生成时间: 2026年06月24日 06:33:05
重点关注论文数量: 0
了解领域论文数量: 2
不相关论文数量: 24
了解领域论文（摘要翻译）
1. lymph 2.0库：面向多面体DG方法的$p$-自适应算法与并行装配策略
The lymph 2.0 library: p-adaptive algorithms and parallel assembly strategies for polytopal DG methods
作者: Paola F. Antonietti, Mattia Corti, Caterina B. Leimer Saglio, Stefano Pagani
类别: math.NA
发布日期: 2026-06-23
arXiv ID: 2606.24646v1
链接: http://arxiv.org/abs/2606.24646v1
摘要翻译
中文标题: lymph 2.0库：面向多面体DG方法的$p$-自适应算法与并行装配策略
摘要翻译: 本工作展示了lymph 2.0库的新版本，这是一个面向一般多面体网格上高阶间断伽辽金离散的开源MATLAB框架。lymph 2.0版本扩展为支持具有逐单元多项式逼近阶数的离散，从而允许基于后验误差指示子的$p$-自适应策略的设计。此外，该库引入了一种统一的装配框架，将离散算子的构造从底层物理模型中抽象出来，提升了代码的模块性、并行性、可维护性和可扩展性。而且，所提出的方法通过专用的并行工具实现了共享内存并行。多个数值算例验证了所提方案在降低计算成本的同时保持逼近精度的有效性。
2. 二维感应加热问题的分析与逼近
Analysis and approximation of a two-dimensional induction heating problem
作者: Gabriel R. Barrenechea, Katherine MacKenzie, Abner Salgado
类别: math.NA
发布日期: 2026-06-23
arXiv ID: 2606.24517v1
链接: http://arxiv.org/abs/2606.24517v1
摘要翻译
中文标题: 二维感应加热问题的分析与逼近
摘要翻译: 本文分析了一个稳态二维感应加热问题解的存在性及其有限元逼近。该问题的主要困难之一在于其右端项乍看之下仅为可积的。利用所涉及偏微分方程的先验正则性结果，证明了问题的自然弱形式是合理的。然后，我们研究了有限元逼近，并证明了标准 Galerkin 有限元法在凸区域及适当的网格条件下收敛。通过将最近提出的保界方法 (BPM) 应用于热方程，我们改进了这一结果，并证明该方法在更宽松的区域与网格条件下收敛到问题的解。由于这些分析未对解的正则性作任何假设，因此有限元法的收敛性也证明了了解的存在性。多个数值实验证实了理论结果，并展示了使用保界方法相比标准有限元方法的改进。
不相关论文（基本信息）
1. 数值积分对 Hamilton-Jacobi-Bellman 方程策略迭代收敛性的影响
The Effect of Quadrature on the Convergence of Policy Iteration for Hamilton-Jacobi-Bellman Equations
作者: Thomas Hall, Iain Smears, Endre Süli, Harry Wells
类别: math.NA
发布日期: 2026-06-23
arXiv ID: 2606.24806v1
链接: http://arxiv.org/abs/2606.24806v1
摘要: Modern finite element libraries allow users to express partial differential equations directly in variational form, with the added convenience of automatic quadrature selection. In the context of Hamilton-Jacobi-Bellman (HJB) equations, automatic quadrature selection can result in nonmatching quadratures between different terms that may lead to loss of convergence of the policy iteration, which is otherwise expected from theory to converge superlinearly. The simple remedy of enforcing matching quadrature recovers the expected superlinear convergence.
2. 精确Sobolev三明治与Radon域$L^p$脊积分空间对ReLU$^k$网络的逼近率
Sharp Sobolev Sandwich and Approximation Rates of Radon-Domain $L^p$ Ridge Integral Spaces for ReLU$^k$ Networks
作者: Juncai He, Zitong Tian
类别: math.NA
发布日期: 2026-06-23
arXiv ID: 2606.24795v1
链接: http://arxiv.org/abs/2606.24795v1
摘要: We develop the $L^p$ space and approximation theory for shallow neural networks with $\mathrm{ReLU}^k$ activations. The central object is the Radon-domain $L^p$ space $\mathcal{R}L^p_k(Ω)$ containing all functions on a bounded domain $Ω$ that admit a ridge integral representation whose coefficient density belongs to $L^p$ in the Radon domain. In the Hilbert case $p=2$, we prove by elementary Fourier analysis that this space recovers the critical Sobolev space $H^{k+(d+1)/2}(Ω)$. For general $1 LAMG+: A Robust Lean Algebraic Multigrid Solver for Graph Laplacians 作者: Oren E. Livne 类别: math.NA 发布日期: 2026-06-23 arXiv ID: 2606.24791v1 链接: http://arxiv.org/abs/2606.24791v1 摘要: Graph-Laplacian systems $Lφ=b$ underlie spectral clustering, semi-supervised learning, finite-element analysis, and network-flow solvers. We present LAMG+, a lean, parameter-free, empirically linear-time algebraic multigrid solver: a Julia re-derivation of Lean Algebraic Multigrid (LAMG) with two targeted refinements. We establish three facts. (1) Benchmarking against approximate-Cholesky (AC) and four other solvers (BoomerAMG, PETSc GAMG, pyAMG, CMG): LAMG+ and AC are complementary peers – AC is faster on social/citation graphs; LAMG+ is faster on finite-element/structural matrices (fastest robust solver, most memory-frugal, $2.2\times$ faster than the robust AC variant on large graphs). Only LAMG+ and AC converge across all 13 test classes; the others fail or slow by an order of magnitude off their home turf. (2) Linear scaling: LAMG+ is empirically $O(m)$ with $m$ nonzeros over the full 1,711-graph SuiteSparse set (100% converged, median 4 cycles, log-log slope 1.01), verified up to $2.4\times 10^8$ nonzeros. (3) Robustness: prior benchmarking reported LAMG non-convergent on certain families; running the unmodified LAMG 2.2.1 under identical conditions establishes full convergence, indicating an evaluation artifact. A Local Fourier Analysis proves a strict interpolation-order deficit on grid-aligned anisotropy. Two lean local refinements – a strength-of-connection aggregation veto and selective caliber-2 interpolation – resolve LAMG’s anisotropy failure (convergence factor $\approx 0.99 \to 0.11$) with negligible overhead. Dirac-Frenkel dynamics with inertia for nonlinearly parametrized solutions of evolution problems 作者: Matteo Raviola, Benjamin Peherstorfer 类别: math.NA, cs.LG 发布日期: 2026-06-23 arXiv ID: 2606.24769v1 链接: http://arxiv.org/abs/2606.24769v1 摘要: Even when Dirac-Frenkel dynamics determine a well-defined evolution in function space, the corresponding parameter dynamics can be non-unique or ill-conditioned for redundant nonlinear parametrizations such as neural networks or mixture models. We propose to add inertia to the Dirac-Frenkel dynamics and show that this allows useful parameter velocity information to persist from the past trajectory in directions that are weakly informed, while well-informed parameter velocity directions continue to follow the Dirac-Frenkel dynamics. We prove that the inertial formulation yields well-posed parameter dynamics and provide a posteriori error bounds. After time discretization, the method requires the solution of the same type of regularized linear least-squares problem as standard Dirac-Frenkel dynamics, but with the previous velocity appearing as an anchor. Numerical experiments demonstrate the increased robustness obtained with inertia. Recursive expansion of the matrix step function using polynomials of degree eight 作者: Emanuel H. Rubensson, Elias Jarlebring, Gustaf Lorentzon 类别: math.NA 发布日期: 2026-06-23 arXiv ID: 2606.24701v1 链接: http://arxiv.org/abs/2606.24701v1 摘要: We consider the problem of efficiently computing the matrix step function of a large dense symmetric matrix. To this end, we introduce a recursive polynomial expansion method in which a composite polynomial of high degree is built recursively from component polynomials of degree eight. The component polynomial used in each iteration is designed to achieve strong amplification of the spectral gap across the step while favorably positioning the updated gap for subsequent iterations. A key ingredient is a novel evaluation scheme for arbitrary matrix polynomials of degree exactly eight requiring only three matrix-matrix multiplications and three matrices in memory. This scheme makes available a substantially larger class of component polynomials than previously possible within a three-multiplication budget, thereby expanding the class of composite polynomials that can be generated. Together with our polynomial selection strategy, this leads to a significant and consistent reduction in the number of matrix-matrix multiplications required to compute the matrix step function compared to existing recursive expansion methods. Extended pseudo-spectral physics-informed neural networks for phase-field models 作者: Callum Marsh, Radek Erban, Andreas Munch 类别: q-bio.QM, cs.LG, math.NA, physics.bio-ph 发布日期: 2026-06-23 arXiv ID: 2606.24660v1 链接: http://arxiv.org/abs/2606.24660v1 摘要: Phase-field models play a central role in the continuum description of phase separation, in which the bulk free-energy density and the interfacial thickness parameter determine pattern formation and microstructural evolution. In practice, these constitutive quantities are rarely known a priori and must be inferred from limited dynamical observations. In this work, an extended pseudo-spectral physics-informed neural network (ESPINN) framework is developed for the inverse identification of phase-field models from transient snapshot data. It enables the simultaneous recovery of both the bulk chemical potential and unknown gradient coefficients. Numerical experiments on the one-dimensional Cahn-Hilliard equation demonstrate accurate and statistically stable reconstruction in the noiseless regime, with substantial constitutive information recoverable from even a single snapshot pair. In the presence of noise, reconstruction accuracy degrades gracefully, and increasing the number of snapshots improves robustness by reducing variance across runs. These results establish ESPINN as a data-efficient and physically consistent approach for learning free-energy structure in continuum models of phase separation. Certified Reachable Sets for Nonlinear Reaction–Diffusion Systems 作者: Mohamed Amine Ouchdiri, Mohamed Maghenem, Saad Benjelloun, Adnane Saoud 类别: math.NA, math.OC 发布日期: 2026-06-23 arXiv ID: 2606.24500v1 链接: http://arxiv.org/abs/2606.24500v1 摘要: Reachability analysis for dynamical systems seeks to compute a set containing all reachable states at a given time. Compared to ordinary differential equations (ODEs), the analysis of nonlinear reaction–diffusion PDEs with parametric uncertainties remains largely underexplored, due to the infinite-dimensional state space and the variety of solutions under different parameters. We address this through a three-step procedure: 1) Finite Element Methods (FEM)s to discretise the space and generate a finite-dimensional FEM-based model, 2) Proper Orthogonal Decomposition (POD) to build a Reduced-Order Model (ROM), and 3) set-based reachability-analysis methods applied to the ROM. We propose a framework that enables us to derive explicit upper bounds on the approximation errors introduced at each stage of the pipeline. In particular, we quantify the discrepancy between trajectories of the original PDE and those of the FEM-based discretization, as well as the error between the FEM-based model and the reduced-order model. Importantly, these bounds are shown to hold uniformly over the considered set of parameters. By combining these error estimates, we obtain an over-approximation of the reachable set of the original PDE. The approach is illustrated on the Allen–Cahn equation and a logistic growth PDE. Randomized Estimation of T-Eigenvalues of T-SPD Tensors: A Two-Sided Bracket 作者: Hemant Sharma, Nachiketa Mishra 类别: math.NA 发布日期: 2026-06-23 arXiv ID: 2606.24491v1 链接: http://arxiv.org/abs/2606.24491v1 摘要: In earlier work \cite{sharma2025} we developed deterministic analytical bounds on the T-eigenvalues of symmetric positive definite (SPD) third-order tensors under the Kilmer–Martin T-product: the trace–determinant (TDet) bounds via the AM–GM inequality, and the trace-dependent (TDep) bounds generalizing Samuelson’s inequality. While these bounds are cheap and guaranteed-valid, their relative gap grows as $\sqrt{d-1}$ in the tensor dimension $d = np$, limiting their usefulness for large tensors.
This paper develops randomized estimators for the extreme T-eigenvalues of T-SPD tensors that complement the deterministic bounds. We adapt the Halko–Martinsson–Tropp framework \cite{halko2011} to the T-product setting and introduce four methods: (i) a randomized power method that produces a lower bound on $λ_1$ with exponential convergence; (ii) a randomized subspace iteration with a tensor-analogue HMT error bound; (iii) a two-sided rigorous bracket combining the randomized lower bound with the deterministic TDep upper bound; and (iv) a Hutchinson-based fully randomized TDep bound for matvec-only settings. New convergence results for Carleman linearization 作者: Michele Boreale, Luisa Collodi 类别: cs.MS, math.NA 发布日期: 2026-06-23 arXiv ID: 2606.24473v1 链接: http://arxiv.org/abs/2606.24473v1 摘要: We prove new error bounds for finite Carleman truncations of polynomial ordinary differential equations. The analysis works directly in the original monomial basis and for selected observables, such as state coordinates. Using a Dyson–Duhamel expansion, we separate the degree-preserving linear part from the degree-raising nonlinear part and track how truncation errors can propagate back to the observable. The resulting bounds are degree-aware and retain logarithmic-norm information from the original linear dynamics. We obtain explicit finite-degree estimates and geometric convergence over certified time horizons. Comparisons with existing bounds, in particular those of Forets–Pouly, are given on the Stuart–Landau and Van der Pol systems. A sparse spectral method on a class of domains bounded by planar algebraic curves 作者: Jiajie Yao, Marco Fasondini, Sheehan Olver 类别: math.NA 发布日期: 2026-06-23 arXiv ID: 2606.24423v1 链接: http://arxiv.org/abs/2606.24423v1 摘要: We develop a sparse spectral method for solving partial differential equations on a class of two-dimensional geometries bounded by algebraic curves. The numerical method uses generalised bivariate Koornwinder polynomials which form a complete orthogonal basis, but one which is not graded in terms of polynomial degree. The polynomials are built from new families of univariate semiclassical orthogonal polynomials whose associated operator matrices (Jacobi matrices, raising matrices and differentiation matrices) are computed with optimal linear complexity in the number of basis functions. When used to discretise partial differential equations the resulting matrices are sparse enabling efficient numerical solution. Moreover, we develop fast transforms from values on a grid to expansion coefficients. The efficiency and accuracy of the resulting spectral method are illustrated through a series of numerical experiments on geometries whose boundaries are smooth and piecewise smooth including non-convex geometries. We observe algebraic convergence for geometries with corners, which accelerates to exponentially fast (spectral) convergence when the boundary is smooth. The 3D Nonconforming Virtual Element Method with enhanced spaces: numerical investigation and implementation guide 作者: Franco Dassi, Manuel Trezzi 类别: math.NA 发布日期: 2026-06-23 arXiv ID: 2606.24306v1 链接: http://arxiv.org/abs/2606.24306v1 摘要: In this paper, we describe the 3D Nonconforming Virtual Element Method, where the local spaces are enhanced in order to build an $L^2$ projection of virtual functions onto polynomials of degree $k$. This higher order projection provides a more accurate representation of the load term and, more importantly, it enables the construction of the reaction term. We conduct a convergence analysis of the $L^2$ norm and $H^1$ seminorm errors with respect to mesh size and approximation degree. We also compare our results with those obtained using the conforming Virtual Element Method. Deep numerical schemes for systems of Ergodic BSDEs with applications to regime-switching forward utilities 作者: Guillaume Broux-Quemerais, Sarah Kaakai, Anis Matoussi, Wissal Sabbagh 类别: math.NA, cs.LG, math.PR 发布日期: 2026-06-23 arXiv ID: 2606.24271v1 链接: http://arxiv.org/abs/2606.24271v1 摘要: In this paper, we introduce two neural-network-based numerical schemes for solving systems of coupled ergodic Backward Stochastic Differential Equations (eBSDEs), motivated by the approximation of optimal strategies within the framework of forward utilities in a regime-switching stochastic factor model. Our approach builds on the representation of such models through systems of eBSDEs introduced in [HLT20]. We first establish a link between the solution of the system of ergodic BSDEs and that of an associated multidimensional BSDE with random terminal time, given by the hitting time of the positive recurrent stochastic factor. Building on this representation, we introduce a locally additive deep learning scheme obtained by minimizing aggregated local error terms. We then present a new Deep Galerkin Method (DGM) inspired algorithm that minimizes the residual of the associated ergodic PDE system, relying on a representation of the ergodic cost. Finally, we apply this framework to regime-switching forward utilities in a stochastic factor model. We first derive a general consistency SPDE that characterizes regime-switching forward utilities and retrieve their representation with systems of ergodic BSDEs in the homothetic case. Numerical experiments demonstrate the performance of the proposed methods, with a particular focus on the impact on forward preferences of taking into account regime switches. Discovery of connectivity-trainability trade-off of IQP Circuits for Hamiltonian Optimization 作者: Quoc Chuong Nguyen 类别: cs.SI, math.NA, physics.data-an, quant-ph 发布日期: 2026-06-23 arXiv ID: 2606.24264v1 链接: http://arxiv.org/abs/2606.24264v1 摘要: Instantaneous Quantum Polynomial-time (IQP) circuits are promising candidates for near-term quantum advantage due to the conjectured classical hardness of their sampling task. However, their capabilities for optimization remain largely unexplored. We present a systematic investigation of the performance and trainability of IQP circuits for Hamiltonian optimization. Our results reveal a trade-off between optimization performance and circuit connectivity, demonstrating that the circuit structure plays a key role in determining the ability of IQP circuits to reach low-energy states. Quasi-Monte Carlo for SDE Simulation: Error Analysis and Dimensionality Reduction 作者: Du Ouyang, Zexin Pan, Zhijian He 类别: math.NA 发布日期: 2026-06-23 arXiv ID: 2606.24207v1 链接: http://arxiv.org/abs/2606.24207v1 摘要: We investigate the numerical simulation of general stochastic differential equations (SDEs) using Quasi-Monte Carlo (QMC) methods. First, we provide a rigorous theoretical analysis of the QMC method applied to the Euler-Maruyama (EM) scheme, establishing that it significantly accelerates the decay of the sampling error and achieves an asymptotically superior convergence rate over the classical Monte Carlo method. Second, the traditional EM scheme exhibits a slow polynomial decay of the discretization error, which necessitates a large number of time steps and leads to a significantly high integration dimension. To address this issue, we propose a Multilevel Stochastic Time Grid (MSTG) method based on Exact Simulation techniques, and we rigorously establish its convergence rate under randomized QMC sampling, proving that it preserves the high-order convergence of the sampling error. In terms of the overall error, the truncation error of the proposed MSTG method exhibits a remarkably fast super-exponential decay. Consequently, to achieve a given accuracy level, our approach requires significantly fewer discretization steps than the EM scheme, thereby drastically reducing the actual integration dimension of the QMC method. This substantial dimensionality reduction strategy greatly enhances the practical efficiency of the QMC algorithm. Numerical experiments fully corroborate the superiority of the proposed approach. Tractability versus curse of dimensionality for geometric $L_p$-discrepancies 作者: Erich Novak, Friedrich Pillichshammer 类别: math.NA, math.NT 发布日期: 2026-06-23 arXiv ID: 2606.24195v1 链接: http://arxiv.org/abs/2606.24195v1 摘要: This paper studies tractability versus the curse of dimensionality for several geometric $L_p$-discrepancies through a unified discrepancy–integration duality framework, where worst case integration errors in suitable function spaces equal the corresponding discrepancies. A general lower bound method for non-negative linear rules in spaces under broad tensor-product assumptions establishes exponential information complexity in the dimension $d$, yielding the curse of dimensionality for the respective discrepancy. We complement this overview by new results on discrepancy–integration duality and the curse of dimensionality for the periodic $L_p$-discrepancy. The current state of research on this general problem is summarized in a clearly laid out table, which also highlights the remaining open questions. Dynamical low-rank methods for the Wigner equation I: separable difference potential 作者: Sihong Shao, Yuehan Shao 类别: math.NA, physics.comp-ph, quant-ph 发布日期: 2026-06-23 arXiv ID: 2606.24190v1 链接: http://arxiv.org/abs/2606.24190v1 摘要: Recent advances in dynamical low-rank approximation (DLRA) have demonstrated its effectiveness in high-dimensional simulations. However, existing DLRA algorithms still face significant challenges when handling systems that involve complex collision terms, including the pseudo-differential operator (${\rm Ψ}$) in the Wigner equation, a representative operator characterized by nonlocality. It is deserving to carry out a series of works to develop the DLRA algorithms for solving the Wigner equation. As the first step in this series of works, we propose an efficient DLRA algorithm for the Wigner equation, using a separable decomposition of the difference potential. We combine this separable assumption with two often-used truncations of ${\rm Ψ}$, namely $\mathcal{K}$-truncation and $\mathcal{Y}$-truncation, to obtain a kind of separated representation of ${\rm Ψ}$. Complexity analysis and several challenging experiments, including harmonic oscillators, Gaussian barrier scattering, electron-electron scattering, and a Helium-like system, all of which satisfy the separable assumption, confirm that the proposed DLRA algorithm has significant advantages, achieving a reduction in computational effort by one to two orders of magnitude in both runtime and memory requirements compared to the full-grid approach. It is worth noting that, even in the absence of a predetermined low-rank structure for the solution, DLRA can still serve as a numerical scheme that balances efficiency and accuracy. E Scheme and Flux-Limiter Scheme, Revisited 作者: Huazhong Tang 类别: math.NA 发布日期: 2026-06-23 arXiv ID: 2606.24186v1 链接: http://arxiv.org/abs/2606.24186v1 摘要: This paper revisits the {\em E scheme} of Osher \cite{Osher-SINUM1984} and the {\em flux-limiter scheme} of Sweby for quasi-linear hyperbolic conservation laws \cite{Sweby-SINUM1984}. Part of existing results will be re-understood and some new results will be presented. For a scalar conservation law, except for the conservative monotone schemes, the E scheme is a type of numerical methods that satisfy the discrete entropy condition for any convex entropy, but numerical entropy flux is not unique. Two-point E flux is monotone flux, but multi-point (three or more points) E flux may not necessarily be monotone flux, and multi-point monotone flux may not necessarily be E flux. Sweby’s flux-limiter scheme for the quasi-linear conservation laws was built on the E flux-based splitting $f_{j+1}-f_j=f_{j+1} { -\hat{f}^{\text{\tiny E}}_{j+\frac12}+\hat{f}^{\text{\tiny E}}_{j+\frac12}}-f_j$ and the LW scheme. It may not be second-order accurate in both space and time. Flexible and Stable Dynamics Discovery with Onsager’s Variational Principle 作者: Anthony Gruber, Ritoban Roy-Chowdhury, Irina Tezaur, Nathan M. Urban 类别: math.DS, math.NA 发布日期: 2026-06-23 arXiv ID: 2606.24106v1 链接: http://arxiv.org/abs/2606.24106v1 摘要: Variational principles specify the dynamics of a physical system via the extremization of associated functional data. Onsager’s variational principle (OVP), which characterizes dissipation-dominated phenomena such as phase separation, admits an unconditionally energy-stable time discretization through the minimization of a Rayleighan functional combining free energy and dissipative effects. The present work considers the case where one or more parts of this functional are empirically approximated or otherwise uncertain. To address this, a novel variational discretization of OVP is introduced which recovers previous work as a special case, and a learning problem is formulated which identifies uncertain terms in the free energy and dissipation potential from observable data. It is shown that the resulting OVP-based models connect directly to previous work in terms of proximal methods, Sobolev and Wasserstein gradient flows, while remaining provably energy-stable under arbitrarily long rollouts. The approach is illustrated on examples including Allen-Cahn, Fokker-Planck, and Cahn-Hilliard system models, where the effects of bulk free-energy densities, nonlocal potentials, and nonstandard boundary conditions are effectively learned with model classes consisting of polynomials, shallow neural networks, and spectral convolution kernels. Computing the minimal monomial basis for multivariate Birkhoff interpolation 作者: Li Yuanhe, Jiang Xue, Li Zhe 类别: math.NA 发布日期: 2026-06-23 arXiv ID: 2606.24054v1 链接: http://arxiv.org/abs/2606.24054v1 摘要: This paper studies algorithms for computing the minimal monomial basis for multivariate Birkhoff interpolation problems. Our approach is built around the notion of a reverse reduced set, which serves as the key tool for bridging the interpolation conditions to the monomial basis, thereby avoiding the construction and evaluation of Vandermonde matrices required in existing algorithms. For the single-node case, we prove that after Gaussian elimination on the incidence matrix, the least monomials of the polynomial set corresponding to the interpolation conditions precisely constitute the minimal monomial basis. To handle the multi-node case, we exploit the one-to-one correspondence between interpolation functionals and formal power series, whereby interpolation conditions at arbitrary nonzero nodes can all be converted to the origin. This provides both a coherent theoretical framework and a constructive algorithm for determining a proper minimal monomial basis for the general multivariate Birkhoff interpolation problem. Numerical examples demonstrate the effectiveness of the proposed algorithm. Diagonal Frog: High-order positivity-preserving FD schemes for anisotropic Fokker-Planck equations 作者: Andrey Itkin 类别: math.NA, physics.bio-ph, physics.comp-ph, q-fin.CP, q-fin.PR 发布日期: 2026-06-22 arXiv ID: 2606.23980v1 链接: http://arxiv.org/abs/2606.23980v1 摘要: The Fokker-Planck equation is fundamental to statistical mechanics, yet in settings with multiple state variables, anisotropic (cross-) diffusion, and jumps, conventional discretizations frequently produce non-physical negative probability densities. Building on the operator approach of “A. Itkin, Pricing derivatives under Levy models. Modern finite difference and pseudo-differential operators approach, Springer, 2017, ISBN 978-1-4939-6792-6”, we introduce a family of “Diagonal Frog” discretizations whose spatial operators are eventually M-matrices (EM-matrices). Although these operators lack a local M-matrix structure, positivity of the directional sub-operators emerges in the spirit of Zeno’s paradox: the matrix exponential, assembled as the limit of infinitely many ever-smaller substeps, is provably nonnegative after a short transient even though no single substep is. For the mixed-derivative block, whose generator is not eventually nonnegative, positivity instead rests on a factorized resolvent solver and holds conditionally, on an explicit step-size window; discrete mass is conserved exactly by the splitting for every step size. The resulting schemes are second-order accurate in time and space and require O(m 2 N + m 3) operations per time step, where m is the dimension of the Krylov subspace used to apply the exponential. As stress tests, we solve a two-dimensional anisotropic Fokker-Planck equation in the strong cross-diffusion regime against an exact Gaussian reference, a Kramers escape problem in a double-well potential, and an advection-dominated problem, and observe that the schemes remain stable, nonnegative, and mass-conservative for a wide range of Pécklet numbers (so, don’t need any flux limiter). Finally, we extend the construction to multidimensional processes and to the backward Kolmogorov equation with jumps. Variational inference and density estimation with non-negative tensor of hierarchical tucker format 作者: Xun Tang, Haoxuan Chen, Lexing Ying 类别: math.NA 发布日期: 2026-06-22 arXiv ID: 2606.23949v1 链接: http://arxiv.org/abs/2606.23949v1 摘要: In this work, we present an efficient method to compress a high-dimensional discrete probability function, i.e., a probability tensor, into a non-negative hierarchical Tucker format. The methodology is a two-stage procedure. In the first stage, we take an existing interpolation method to compress the target tensor into a hierarchical Tucker (HT) in a manner similar to the CUR decomposition for low-rank matrix reconstruction. In the second stage, we fit the first-stage output against a non-negative hierarchical Tucker ansatz using a second-order method tailored specifically for this setting. When the tensor is of order (d), both stages admit an (\mathcal{O}(d)) computational complexity, and therefore the proposed methodology readily extends into high-dimensional settings. Numerical experiments show success in compressing various high-dimensional probability tensors. Constrained Variable Projection for Structured Problems 作者: Emanuele Zangrando, Sara Venturini, Francesco Rinaldi, Francesco Tudisco 类别: math.OC, cs.LG, math.NA 发布日期: 2026-06-22 arXiv ID: 2606.23939v1 链接: http://arxiv.org/abs/2606.23939v1 摘要: Variable projection is a classical technique for separable nonlinear least-squares problems, in which variables that enter linearly are eliminated exactly, yielding a reduced nonlinear problem. By expressing this framework as a particular instance of a broader class of bilevel optimization problems, we develop a constrained variable-projection framework for data-science models, where the remaining variables are subject to convex constraints and the eliminated variables arise from a lower-level least-squares problem. In particular, by interpreting variable projection as a collapsed bilevel optimization problem, we derive exact reduced-gradient formulas compatible with automatic differentiation and propose a conditional-gradient algorithm for the resulting constrained reduced problem. We establish convergence guarantees under standard smoothness and compactness assumptions, and discuss extensions to structured lower-level variables. Numerical experiments on sparse autoencoding, dictionary learning, blind deconvolution, and few-shot learning suggest that the method can improve wall-clock efficiency and data efficiency relative to natural joint-optimization baselines. Hessian-augmented Supervised Learning for Hamilton-Jacobi-Bellman PDEs 作者: Matías Gómez-Aedo, Behzad Azmi, Yuyang Huang, Dante Kalise, Karl Kunisch 类别: math.OC, cs.LG, math.NA 发布日期: 2026-06-22 arXiv ID: 2606.23827v1 链接: http://arxiv.org/abs/2606.23827v1 摘要: A data-driven method is developed for approximating value functions in deterministic optimal control problems with nonlinear control-affine dynamics. The Pontryagin Maximum Principle optimality system is solved from multiple initial conditions to generate training data consisting of values, gradients, and Hessians of the value function, where Hessian information is obtained from a matrix Riccati equation along optimal trajectories. These quantities augment a weighted least-squares regression over sparse polynomial bases on hyperbolic cross index sets, with gradients and Hessians contributing additional linear equations per sample and substantially reducing sample complexity compared to value-only regression. Feedback laws are recovered analytically from the learned value function. In high dimensions, a partial Hessian strategy controls the cost of data generation. The approach is validated on problems of increasing state dimension, where second-order data augmentation is shown to improve approximation accuracy and closed-loop performance, with up to an order-of-magnitude reduction in the number of training samples required relative to lower-order methods. Ten Digits on a Train: AI-Assisted Verification of Two Eigenvalue Problems 作者: Matthew J. Colbrook 类别: math.NA, cs.AI, cs.HC, math.SP 发布日期: 2026-06-22 arXiv ID: 2606.23821v1 链接: http://arxiv.org/abs/2606.23821v1 摘要: Accurate numerical eigenvalues are often difficult to certify, especially in singular or non-normal settings. This article reports a human–AI collaboration on two such computations. For a singular self-adjoint Schrödinger operator, a verified zero count and Dirichlet–Neumann bracketing certify the complete negative spectrum to ten decimal places. For a delicate non-normal atom–molecule benchmark, a previously unresolved resonance pair is separated, with each member enclosed to ten digits. The second result is achieved not by increasing the precision of one-way shooting, but by reformulating the problem as a global matching system for projective solution lines. The infinite tail is encoded as uncertainty in the terminal projective data, and a componentwise, tail-robust Krawczyk–Brouwer inclusion supplies the certificate. This gives a reusable architecture for analytic boundary-value systems with ill-conditioned propagation and uncertain asymptotic data. The collaboration also exposes the strengths and limits of AI assistance. AI rapidly produced accurate candidates and plausible proof strategies, but several failed, including one apparently complete tail argument that omitted the componentwise check required by a nonuniform polydisc. Validated computation is a stringent test of AI-assisted mathematics: the output is not merely a number, but a number with a proof. These examples show why the proof object matters, and why human mathematical judgment remained decisive. More broadly, as AI makes code, exposition, and plausible numerical claims inexpensive, standards for verification, attribution, peer review, and training must adapt. The implications are unsettling; the opportunity is extraordinary.3. LAMG+: 一种用于图拉普拉斯算子的稳健精简代数多重网格求解器
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