinates. We present a numerical procedure for computing and continuing one-parameter families of RPOs of the planar Newtonian three-body problem. The method exploits consecutive syzygies, understood here as configurations in which the three bodies are aligned and their velocities satisfy the corresponding symmetry conditions. Matching the positions and momenta at two consecutive alignments reduces the computation of RPOs to a low-dimensional nonlinear problem. Its solutions are then numerically continued, and linear stability is determined from the nontrivial eigenvalues of the rotated monodromy matrix after removing the neutral directions associated with conserved quantities and continuous symmetries. The procedure is applied to several mass distributions and initial configurations, producing families of Poincaré, Hill, and binary-type solutions. These families exhibit transitions from nearly circular to highly eccentric motion, changes of stability near resonances and turning points, and absolute periodic solutions when the rotation angle is a rational multiple of 2π. In the Hill families, the continuation connects satellite configurations with circumstellar motion as the smallest body loses its gravitational binding to the intermediate body. Circumbinary and circumstellar configurations are also obtained in the binary regime. The results illustrate the dynamical diversity of RPOs and provide coherent three-body motions that can be used as prescribed trajectories in restricted four-body models.

math.DS↗

arXiv2026

Separable Nonnegative Matrix Factorization Using Powered Ratio-of-Norms Regularization

Separable nonnegative matrix factorization (SNMF) has been widely used for low-rank representation and clustering of nonnegative data, owing to its ability to produce part-based and interpretable decompositions. In particular, SNMF is closely related to graph clustering and community detection. To enhance sparsity and identifiability of the learned factors, we propose an $\ell_1^p/\ell_2$-regularized SNMF model based on a powered ratio-of-norms regularizer. The resulting formulation is nonconvex and nonsmooth, which poses significant challenges for optimization. To address this, we develop efficient algorithms based on the difference-of-convex function algorithm (DCA) and the alternating direction method of multipliers (ADMM). The proposed methods decompose the original problem into tractable subproblems, leveraging closed-form proximal operators associated with the powered norm terms. We establish descent and limiting criticality properties for the DCA scheme and convergence under standard assumptions for the ADMM scheme. Extensive numerical experiments on synthetic datasets and hand gesture classification tasks demonstrate that the proposed approach achieves competitive or improved performance in anchor identification and classification accuracy compared with existing SNMF methods, while maintaining competitive computational efficiency.

math.NA↗

arXiv2026

Sharp Mixed Spectral Barron Regularity of Coulombic Many-Electron Wave Functions

We establish sharp mixed spectral Barron regularity for eigenfunctions of molecular Coulomb Hamiltonians. The mixed norm is a Fourier $L^1$ norm with one isotropic weight and coordinate-product weights, and therefore detects regularity invisible to the isotropic Barron scale. For a nonempty set $I$ of electron indices on which the wave function is antisymmetric, we derive an explicit admissible region for the isotropic order $s$ and the coordinate orders $α,β$. This region is optimal as a uniform statement over the class of clamped-nuclei Coulomb Hamiltonians. For fixed-spin components with two occupied spin blocks, it reduces to $s+α+β<1$; in the fully spin-polarized class it reduces to $s+α<1$. In particular, if $\mathcal I_σ$ denotes the family of occupied same-spin blocks determined by $σ$, then every fixed-spin spatial component $ψ_σ$ satisfies, for every $0\leqα<1$, \[ \left(\sum_{I\in\mathcal I_σ}\prod_{i\in I}\langleξ_i\rangle^α\right)\widehat{ψ_σ}\in L^1(\mathbb{R}^{3N}). \] For a fully spin-polarized state, $\mathcal I_σ=\{\{1,\ldots,N\}\}$.

math.AP↗

arXiv2025

Unconditionally optimal error Estimate of a linearized Second-order Fully Discrete Finite Element Method for the bioconvection flows with concentration dependent viscosity

In this paper, the coupled and decoupled BDF2 finite element discrete schemes are obtained for the time-dependent bioconvection flows problem with concentration dependent viscosity, which consisting of the Navier-Stokes equation coupled with a linear convection-diffusion equation modeling the concentration of microorganisms in a culture fluid. The unconditionally optimal error estimate for the velocity and concentration in $L^2$-norm and $H^1$-norm are proved by using finite element approximations in space and finite differences in time. Finally, the numerical results for different viscosity are showed to support the theoretical analysis.

math.NA↗

arXiv2024

Convergence of a Ramshaw-Mesina Iteration

In 1991 Ramshaw and Mesina introduced a clever synthesis of penalty methods and artificial compression methods. Its form makes it an interesting option to replace the pressure update in the Uzawa iteration. The result, for the Stokes problem, is \begin{equation} \left\{ \begin{array} [c]{cc} Step\ 1: & -\triangle u^{n+1}+\nabla p^{n}=f(x),\ {\rm in}\ Ω,\ u^{n+1}|_{\partialΩ}=0,\\ Step\ 2: & p^{n+1}-p^{n}+β\nabla\cdot(u^{n+1}-u^{n})+α^{2}\nabla\cdot u^{n+1}=0. \end{array} \right. \end{equation} For saddle point problems, including Stokes, this iteration converges under a condition similar to the one required for Uzawa iteration.

math.NA↗

arXiv2026

Assessment of Numerical Lift Coefficient Data for a Circular Cylinder with Application to Bladeless Turbines

We assess computed lift coefficient data for flow past a circular cylinder to evaluate their suitability for practical applications. Specifically, we consider lift coefficient data for a circular cylinder over Reynolds numbers from 120 to 8000. The results are obtained from two-dimensional finite element simulations of the incompressible Navier-Stokes equations using pressure robust discretizations. We compare the computed lift coefficients with published experimental and numerical results, finding good agreement in some cases but significant disagreement in others. Because lift fluctuations are central to vortex-induced vibration concepts, these data therefore provide input for the analysis and preliminary design of bladeless turbines.

physics.flu-dyn↗

arXiv2026

ENPINN: Energy-Norm-Guided Gradient-Enhanced PINNs for Generalized Transport Problems with Sharp Gradients

Physics-informed neural networks (PINNs) have emerged as a meshless alternative to conventional numerical methods for solving partial differential equations (PDEs). However, their limited ability to capture sharp gradients can lead to substantial errors when resolving boundary and interior layers. Here, we introduce an energy-norm-enhanced PINN (ENPINN) that incorporates gradient information and variational structure into the loss function to improve the resolution of layer-dominated solutions. We first examine two related formulations: weak-loss PINNs (WLPINNs), which incorporate test functions into the conventional PINN residual, and gradient-enhanced PINNs (gPINNs), which augment the loss with spatial derivatives of the PDE residual. By analyzing these formulations, we identify their limitations in resolving steep solution gradients and motivate the systematic construction of ENPINN. We establish theoretically how the energy-norm error depends on the ENPINN loss and show that a suitably modified residual-derivative term is essential for accurately capturing boundary layers. We further establish the existence of neural-network approximations with arbitrarily small energy error and derive corresponding derivative bounds, providing a theoretical foundation for the proposed framework. The performance of ENPINN is assessed through systematic comparisons with existing PINN variants for convection-diffusion-reaction problems exhibiting steep gradients. Numerical experiments include a combustion model, a coupled multi-scale system, a two-dimensional Burgers equation with an interior layer, and a three-dimensional time-dependent problem.

math.NA↗

arXiv2026

Nechvile-Transformed Spacecraft Dynamics and Propellant Computation in the 3-Body Problem

The uncontrolled equations of motion in the Nechvile frame for the restricted three-body problem have been well-known since at least the 1960s. It would seem that adding an external force to these equations is quite trivial: simply add an external force per mass term to the acceleration equations. Here we show that the last statement is not true. In fact, we show that the additive generic external force must be multiplied by the inverse of $(1 + e \cosθ)^3$ where $e$ is the relative eccentricity of the primaries and $θ$ is the true anomaly of the rotating frame located at the barycenter. Furthermore, when this result is combined with the mass flow rate equation, it generates several surprising results due to the mismatch between the resulting quadratic term and the cubic term in the equations of motion. This leads to a corresponding modification of the rocket equation itself. A Birkhoff-theoretic solution to an illustrative cislunar space mission problem shows propellent savings of 80% with the use of the correct cost functional. The popular quadratic cost utilizes more than $2X$ the minimum propellant consumption.

math.OC↗

arXiv2026

Estimating systematic errors in Bayesian inversion using transport maps

In indirect measurements, the sought parameters have to be determined by solving an inverse problem, typically in a Bayesian framework. Often, the accurate numerical simulation of the measuring process is computationally demanding, making it necessary to rely on approximate models. These surrogates, however, introduce an additional model error and thus may distort the resulting parameter distribution. Moreover, even with the additional speed granted by the surrogate, posterior determination through conventional means such as Markov chain Monte Carlo might be cost intensive, specifically for complicated posterior shapes. In this paper, we propose a unified framework that combines Bayesian inference, model error correction and a transport-based sampling scheme to address these issues. To train the transport scheme, we investigate two different losses: one equivalent to the Kullback-Leibler divergence associated to the transport problem and one based on an upper bound of this loss, generally known as the evidence lower bound. We demonstrate that training the transport based on the latter changes the optimisation landscape drastically, potentially introducing an undesired bias in approximating the target posterior. We compare the computational cost of our approach with established methods and underline the theoretical results with numerical examples.

stat.ME↗

arXiv2026

A class of low-rank short recurrences for nonsymmetric linear matrix equations

We propose a new class of short matrix recurrences for the solution of nonsymmetric linear equations of the type $\mathbf{A}_1\mathbf{X}\mathbf{B}_1+\ldots+\mathbf{A}_p\mathbf{X}\mathbf{B}_p=CD^T$. Building on ideas underpinning the recently introduced subspace conjugate gradient algorithm, we derive low-rank short recurrences that generalize one-dimensional subspace projection methods to the nonsymmetric matrix equation setting. To limit memory consumption and maximize computational efficiency, rank truncation strategies and modern randomization procedures are incorporated into the proposed algorithms. Computational experiments on a benchmark problem as well as a challenging discretized mixed formulation of a diffusion equation with random inputs illustrate the potential of the proposed methodology.

math.NA↗

arXiv2026

Inverse Source Problem for a Time-Fractional Diffusion-Wave Equation with a Singular Inverse-Square Potential

This paper investigates an inverse source problem for a time-fractional diffusion-wave equation with a singular inverse-square potential. The source term is assumed to consist of a known temporal factor and an unknown spatial component, which is to be recovered from terminal-state measurements. The well-posedness and regularity of the forward problem are established within an appropriate energy framework by exploiting Hardy-type inequalities and the spectral properties of the associated singular elliptic operator. The terminal observation operator is then shown to be compact, and uniqueness of the spatial source is established under a suitable nondegeneracy condition on the temporal factor. To stabilize the resulting ill-posed inverse problem, a Tikhonov regularization approach is introduced. The gradient of the regularized functional is derived through an adjoint problem involving a right-sided fractional derivative, leading to an adjoint-based conjugate gradient method with an exact line search for the numerical reconstruction of the unknown source. Numerical experiments are conducted on both one-and two-dimensional spatial domains, using both exact and noisy terminal data, to demonstrate the effectiveness and stability of the proposed source reconstruction method.

math.NA↗

arXiv2026

A coercive space-time variational approach to fractional diffusion problems

We consider a fractional diffusion problem with temporal nonlocality acting on the diffusive flux. A coercive space--time variational formulation in Bochner-valued fractional Sobolev spaces is derived and the existence, uniqueness, and regularity of solutions are established. We further develop a conforming tensor-product Galerkin discretization and prove quasi-optimal error estimates in the anisotropic energy norm and improved convergence rates in weaker norms using duality arguments. In contrast to some space-time formulations for classical diffusion, the method preserves the causal structure of the evolution problem and leads to a time-stepping procedure with memory terms. On uniform time grids, the discrete history operator has a lower-triangular Toeplitz structure which enables an efficient implementation using fast recursive convolution techniques.

math.NA↗

arXiv2026

Random attractors and almost-sure stability under discretization of a stochastic autoparametric system

For a stochastic autoparametric block-and-pendulum system, the long-time dynamics exhibit two fundamental features: the almost-sure stability of the single mode solution, characterized by its Lyapunov exponent, and the global asymptotic dynamics when this single mode solution loses stability. This naturally raises the question of whether these dynamical features are preserved under discretization, since such preservation is essential for the resulting discrete system to faithfully capture the qualitative behavior of the continuous system. To address this question, we first establish the existence of a random attractor for the continuous system subject to multiplicative stochastic excitation, providing a rigorous characterization of the global asymptotic dynamics. We then propose a numerical discretization that induces a discrete random dynamical system and prove the convergence of its random attractor to the continuous one as the step size tends to zero. In addition, we show that the numerical Lyapunov exponent of the single mode solution has the same sign as its continuous counterpart for sufficiently small step sizes, thus preserving the corresponding almost-sure stability or instability classification. These results demonstrate that the proposed discretization captures both the global asymptotic dynamics and the stability characteristics of the underlying stochastic autoparametric system.

math.DS↗

arXiv2026

Computational Oncology of Chemotaxis-Driven Tumour--Immune Spatial Patterning and Stability

We develop a reaction--diffusion--chemotaxis model for spatial tumour--immune--chemokine dynamics that couples logistic tumour growth, immune-mediated killing, chemokine-dependent immune recruitment, chemotactic migration, and signal production. For the nondimensional system, we establish local classical solvability, nonnegativity, a uniform tumour-density bound, and global mass estimates for the immune and chemokine components. The tumour-free equilibrium is stable precisely when the baseline immune-control index satisfies $σ_0/δ>1$, whereas positive homogeneous coexistence is characterized by a scalar nonlinear equation. Linearization in the Neumann Laplacian eigenbasis yields a mode-dependent cubic dispersion relation, showing that chemotaxis does not alter the tumour-invasion threshold but can destabilize homogeneous coexistence through a finite-wavelength oscillatory instability above a critical sensitivity $ξ_c$. A conservative finite-volume discretization with upwind chemotactic fluxes and implicit backward differentiation formula time integration is used to test these predictions. Numerical experiments recover the analytical equilibria and growth rates, identify the dominant unstable mode, reproduce the transition to spatial heterogeneity, and quantify the effects of immune recruitment, decay, and diffusion on the stability boundary. Grid-refinement, mass-balance, residual, and nonnegativity diagnostics support the computational reliability of the results.

math.AP↗

arXiv2026

Analysis of a finite element method for second order uniformly elliptic PDEs in non-divergence form

We propose one finite element method for both second order linear uniformly elliptic PDE in non-divergence form and the uniformly elliptic Hamilton-Jacobi-Bellman (HJB) equation. For both linear elliptic PDE in non-divergence form and the HJB equation, we prove the well-posedness of strong solution in $W^{2,p}(Ω)$ and optimal convergence in discrete $W^{2,p}$-norm of the finite element approximation to the strong solution for $1<p\leq 2$ on convex polyhedra in $\mathbb{R}^{d}$ ($d=2,3$). If the domain is a two dimensional non-convex polygon, $p$ is valid in a more restricted region. Furthermore, we relax the assumptions on the continuity of coefficients of the HJB equation, which have been widely used in literature.

math.NA↗

arXiv2026

Adjoint DSMC Method for Spatially Inhomogeneous Boltzmann Equation with General Boundary Conditions

We develop adjoint Direct Simulation Monte Carlo (DSMC) formulations for the spatially inhomogeneous Boltzmann equation with periodic, specular reflecting, diffuse thermal, and prescribed inflow boundary conditions. Periodic and specular boundaries are treated using a pathwise particle adjoint conditional on the realized event history. For diffuse thermal boundaries, we introduce a randomized-time regularization of wall-crossing events and use score-function terms to differentiate the resulting boundary probabilities. Reparameterization of the outgoing half-Maxwellian samples provides sensitivities with respect to wall temperatures and tangential wall velocities. Prescribed inflow requires a different construction because perturbations of incoming particles affect subsequent cell populations, local collision frequencies, and collision schedules. We therefore derive an ensemble adjoint based on the locally linearized Boltzmann collision operator and evaluate boundary sensitivities using local inflow-source scores, with injection counts held fixed. For a scalar objective, the dominant adjoint cost is largely independent of the number of parameters. Numerical experiments for Maxwell molecules validate the formulations against centered finite differences for thermal, mixed thermal-specular, two-sided inflow, and high-Mach Couette-flow configurations. The results demonstrate consistent gradient estimates, Monte Carlo convergence, stability with respect to the thermal regularization parameter, and accurate sensitivity calculation in a regime with limited relative statistical noise.

math.NA↗

arXiv2026

A Reduced Magnetic Vector Potential Approach with Higher-Order Splines

This work presents a high-order isogeometric formulation for magnetoquasistatic eddy-current problems based on a decomposition into Biot-Savart-driven source fields and finite-element reaction fields. Building upon a recently proposed surface-only Biot-Savart evaluation, we generalize the reduced magnetic vector potential framework to the quasistatic regime and introduce a consistent high-order spline discretization. The resulting method avoids coil meshing, supports arbitrary winding paths, and enables high-order field approximation within a reduced computational domain. Beyond establishing optimal convergence rates, the numerical investigation identifies the requirements necessary to recover high-order accuracy in practice, including geometric regularity of the enclosing interface, accurate kernel quadrature, and compatible trace spaces for the source-reaction coupling.

math.NA↗