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arXiv2026

Model reduction of port-Hamiltonian systems via neural networks

In this paper, we consider structure-preserving model reduction of port-Hamiltonian (pH) systems which extend classical Hamiltonian systems with dissipation and an input-output port. These pH systems are often used in multi-physics systems, as the interconnection of one or more \pH systems results again in a pH system. If particularly the system matrices associated with the interconnection and/or dissipation of a pH system are state-dependent, then the evaluation of standard reduced-order models (ROMs) may depend on the dimension of the original full-order model, resulting in high computational costs. To circumvent these high costs, we propose to use structure-preserving neural networks. In particular, we perform two steps: (1) we use the generalized manifold Galerkin projection to project the pH system onto the reduced space; then (2) we train a neural network to learn the map from the reduced-order state to the reduced-order interconnection and dissipation system matrices. To ensure that the resulting ROM is again a pH system, the architecture of the neural network is chosen such that the skew-symmetry and positive semi-definiteness of the reduced-order systems matrices are maintained. In a numerical example, we consider a nonlinear mass-spring-damper system with state-dependent system matrices. The numerical results show that the proposed method achieves a significant computational speed-up compared to the original \ROM with comparable accuracy.

math.NA↗

arXiv2026

Well-conditioned iterative methods for large open quantum systems

Markovian open quantum systems are well modeled by the Lindblad Master Equation (ME) $\frac{\mathrm{d}}{\mathrm{d} t} ρ_t = \mathcal{L} ρ_t$, where $\mathcal{L}$ is a linear (super-)operator and $ρ_t$ is the system state, a positive matrix. When designing or characterizing a quantum system, one is usually interested in the steady state $ρ_\infty$ (such that $\mathcal{L} ρ_\infty = 0$), the first few excited states, and trajectories $t\mapsto ρ_t$. In finite dimension, $ρ_t$ is an $n\times n$ matrix, $\mathcal{L}$ thus typically costs $n^4$ to store explicitly as a dense matrix, and $O(n^6)$ to diagonalize or invert exactly, making standard linear algebraic techniques expensive for large systems. However, $\mathcal{L}$ usually costs only $O(n^3)$ to apply. This makes iterative methods appealing, but they do not work without a good preconditioner. In this article, our main observation is that a part of the Lindblad equation, corresponding to the so-called no-jump evolution $\mathcal{S}$, can be inverted efficiently. Using this inverse map, we introduce an auxiliary completely positive trace-preserving (CPTP) map $Φ$ whose fixed point is directly related to $ρ_\infty$, all the other eigenvalues having smaller magnitude. The map $Φ$ is thus well suited to iterative methods, and $ρ_\infty$ can be found in a few Arnoldi iterations. Using the same inverse map $\mathcal{S}^{-1}$ as preconditioner, we compute the low-lying spectrum efficiently via shift-invert Arnoldi, and, as a proof of concept, build an implicit time integrator that is competitive on stiff systems in the low-precision regime. For the steady-state and low excited states problems, our methods scale like $O(n^3)$ per iteration and offer state-of-the-art performance on CPU and GPU.

quant-ph↗

arXiv2026

Constrained minimax approximation for quantum signal processing

Quantum signal processing (QSP) provides a simple and efficient framework for implementing polynomial transformations using quantum circuits. Its classical design stage leads to a constrained minimax approximation problem: find a polynomial of prescribed parity that approximates a target function uniformly on a fitting set while remaining bounded in magnitude by one on the domain $[0,1]$, which can be viewed as a semi-infinite constraint. Discretization converts the problem into a linear program, but feasibility at a set of finitely many sampled points does not ensure feasibility on the whole domain, especially when an optimal approximant reaches the boundary of the feasible set. We investigate two approaches to address this difficulty. A Remez exchange method combined with active-set constraint enforcement is efficient on many tested instances, but its stability depends on the target and problem geometry. We then introduce nonlinear Fourier retraction, which uses QSP completion and phase synthesis to turn a nearly feasible polynomial into phase factors for a feasible QSP polynomial without increasing the degree. Across representative problems, retraction largely preserves approximation accuracy and remains effective on instances where the Remez heuristic is unstable. The resulting workflow connects classical minimax approximation and semi-infinite optimization with nonlinear Fourier analysis, and is implemented in the qsppack software package.

quant-ph↗

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

Solving the Incompressible Navier-Stokes Equations on Oriented Curved Surfaces Discretized by Point Clouds

We present a meshfree numerical solver for the incompressible Navier-Stokes equations on oriented curved surfaces that are represented by surface point clouds. On curved surfaces, numerical challenges pertaining to stiffness and pressure-velocity coupling are exacerbated. Moreover, vector calculus on curved surfaces differs from its Euclidean counterpart. The presented method operates on surface point clouds in an Eulerian frame of reference without requiring a computational grid or mesh. It achieves consistent approximation in space and time with high order of accuracy; we demonstrate up to order six. The incompressibility constraint is locally imposed as a weak artificial compressibility approximation, avoiding global matrix inversion. We show that the method provides consistent and convergent approximations of surface vector fields and differential operators. We study the relationship between error, spatial resolution, and artificial Mach number and characterize the frequency spectrum of the artificial oscillations. We provide numerical solutions of the incompressible Navier-Stokes equations on symmetric surfaces, such as the sphere and torus, and on parametric and non-parametric asymmetric surfaces. Since the proposed method works directly on unstructured surface point clouds, it provides a promising approach for simulations on image-derived geometries, such as in biological morphogenesis from microscopy videos.

math.NA↗

arXiv2026

Modeling of Mobility and Energy Policies in an Agent-Based Framework: Case Studies for Chicago Region in 2050

Metropolitan regions are simultaneously pursuing several interventions to improve mobility, accessibility, and energy efficiency, necessitating integrated tools to understand how these policies interact to affect travel behavior, energy use, and infrastructure needs. This paper evaluates the combined impacts of electrification, freight demand management, road pricing, parking reform, and transit expansion on the Chicago metropolitan transportation system in 2050, using a business-as-usual (BAU) scenario as the baseline. We employ POLARIS, a large-scale agent-based modeling framework calibrated to 2019 conditions, to simulate nine policy scenarios for the seven-county northeastern Illinois region. The framework co-simulates activity-based passenger demand, endogenous freight generation, multimodal traffic assignment, and transit operations, with charging infrastructure and freight operations optimized for each case. Our findings reveal that under the high electrification scenario, total fuel mass declines by 68% while total charging energy increases by approximately 4-8x from BAU, resulting in a peak power demand near 4 GW concentrated in the urban core. Furthermore, freight management policies reduce freight VMT by increasing trip frequency but shortening distances, smart road pricing most effectively reduces auto VMT, and transit expansion boosts ridership by 18% relative to BAU. By presenting the first integrated, agent-based scenario framework for Chicago that jointly evaluates these interventions, this study provides actionable insights for regional transportation planning, grid infrastructure investment, and emissions reduction, highlighting the value of targeted charger upgrades and coordinated policy bundles.

physics.soc-ph↗

arXiv2026

Latent-Space No-Arbitrage Geometry of Generative Models for Implied Volatility Surfaces

Generative models for implied volatility surfaces must produce outputs that satisfy static no-arbitrage constraints. We study these constraints in latent space. For a fixed generator, we assign each latent code a scalar margin determined by the no-arbitrage conditions of the generated surface. The codes with nonnegative margin form the admissible latent set. We establish conditions under which strictly admissible codes remain admissible under small perturbations and the boundary of the admissible set is characterized by zero margin. For regular boundary components, we formulate a level-set equation whose local dynamics are directed toward the zero-margin set. The analysis treats the generator as a map from latent variables to surfaces and is therefore not restricted to a particular architecture. It applies to variational autoencoders, generative adversarial networks, and other generative models with a deterministic realization map. Numerical tests recover known boundaries in analytic examples. Experiments with a variational autoencoder trained on Heston surfaces show that similar reconstruction errors can correspond to different admissible regions and that the latent prior may be concentrated inside such a region. The computed boundary can also be used to modify latent codes that generate violating surfaces.

q-fin.CP↗

arXiv2026

Single- and Multilevel Quadrature with Error Control for Fourier Pricing under the Rough Heston Model

Unlike the classical Heston model, Fourier pricing under the rough Heston model requires solving a fractional Riccati equation at every quadrature point. Since the required resolution varies with model parameters and quadrature point, a single uniform time discretization can be inefficient. We develop single- and multilevel Gauss-Laguerre quadrature methods that balance the time discretization and Fourier quadrature errors. Both methods scale the laguerre weight to the estimated Fourier integrand decay. The single-level method allocates a prescribed tolerance between the two errors. The multilevel method splits the integrand into a level-zero term and level differences, selecting quadrature points separately at each level. Suppose that the Fourier integrand discretization error is $O(Δt^p)$, that evaluating the characteristic function once costs $O(Δt^{-β})$, and that the algebraic Gauss-Laguerre quadrature error is $O(N^{-s_{SL}/2})$, where $s_{SL}$ is the smoothness index. Under this estimate and assumptions on the regularity and decay of level differences, we prove that the proposed single-level method requires $O(ε^{-(β/p+2/s_{SL})})$ computational work to achieve accuracy $ε$, whereas the proposed multilevel method requires $O(ε^{-β/p})$ computational work. We also study root-exponential Gauss-Laguerre error models for practical multilevel quadrature allocation. Numerical experiments support the observed fractional Riccati and Fourier integrand convergence rates and root-exponential quadrature behavior, and show substantial reductions in quadrature cost from the proposed scaling. The multilevel method provides clear computational savings over the single-level method. We further benchmark the multilevel fractional Riccati method against the BL2 Markovian approximation and report lower total CPU time in the tested configurations.

q-fin.CP↗

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

Higher Order Multidimensional Slope Limiters with Local Maximum Principles

Higher-order numerical methods are used to find accurate numerical solutions to hyperbolic partial differential equations. Limiting is required to either converge to the correct type of solution or to adhere to physically motivated local maximum principles and less restrictive limiting procedures are required so as to not severely decrease the accuracy. In this paper, we adapt the existing slope limiter framework introduced in [Zhang \& Shu, J. Comput. Phys., 229(9):3091-3120, 2010] to achieve distinct local boundedness principles. We conclude that quadrature points contributing to numerical fluxes on either side of a face can be limited based on shared face-defined maximum principles and the resulting cell mean at the next timestep satisfies a cell mean maximum principle. Furthermore additional points arising in a decomposition of a cell mean must be limited locally when going beyond piecewise linear reconstructions. This allows the design of new multidimensional limiters which at second order can attain the same cell mean maximum principle as existing slope limiters, but allows more of the higher order flux to be used, generalises beyond second order schemes and can be modified for user specified local maximum principles.

math.NA↗

arXiv2026

Parameter optimization for restarted mixed precision iterative sparse solver

The problem of optimal precision switching for the conjugate gradient (CG) method applied to sparse linear systems is considered. A sparse matrix is defined as an $n\!\times\!n$ matrix with $m\!=\!O(n)$ nonzero entries. The algorithm first computes an approximate solution in single precision with tolerance $\varepsilon_1$, then switches to double precision to refine the solution to the required stopping tolerance $\varepsilon_2$. Based on estimates of system matrix parameters -- computed in time which does not exceed $1\%$ of the time needed to solve the system in double precision -- we determine the optimal value of $\varepsilon_1$ that minimizes total computation time. This value is obtained by classifying the matrix using the $k$-nearest neighbors method on a small precomputed sample. Classification relies on a feature vector comprising: the matrix size $n$, the number of nonzeros $m$, the pseudo-diameter of the matrix sparsity graph, and the average rate of residual norm decay during the early CG iterations in single precision. We show that, in addition to the matrix condition number, the diameter of the sparsity graph influences the growth of rounding errors during iterative computations. The proposed algorithm reduces the computational complexity of the CG -- expressed in equivalent double-precision iterations -- by more than $17\%$ on average across the considered matrix types in a sequential setting. The resulting speedup is at most $1.5\%$ worse than that achieved with the optimal (oracle) choice of $\varepsilon_1$. While the impact of matrix structure on Krylov subspace method convergence is well understood, the use of the sparsity graph diameter as a predictive feature for rounding error growth in mixed-precision CG appears to be novel. To the best of our knowledge, no prior work employs graph diameter to guide precision switching in iterative linear solvers.

math.NA↗

arXiv2026

Structural Packing and Dyadic Factorization of Sparse Positive Definite Matrices

Efficient inversion of large sparse positive definite matrices requires exploiting sparsity patterns beyond those captured by conventional bandwidth reduction. In this work, we recast nested dissection, a prominent alternative, as a two-stage framework. The matrix was first packed into block-tridiagonal or dyadic form, followed by sparse Gram-Schmidt orthogonalization. This decomposition provided a unified perspective on sparse matrix factorization and inversion and identified dyadic structure as a fundamental component of sparse Cholesky factorization. For the first stage, we introduced a packing algorithm that recovered block-tridiagonal and dyadic patterns using a novel $\ell_1$ criterion. Using approximate distances obtained through classical multidimensional scaling, the method was effective when the target structure was sufficiently represented among the nonzero entries. Iterative application could also remove structural noise and reveal hidden dyadic organization, corresponding to separator identification in nested dissection. For the second stage, we developed the theory of dyadically structured matrices. We derived sparse factorization and inversion procedures, analyzed their computational complexity, and obtained an efficient inversion algorithm. A modified version reduced the cost of inverting block-tridiagonal matrices, demonstrating the benefit of exploiting their structure directly rather than treating them as generic band matrices.

math.NA↗

arXiv2026

Code Generation for Near-Roofline Finite Element Actions on GPUs from Symbolic Variational Forms

We present a novel parallelization strategy for evaluating Finite Element Method (FEM) variational forms on GPUs, focusing on those that are expressible through the Unified Form Language (UFL) on simplex meshes. We base our approach on code transformations, wherein we construct a space of scheduling candidates and rank them via a heuristic cost model to effectively handle the large diversity of computational workloads that can be expressed in this way. We present a design of a search space to which the cost model is applied, along with an associated pruning strategy to limit the number of configurations that need to be empirically evaluated. The goal of our design is to strike a balance between the device's latency-hiding capabilities and the amount of state space, a key factor in attaining near-roofline performance. To make our work widely available, we have prototyped our parallelization strategy within the Firedrake framework, a UFL-based FEM solver. We evaluate the performance of our parallelization scheme on three generations of Nvidia GPUs, specifically the H200, Titan V and Tesla K40c, across a range of operators commonly used in applications, including fluid dynamics, wave propagation, and structural mechanics, in 2D and 3D geometries. Our results demonstrate that our proposed algorithm achieves more than $50\%$ roofline performance in $60\%$ of the test cases.

cs.DC↗

arXiv2026

Easy-to-Implement One-Step Schemes for Stochastic Integration

Convenient, easy to implement stochastic integration methods are developed on the basis of abstract one-step deterministic order $p$ integration techniques. The abstraction as an arbitrary one step map allows the inspection of easy to implement stochastic exponential time differencing Runge-Kutta (SETDRK), stochastic integrating factor Runge-Kutta (SIFRK) and stochastic RK (SRK) schemes. Such schemes require minimal modifications to existing deterministic schemes and converging to the Stratonovich SDE. These schemes capture all symmetric terms in the Stratonovich-Taylor expansion, are order $p$ in the limit of vanishing noise, can attain at least strong order $p/2$ or $p/2-1/2$ (parity dependent) for drift commutative noise, strong order $1$ for commutative noise, and strong order $1/2$ for multidimensional non-commutative noise. Numerical convergence is demonstrated using different bases of noise for 2nd, 3rd and 4th order SETDRK, SIFRK and SRK schemes.

math.NA↗

arXiv2026

A Practical Mode-parallel Implementation of the (H-)Tucker Decomposition via Randomization

In the last decades, tensors have emerged as the right tool to represent multidimensional data in a compact yet informative manner. Moreover, it is well-known that by performing low-rank factorizations of such tensors one is often able to effectively unveil possible hidden structure in data, mainly due to unexpected dependencies among the different variables encoded in the given tensor. However, computing these factorizations is extremely energy-consuming and memory-demanding, especially for high-dimensional tensors, namely those with a large number of modes. In this paper we focus on two state-of-the-art tensor decompositions: the Tucker and H-Tucker decompositions. We propose novel numerical strategies able to perform these factorizations in a mode-parallel fashion, that is the operations required by the algorithm along all modes are performed in parallel. This is in contrast to what is achieved by many procedures available in the literature that parallelize some of the operations along each mode, e.g., tensor-times-matrix steps, while still visiting one mode at the time in a sequential manner. Our strategies make use of cutting-edge randomization techniques comprising fiber sampling and randomized range-finding steps. In case of Tucker decomposition, we provide upper bounds on the expected value of the error provided sufficiently large sampling parameters have been adopted. A panel of numerical results showcases the potential of our approach in reducing both the running time and the storage demand of computing Tucker and H-Tucker decompositions. Moreover, experiments carried out in HPC environments illustrate the good scaling of our mode-parallel approach.

math.NA↗

arXiv2026

Five-Structures Preserving Algorithm for charge dynamics model

This paper develops a family of fast, structure-preserving numerical algorithms for the nonlinear Maxwell-Ampere Nernst-Planck equations. For the first-order scheme, the Slotboom transformation rewrites the Nernst-Planck equation to enable positivity preservation. The backward Euler method and centered finite differences discretize the transformed system. Two correction strategies are introduced: one enforces Gauss's law via a displacement correction, and the other preserves Faraday's law through potential reconstruction. The fully discrete scheme exactly satisfies mass conservation, concentration positivity, energy dissipation, Gauss's law, and Faraday's law, with established error estimates. The second-order scheme adopts BDF2 time discretization while retaining the same structure-preserving strategies, exactly conserving mass, Gauss's law, and Faraday's law. Numerical experiments validate both schemes using analytical solutions, confirming convergence orders and positivity preservation. Simulations of ion transport with fixed charges demonstrate exact preservation of Gauss's and Faraday's laws over long-time evolution, reproducing electrostatic attraction, ion accumulation, and electric field screening. The results fully support the theoretical analysis and the schemes' stability and superior performance.

math.NA↗

arXiv2026

Phase-field digital image correlation for integrated displacement and damage measurements

This work presents a novel digital image correlation (DIC) framework for full-field measurements of displacement, strain, and damage, based on a phase field (PF) approach. The idea is to take advantage of the ability of the PF method to track complex crack morphologies and to provide a natural way in DIC to perform damage and crack measurements from experimental speckle images, in addition to displacement and strain fields. Moreover, incorporating the damage variable into DIC can improve the displacement accuracy near the crack tip, and can avoid the need of user-defined masks when dealing with cracked samples, which is advantageous when cracks become complex and the manual application of masks becomes challenging. The theoretical formulation of the proposed framework, namely PF-DIC, was presented in detail in the paper, along with a finite element implementation. Numerical examples have demonstrated the capability of the proposed PF-DIC in terms of capturing different types of cracks while providing similar measurement accuracy to that of masked DIC. Additionally, it is shown that the PF-DIC can be easily adapted to selectively identify critical cracks under specific loading conditions or mechanisms for damage assessment and diagnostic purposes. The proposed DIC framework can be used to characterize material defects, support structural health monitoring, and enable a potential unification of PF simulations and experimental fracture measurements

math.NA↗

arXiv2026

3D Uncertainty Quantification for Photoacoustic Tomography

Photoacoustic tomography (PAT) is a promising modality for high-resolution biomedical imaging, motivating the need for reliable uncertainty quantification (UQ) of reconstructed images. Bayesian approaches provide a rigorous framework for UQ but remain computationally challenging for realistic three-dimensional PAT and are sensitive to numerical approximations in the governing wave equation. We develop a finite-element Bayesian UQ framework for PAT that accommodates complex computational domains and detector geometries while enabling large-scale three-dimensional inference. The proposed methodology reformulates the randomize-then-optimize (RTO) sampling strategy as a matrix-free algorithm that generates independent posterior samples using only forward and adjoint wave propagations. Particular attention is given to constructing an adjoint discretization that forms an exact transpose pair with the discrete forward operator while remaining consistent with the continuous PAT adjoint, enabling efficient least-squares solvers within the sampling procedure. We investigate the influence of temporal discretization, artificial boundary conditions, and adjoint consistency on posterior uncertainty and identify discretization strategies that avoid numerical artifacts. The framework is validated against exact posterior statistics, existing Bayesian PAT methods, and Hamiltonian Monte Carlo using the No-U-Turn Sampler (NUTS), and is demonstrated on a three-dimensional problem with approximately $2\times 10^5$ unknowns on a general finite-element domain. To the best of our knowledge, this is the first large-scale Bayesian PAT study on general three-dimensional finite-element geometries, and the methodology extends naturally to a broad class of linear PDE-constrained inverse problems.

math.NA↗