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arXiv2026

Weighted Laplace Spaces for Spectral Measures and Rational Approximation

We introduce the weighted Laplace space $H_w$, an RKHS of Laplace transforms on $(0,\infty)$, and study spectral measures in its dual space $H'_w$. For conforming FEM discretizations of the Dirichlet Laplacian on bounded Lipschitz domains, we prove the dual-norm inequality $\|μ_h\|_{H'_w} \leq \|μ\|_{H'_w}$, where $μ= \sum_kδ_{λ_k}$ and $μ_h = \sum_k δ_{λ_{k,h}}$. The proof combines min-max monotonicity of FEM eigenvalues with a heat-trace representation of the dual norm. We then analyze $H_w$-adapted rational approximation of shifted symbols $ϕ(x)=(x+κ^2)^{-β}$ and give a conditional transfer principle for estimates proved in the corresponding weighted Laplace pre-image norm. Via dual pairing, the norm inequality yields uniform bounds for finite spectral sums and related transformed observables.

math.NA↗

arXiv2026

Effects of Trotter Error, Digitization Error, and Initial State Overlap on Tapered Quantum Phase Estimation for Minimum Eigenvalue Computation

We numerically implement full quantum circuits of optimal tapered Quantum Phase Estimation (in the form of an implementable approximate bandwidth-limited DPSS taper) for the purpose of minimum eigenvalue computation, where the Hamiltonian time-evolution unitary is implemented using standard Trotterization, and the taper state is constructed using an optimized bandwidth-limited taper. We illustrate, on representative quantum Hamiltonians, with tractable exact classical numerical quantum circuit simulations, how tapered QPE performs with respect to optimal phase sampling rate, absolute error, Shannon entropy of the optimal phase distribution, when the tQPE algorithm parameters are changed. Those algorithm parameters are Trotter error, finite sampling, imperfect initial states, total evolution time, and digitization error (number of phase qubits). The reported numerical experiments include up to $\ell=10$ phase qubits of precision with $m=3$ additional phase-register error suppression qubits, on a $4$-qubit Heisenberg quantum magnet model Hamiltonian, using up to 12th-order Trotterization. We numerically show i) the steady-state optimal phase sampling rate is determined by the initial state overlap with the ground-state, ii) with respect to phase sampling probability, or overall eigenvalue sampling error rate, there is no strong evolution time dependence for tQPE unless there is high Trotter error, iii) the approximated time-evolution unitaries in tQPE, like in standard QPE, can result in substantial ringing, which leads to non-physical eigenvalue estimates.

quant-ph↗

arXiv2026

Convergence rates of randomly pivoted methods for low-rank approximation

Randomly pivoted Cholesky, QR, and LU are iterative algorithms that form structured low-rank approximations of a matrix by sampling columns, or rows and columns, from the residual. We give convergence rates for these methods that depend on the decay of the singular values of the original matrix. The rates hold, up to a constant, when pivots are drawn from an approximation to the exact distribution. This extra degree of freedom can be used to trade exact sampling for cheaper approximations. We use it to establish rates for variants of randomly pivoted LU, including one that picks pivots using a sketch of the input matrix.

math.NA↗

arXiv2026

A Preliminary Study on the Dimensional Stability Classification of Polynomial Spline Spaces over T-meshes

This paper studies dimensional stability of polynomial spline spaces over T-meshes with highest-order smoothness. Dimensional stability is defined as the invariance of the dimension of the spline space over structurally isomorphic T-meshes, where a structurally isomorphic map preserves edge intersections and the mutual positions of all edges. Absolute stability, a stronger notion, is introduced via structurally similar maps that depend solely on the topology of T-connected components. It is proved that every T-connected component admits a unique decomposition into a diagonalizable part and a completely non-diagonalizable component (CNDC). This reduces stability to the rank stability of the conformality matrix associated with the CNDC. For diagonalizable T-meshes, the conformality vector space decomposes as a direct sum over independent T $l$-edges, supporting basis function construction per T $l$-edge. These results provide a systematic framework for classifying dimensional stability and a rigorous theoretical foundation for constructing basis functions of spline spaces over T-meshes.

math.NA↗

arXiv2026

Sloshing reduction in a swaying tank with porous baffles using scaled boundary finite element method

Sloshing is an inevitable phenomenon in an ocean-going vessel that can have adverse effects. In this work, the mitigation of sloshing is investigated using multiple thin porous baffles of various configurations in a partially filled swaying tank. The boundary value problem is solved within the framework of a linearized potential flow theory using the scaled boundary finite element method (SBFEM). The flow through the thin porous baffles is assumed to follow Darcy's law. The computational domain is divided into a minimum number of subdomains due to the presence of porous baffles and to ensure star convexity. Higher-order polynomials are used along each subdomain edge to represent the unknown field, i.e., velocity potential. The developed numerical model is validated with the known results in the literature. Subsequently, various results, such as the amplification factor and the forces of the tank wall, are presented and discussed for the effect of configuration, porosity, slosh tank width, depth of baffle submergence and the space between adjacent baffles. From the parametric study, it is observed that top-mounted baffles enhance sloshing suppression by $50\%$ compared to bottom-mounted vertical baffles, considering all sloshing modes. Assessing the overall effectiveness, top-mounted convex baffle configuration emerges as the most efficient configuration for sloshing suppression, achieving a well-balanced reduction across all modes.

physics.flu-dyn↗

arXiv2026

The Sample Complexity of Learning Lipschitz Operators with respect to Gaussian Measures

Operator learning, the approximation of mappings between infinite-dimensional function spaces using machine learning, has gained increasing research attention in recent years. Operator approximations can serve as efficient surrogate models for problems in computational science and engineering, complementing traditional methods. However, despite their empirical success, our understanding of the underlying mathematical theory is in large part still incomplete. In this paper, we study the approximation of Lipschitz operators with respect to Gaussian measures. We prove higher Gaussian Sobolev regularity of Lipschitz operators and establish lower and upper bounds on the Hermite polynomial approximation error. We then study general reconstruction strategies of Lipschitz operators from $m$ arbitrary (potentially adaptive) linear samples. As a key finding, we tightly characterize the corresponding sample complexity, that is, the smallest achievable worst-case error among all possible choices of (adaptive) sampling and reconstruction strategies, in terms of $m$. As a consequence, we identify an inherent curse of sample complexity: No method to approximate Lipschitz operators based on $m$ linear samples can achieve algebraic convergence rates in $m$. On the positive side, we prove that a sufficiently fast spectral decay of the covariance operator of the underlying Gaussian measure guarantees convergence rates which are arbitrarily close to any algebraic rate. Overall, by tightly characterizing the sample complexity, our work confirms the intrinsic difficulty of learning Lipschitz operators, regardless of the data or learning technique.

cs.LG↗

arXiv2026

A Thermodynamically Consistent High-Order Framework for Staggered Lagrangian Hydrodynamics

We present a consistent high-order staggered Lagrangian hydrodynamics framework designed to reconcile an underlying disparity in existing curvilinear formulations: the mismatch between quadrature-based "strong" mass conservation and the discrete degrees of freedom (DOFs) of thermodynamic variables. By mathematically coupling the numerical quadrature rule with the density representation, our approach ensures rigorous point-wise consistency between density, internal energy, and pressure. This synchronization eliminates the ambiguity of equation-of-state (EOS) updates inherent in previous high-order staggered methods. To stabilize the discretization, we develop a high-order generalization of the subzonal pressure method by conceptually enriching the pressure field from the $Q^{m-1}$ to the $Q^m$ finite element space. We prove that evaluating this enriched field using a high-order quadrature rule naturally generates a restorative anti-hourglass force, which exactly recovers the classical $Q^1-P^0$ compatible hydrodynamics algorithm as a limiting case for $m=1$. Furthermore, we introduce a concise, algorithmic formulation of tensor artificial viscosity that streamlines implementation and significantly reduces computational overhead in high-order settings. The resulting framework yields strictly diagonal mass matrices for both momentum and energy equations, enabling highly efficient, fully explicit time integration without global linear solves. Extensive numerical benchmarks, including smooth convergence tests and complex shock-dominated flows, demonstrate that the proposed method achieves optimal high-order accuracy while maintaining superior geometric robustness.

math.NA↗

arXiv2026

Exact affine conditioning beyond Gaussians: a unique characterization of the ensemble Kalman update

The analysis step of the stochastic ensemble Kalman filter, called the ensemble Kalman update (EnKU), is widely used for approximating posterior distributions in inverse problems and data assimilation. The EnKU approximates the posterior distribution $π_{X\mid Y=y_\star}$ by pushing forward the joint distribution $(X,Y)\simπ$ through an affine map $L^{\mathrm{EnKU}}_{π,y_\star}(x,y)$ that depends only on the covariance structure of $π$ and the observation $y_\star$. While the EnKU yields the exact posterior for Gaussian $π$ in the mean-field, this property alone does not uniquely determine the EnKU. In fact, there are infinitely many affine maps $L_{π, y_\star}$ that achieve such exact conditioning. In this paper, we offer a novel characterization of the EnKU among all such affine maps. We first exhaustively characterize the set ${E}^{\mathrm{EnKU}}$ of joint distributions for which the EnKU yields exact conditioning, showing that it is much larger than the set of Gaussians. Next, we show that except for a small class of highly symmetric distributions within ${E}^{\mathrm{EnKU}}$, the EnKU is the {unique} exact affine conditioning map. Further, we characterize the largest possible set of distributions ${F}$ for which a distribution-dependent, weakly observation-dependent, affine map exists, a class of transports that naturally includes the EnKU. We show that ${F}={E}^{\mathrm{EnKU}}\cup{S}_{\mathrm{nl-dec}}$ with a small symmetry class ${S}_{\mathrm{nl-dec}}$, meaning that for affine conditioning beyond the Gaussian setting, the EnKU has an exact set that is essentially maximally large.

math.ST↗

arXiv2026

Accelerated Sampling with the Inertial Langevin Algorithm

We consider the \emph{inertial Langevin algorithm} (ILA), a simple momentum-based method for sampling from Gibbs distributions of the form $π(x) \propto \exp(-U(x))$. ILA augments the unadjusted Langevin algorithm with an inertia term and a matching noise rescaling, yielding a sampling analogue of Polyak's heavy-ball method from optimization. This modification provably accelerates convergence. For Gaussian targets, ILA attains Wasserstein-2 accuracy $δ$ in $\tilde{\mathcal O (\sqrtκ/\sqrtδ)$ iterations, where $κ= L/m$ denotes the condition number and $\tilde{\mathcal O}$ hides logarithmic factors, thus reaching the ballistic $\sqrtκ$ complexity known from accelerated optimization and proven to be a lower bound for sampling from Gaussians. Moreover, we show that the method's bias can be fully removed and, thus, complexity reduced to $\tilde \Oc(\sqrtκ)$ by considering the sequence consisting of the average of ever two consecutive ILA iterates, which is shown to be equivalent to the well-known BAOAB splitting scheme in the linear setting. Beyond Gaussians, for $L$-smooth potentials with sufficient growth at infinity, we prove geometric ergodicity of ILA and of its continuous-time analogue, the underdamped Langevin dynamics, together with convergence of the discretization bias to zero as the step size decreases. These guarantees hold under considerably simpler parameter restrictions than previously available in the literature, in particular, not imposing a lower bound on the friction, which is conjectured to be crucial for acceleration. We underpin our theoretical findings with numerical experiments covering ill-conditioned Gaussian distributions, total variation image denoising, and the challenging task of maximum likelihood learning of an energy-based model for molecular structure generation.

math.NA↗

arXiv2026

Collagen and myocyte interplay in cardiac volume overload: a multi-constituent growth and remodeling framework

Hearts subjected to volume overload (VO) are prone to detrimental anatomical and functional changes in response to elevated mechanical loading, ultimately leading to heart failure. Experimental findings now emphasize that organ-scale changes following VO cannot be explained by myocyte growth alone, as traditionally proposed in the literature. Collagen degradation, in particular, has been associated with VO and assumed to play a central role in both its acute and chronic stages. This hypothesis, however, remains to be substantiated by comprehensive mechanistic evidence, and each constituent contribution to myocardial growth and remodeling (G&R) processes is yet to be quantified. In this work, we present a multi-constituent G&R framework that integrates a mixture-based constitutive model within the kinematic growth formulation. This framework enables us to mechanistically assess the relative contributions of collagen and myocyte changes to alterations in tissue properties, ventricular dimensions, and growth phenotype. Our numerical results confirm that collagen remodeling affects the passive mechanical response of the myocardium, whereas myocytes predominantly influence the extent and phenotype of VO-induced growth. Importantly, collagen degradation exacerbates myocyte hypertrophy, revealing a synergistic interplay that accelerates the left ventricular eccentric growth and thereby promotes systolic dysfunction. This work constitutes an important step towards an integrated characterization of the early compensatory stages of VO-induced cardiac G&R.

physics.med-ph↗

arXiv2026

An Euler scheme for BSDEs via the Wiener chaos decomposition

The Euler scheme is a standard time discretization for BSDEs, but its implementation hinges on approximating conditional expectations and the associated martingale terms at each time step. We propose an implementation based on the Wiener chaos decomposition to approximate these quantities. In contrast to many numerical schemes that rely on a finite-dimensional Markovian representation, our approach accommodates arbitrary $\mathcal{F}_T$-measurable square-integrable terminal conditions. We provide a comprehensive convergence analysis under additional Malliavin regularity assumptions and illustrate the method on several numerical examples, including genuinely non-Markovian problems arising, for instance, in the pricing and hedging of contingent claims under rough-volatility models.

math.NA↗

arXiv2026

A randomized global GMRES method for matrix equations

In this paper, we develop a new Randomized Global Generalized Minimum Residual (RGlGMRES) algorithm for efficiently computing solutions to large scale linear systems with multiple right hand sides. The proposed method builds on a recently developed randomized global Gram Schmidt process, in which sketched Frobenius inner products are employed to approximate the exact Frobenius inner products of high dimensional matrices. We give some new convergence results of the RGlGMRES method for multiple linear systems. In the case where the coefficient matrix A is diagonalizable, we derive new upper bounds for the randomized Frobenius norm of the residual. In this paper, we study how to introduce matrix sketching in this algorithm. It allows us to reduce the dimension of the problem in one of the main steps of the algorithm. We also show how to apply the RGlGMRES method for solving the Lyapunov matrix equation. To validate the effectiveness and practicality of this approach, we conduct several numerical experiments, which demonstrate that our RGlGMRES method is competitive with the global GMRES method for solving large scale multiple linear systems and Lyapunov matrix equation.

math.NA↗

arXiv2026

Computing accurate singular values using a mixed-precision one-sided Jacobi algorithm

We present a relative forward error analysis of a mixed-precision preconditioned one-sided Jacobi algorithm, analogous to a two-sided version introduced in [N. J. Higham, F. Tisseur, M. Webb and Z. Zhou, SIAM J. Matrix Anal. Appl. 46 (2025), pp. 2423-2448], which uses low precision to compute the preconditioner, applies it in high precision, and computes the singular value decomposition using the one-sided Jacobi algorithm at working precision. Our analysis yields relative forward error bounds involving the scaled condition number of the preconditioned matrix rather than that of the input matrix, potentially improving accuracy when the former is smaller. We present and analyze two approaches for constructing effective preconditioners. Our numerical experiments demonstrate that our algorithm achieves smaller relative forward errors than the LAPACK routines $\texttt{DGESVJ}$ and $\texttt{DGEJSV}$, as well as the MATLAB function $\texttt{svd}$, particularly for ill-conditioned matrices. Timing tests show that our approach accelerates the convergence of the Jacobi iterations and that the dominant cost arises from a single high-precision matrix-matrix multiplication. With improved software or hardware support for this bottleneck, our algorithm would be faster than the LAPACK one-sided Jacobi algorithm $\texttt{DGESVJ}$ and comparable in speed to the state-of-the-art preconditioned one-sided Jacobi algorithm $\texttt{DGEJSV}$, but much more accurate.

math.NA↗

arXiv2026

Quadratization of Autonomous Partial Differential Equations: Algorithmic Solutions

Quadratization for partial differential equations (PDEs) is a process that formally transforms a PDE with a nonquadratic right-hand side into a quadratic form by introducing auxiliary variables. Even though the existence and uniqueness of the solution of this quadratic form are, as of yet, unknown in the general case, this symbolic transformation has been used in diverse fields to simplify the analysis, simulation, and control of PDE models. This paper presents a rigorous definition of PDE quadratization, a sample case study on the solutions of quadratic representations, and theoretical contributions for the PDE quadratization problem of spatially one-dimensional PDEs, including results on existence and complexity. Its main focus, however, is introducing and analyzing QuPDE, an algorithm based on symbolic computation and discrete optimization that outputs a quadratization for any spatially one-dimensional polynomial or rational PDE. This algorithm is the first computational tool to find quadratizations for PDEs to date. We demonstrate QuPDE's performance by applying it to fourteen nonquadratic PDEs in diverse areas such as fluid mechanics, space physics, chemical engineering, and biological processes. QuPDE delivers a low-order quadratization in each case, uncovering quadratic transformations with fewer auxiliary variables than those previously discovered in the literature for some examples, and finding quadratizations for systems that had not been transformed to quadratic form before.

cs.SC↗

arXiv2026

On a blended Anderson / NGMRES acceleration for the Navier-Stokes Picard iteration: convergence analysis and adaptive depth selection

Anderson acceleration (AA) and nonlinear GMRES (NGMRES) have been successfully applied to improve the efficiency and robustness of many nonlinear solvers, with each being shown to have certain advantages over the other. There are two differences between the AA and NGMRES algorithms: the extrapolations are different, and AA uses fixed point residuals in the optimization problem while NGMRES uses true nonlinear residuals. Motivated by recent AA and NGMRES analyses and numerical testing, we consider a blend of AA and NGMRES that uses the AA extrapolation and the true nonlinear residuals for the optimization problem. To focus our study, we consider the blended method applied to the Picard iteration for the Navier-Stokes equations, but many of the ideas developed herein are extendable to a wide variety of nonlinear systems and solvers. We provide a convergence analysis and proof of acceleration of the blended method, provide a detailed numerical comparison of AA, NMGRES and the blended method for several benchmark tests, and develop an effective adaptive depth strategy.

math.NA↗

arXiv2026

Second-order consistency for learning chaotic dynamics via randomized Jacobian matching

Short-horizon accuracy does not ensure that a learned chaotic system has correct long-time dynamics. Trajectory (zeroth-order) matching constrains vector-field values, and Jacobian (first-order) matching constrains local tangent dynamics, but neither determines how the Jacobian varies away from supervised states, so a model can be locally accurate while drifting toward spurious attractors and distorting long-time statistics. We show that second-order supervision mitigates these failures. Because forming full Hessian tensors is computationally prohibitive in high dimensions, we propose model-constrained randomized Jacobian matching, which compares the Jacobians of the true and learned vector fields at randomly perturbed inputs. A Taylor expansion shows that the expected randomized Jacobian loss decomposes into the Jacobian mismatch plus a Hessian mismatch scaled by the noise variance, implicitly enforcing second-order consistency at $O(d^2)$ memory cost without forming the $O(d^3)$ Hessian tensor. In Lorenz 63 with minimal temporal supervision, second-order supervision reduces invariant-measure error and Lyapunov-spectrum MSE, and recovers the constant Hessian norm of the true bilinear field. Across five training seeds, explicit Hessian matching produces catastrophic Lyapunov outliers for four of five seeds, whereas randomized Jacobian matching produces none among 5,000 on-attractor rollouts, and it attains the largest threshold in a directional capture scan. In coupled Lorenz 96, first-order methods enter spurious high-amplitude regimes as forcing increases, while second-order methods retain accurate marginals. Randomized Jacobian matching costs about the same as explicit Hessian matching on Lorenz 63 and 40% less on Lorenz 96, with no reference Hessian evaluations during training.

math.NA↗

arXiv2026

Golden Ruler: A Numeric Format Catalog with Bit-Exact Conformance Vectors for FP8, BF16, MXFP4, and Microscaling Formats

Numeric format proliferation in machine learning hardware -- FP8 (E4M3 and E5M2), BF16, MXFP4, microscaling block formats, and dozens of research variants -- has outpaced the availability of vendor-neutral, bit-exact reference material. Engineers porting models across accelerators encounter silent divergences that are difficult to diagnose without a shared ruler. This paper describes a catalog of 109 numeric formats spanning 12 clusters (83 at v2; the count is a catalog invariant, not a fixed number), a suite of six bit-exact conformance packs covering GF16, MXFP4 element, BF16, FP8 E4M3, FP8 E5M2, and E8M0 block scale, and an IEEE P3109 v3.2.0 cross-walk that maps each pack to its corresponding standards-track configured format. Each pack is a self-contained JSON document with a SHA-256 fingerprint, a shared row schema, and an anchor vector that encodes 3.0 -- the identity phi^2 + 1/phi^2 = 3 -- as a cross-pack sanity check. Packs are cross-validated against ml_dtypes 0.5.4 (Google/JAX); any divergence is documented explicitly and interpreted as a spec-permitted interpretation gap rather than hidden. The work is framed as registry filling: it does not propose new formats, make model-accuracy claims, or assert superiority over any vendor's implementation. All artifacts are publicly available at https://github.com/gHashTag/t27 under an open license.

cs.AR↗

arXiv2026

NGMRES convergence analysis and proof of acceleration for contractive and noncontractive iterations

This paper gives the first convergence analysis and proof of acceleration for nonlinear GMRES (NGMRES) applied to contractive and noncontractive fixed point iterations (FPIs) for solving general nonlinear systems. Our main results are that in both the contractive and noncontractive cases, the ratio gain of the optimization problem is the mechanism responsible for accelerating (or enabling) convergence. Our analysis also reveals a second important quantity related to the optimization problem, which directly predicts the linear convergence rate at each iteration and proves it is at most 1; hence only higher order terms are responsible for NGMRES non-convergence. Numerical results for several challenging nonlinear test problems are given that illustrate the theory, show how the acceleration improves convergence, show that the quantity predicting the linear convergence rate is remarkably accurate and moreover can be useful for adaptively choosing NGMRES depth, show how restarts can improve convergence in noncontractive iterations, show how NGMRES is naturally suited for finding distinct solutions of a multi-solution PDE, and that NGMRES can perform better than Anderson acceleration when applied to superlinear FPIs.

math.NA↗