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Riemannian Proximal Sampler for High-accuracy Sampling on Manifolds

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arXiv 2025年

作者： Guan, Yunrui Balasubramanian, Krishnakumar Ma, Shiqian Department of Computational Applied Mathematics and Operations Research Rice University United States Department of Statistics University of California Davis United States

We introduce the Riemannian Proximal Sampler, a method for sampling from densities defined on Riemannian manifolds. The performance of this sampler critically depends on two key oracles: the Manifold Brownian Increments (MBI) oracle and the Riemannian Heat-kernel (RHK) oracle. We establish high-accuracy sampling guarantees for the Riemannian Proximal Sampler, showing that generating samples with Ε-accuracy requires O(log(1/Ε)) iterations in Kullback-Leibler divergence assuming access to exact oracles and O(log2(1/Ε)) iterations in the total variation metric assuming access to sufficiently accurate inexact oracles. Furthermore, we present practical implementations of these oracles by leveraging heat-kernel truncation and Varadhan’s asymptotics. In the latter case, we interpret the Riemannian Proximal Sampler as a discretization of the entropy-regularized Riemannian Proximal Point Method on the associated Wasserstein space. We provide preliminary numerical results that illustrate the effectiveness of the proposed methodology. © 2025, CC BY.

关键词： Numerical methods

An artificial viscosity approach to high order entropy stable discontinuous Galerkin methods

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arXiv 2025年

作者： Chan, Jesse Department of Computational Applied Mathematics and Operations Research Rice University 6100 Main St HoustonTX77005 United States

Entropy stable discontinuous Galerkin (DG) methods improve the robustness of high order DG simulations of nonlinear conservation laws. These methods yield a semi-discrete entropy inequality, and rely on an algebraic flux differencing formulation which involves both summation-by-parts (SBP) discretization matrices and entropy conservative two-point finite volume fluxes. However, explicit expressions for such two-point finite volume fluxes may not be available for all systems, or may be computationally expensive to compute. This paper proposes an alternative approach to constructing entropy stable DG methods using an artificial viscosity coefficient based on the local violation of a cell entropy inequality and the local entropy dissipation. The resulting method recovers the same global semi-discrete entropy inequality that is satisfied by entropy stable flux differencing DG methods. The artificial viscosity coefficients are parameter-free and locally computable over each cell, and the resulting artificial viscosity preserves both high order accuracy and a hyperbolic maximum stable time-step size under explicit time-stepping. Copyright © 2025, The Authors. All rights reserved.

关键词： Viscosity

Comparing the p -independence number of regular graphs to the q -independence number of their line graphs

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Discrete applied mathematics 2025年 373卷 316-326页

作者： Yair Caro Randy Davila Ryan Pepper Department of Mathematics University of Haifa–Oranim Tivon 36006 Israel Department of Computational Applied Mathematics & Operations Research Rice University Houston TX 77005 USA Department of Mathematics and Statistics University of Houston–Downtown Houston TX 77002 USA

Let G be a simple graph and let L ( G ) denote the line graph of G . A p - independent set in G is a set of vertices S ⊆ V ( G ) such that the subgraph induced by S has maximum degree at most p . The p - independence number of G , denoted by α p ( G ) , is the cardinality of a maximum p -independent set in G . In this paper, and motivated by the recent result that independence number is at most matching number for regular graphs Caro et al., (2022), we investigate which values of the non-negative integers p , q , and r have the property that α p ( G ) ≤ α q ( L ( G ) ) for all r-regular graphs. Triples ( p , q , r ) having this property are called valid α -triples . Among the results we prove are: • ( p , q , r ) is valid α -triple for p ≥ 0 , q ≥ 3 , and r ≥ 2 . • ( p , q , r ) is valid α -triple for p ≤ q < 3 and r ≥ 2 . • ( p , q , r ) is valid α -triple for p ≥ 0 , q = 2 , and r even. • ( p , q , r ) is valid α -triple for p ≥ 0 , q = 2 , and r odd with r = max { 3 , 17 ( p + 1 ) 16 } . We also show a close relation between undetermined possible valid α -triples, the Linear Aboricity Conjecture, and the Path-Cover Conjecture.

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An Adaptive Collocation Point Strategy For Physics Informed Neural Networks via the QR Discrete Empirical Interpolation Method

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arXiv 2025年

作者： Celaya, Adrian Fuentes, David Riviere, Beatrice Department of Computational Applied Mathematics and Operations Research Rice University HoustonTX United States Department of Imaging Physics The University of Texas MD Anderson Cancer Center HoustonTX United States

Physics-informed neural networks (PINNs) have gained significant attention for solving forward and inverse problems related to partial differential equations (PDEs). While advancements in loss functions and network architectures have improved PINN accuracy, the impact of collocation point sampling on their performance remains underexplored. Fixed sampling methods, such as uniform random sampling and equispaced grids, can fail to capture critical regions with high solution gradients, limiting their effectiveness for complex PDEs. Adaptive methods, inspired by adaptive mesh refinement from traditional numerical methods, address this by dynamically updating collocation points during training but may overlook residual dynamics between updates, potentially losing valuable information. To overcome this limitation, we propose an adaptive collocation point selection strategy utilizing the QR Discrete Empirical Interpolation Method (QR-DEIM), a reduced-order modeling technique for efficiently approximating nonlinear functions. Our results on benchmark PDEs, including the wave, Allen-Cahn, and Burgers’ equations, demonstrate that our QR-DEIM-based approach improves PINN accuracy compared to existing methods, offering a promising direction for adaptive collocation point strategies. © 2025, CC BY.

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Surrogate-based multilevel Monte Carlo methods for uncertainty quantification in the Grad-Shafranov free boundary problem

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arXiv 2025年

作者： Elman, Howard C. Liang, Jiaxing Sánchez-Vizuet, Tonatiuh Department of Computer Science Institute for Advanced Computer Studies University of Maryland College Park United States Department of Computational Applied Mathematics & Operations Research Rice University United States Department of Mathematics The University of Arizona United States

We explore a hybrid technique to quantify the variability in the numerical solutions to a free boundary problem associated with magnetic equilibrium in axisymmetric fusion reactors amidst parameter uncertainties. The method aims at reducing computational costs by integrating a surrogate model into a multilevel Monte Carlo method. The resulting surrogate-enhanced multilevel Monte Carlo methods reduce the cost of simulation by factors as large as 104 compared to standard Monte Carlo simulations involving direct numerical solutions of the associated Grad-Shafranov partial differential equation. Accuracy assessments also show that surrogate-based sampling closely aligns with the results of direct computation, confirming its effectiveness in capturing the behavior of plasma boundary and geometric *** Codes 65Z05, 65C05, 62P35, 35R35, 35R60 Copyright © 2025, The Authors. All rights reserved.

关键词： Stochastic systems

THE ELASTIC RAY TRANSFORM

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arXiv 2025年

作者： Ilmavirta, Joonas Kykkänen, Antti Saksala, Teemu Department of Mathematics and Statistics University of Jyväskylä Jyväskylä Finland Department of Computational Applied Mathematics and Operations Research Rice University HoustonTX United States Department of Mathematics NC State University RaleighNC United States

We introduce and study a new family of tensor tomography problems. At rank 2 it corresponds to linearization of travel time of elastic waves, measured for all polarizations. We provide a kernel characterization for ranks up to 2. The kernels consist of potential tensors, but in an unusual sense: the associated differential operators have degree 2 instead of the familiar 1. The proofs are based on Fourier analysis, Helmholtz decompositions, and cohomology. Copyright © 2025, The Authors. All rights reserved.

关键词： Tensors

Learning algorithms for mean field optimal control

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arXiv 2025年

作者： Soner, H. Mete Teichmann, Josef Yan, Qinxin Department of Operations Research and Financial Engineering Princeton University PrincetonNJ08540 United States Department of Mathematics ETH Zürich Switzerland Program in Applied and Computational Mathematics Princeton University PrincetonNJ08540 United States

We analyze an algorithm to numerically solve the mean-field optimal control problems by approximating the optimal feedback controls using neural networks with problem specific architectures. We approximate the model by an N-particle system and leverage the exchangeability of the particles to obtain substantial computational efficiency. In addition to several numerical examples, a convergence analysis is provided. We also developed a universal approximation theorem on Wasserstein *** Codes 35Q89, 35D40, 49L25, 60G99 © 2025, CC BY-NC-ND.

关键词： Optimal control systems

Independence, induced subgraphs, and domination in K1,r-free graphs

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arXiv 2025年

作者： Caro, Yair Davila, Randy Henning, Michael A. Pepper, Ryan Department of Mathematics University of Haifa–Oranim Tivon36006 Israel Department of Computational Applied Mathematics & Operations Research Rice University HoustonTX77005 United States Department of Mathematics and Applied Mathematics University of Johannesburg Auckland Park2006 South Africa Department of Mathematics and Statistics University of Houston–Downtown HoustonTX77002 United States

Let G be a graph and F a family of graphs. Define αF(G) as the maximum order of any induced subgraph of G that belongs to the family F. For the family F of graphs with chromatic number at most k, we prove that if G is K1,r-free, then αF(G) ≤ (r − 1)kγ(G), where γ(G) is the domination number. When F is the family of empty graphs, this bound simplifies to α(G) ≤ 2γ(G) for K1,3free (claw-free) graphs, where α(G) is the independence number of G. For d-regular graphs, this is further refined to the bound α(G) ≤ 2 (d+1/d+2) γ(G), which is tight for d ∈ {2, 3, 4}. Using Ramsey theory, we extend this framework to edge-hereditary graph families, showing that for K1,r-free graphs, we have αF(G) ≤ r(Kr, F∗)γ(G), where F∗ is the set of graphs not in F. Specializing to Kq-free graphs, we show αF(G) ≤ (r(Kq, Kr) − 1)γ(G). Finally, for the k-independence number αk(G), we prove that if G is K1,r-free with order n and minimum degree δ ≥ k + 1, (formula presented) and this bound is sharp for all parameters. Copyright © 2025, The Authors. All rights reserved.

关键词： Graph theory

A Priori Generalizability Estimate for a CNN

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arXiv 2025年

作者： Balsells, Cito Riviere, Beatrice Fuentes, David Department of Computational Applied Mathematics and Operations Research The Ken Kennedy Institute Rice University 6100 Main St HoustonTX77005 United States Department of Imaging Physics The University of Texas MD Anderson Cancer Center 1515 Holcombe Blvd HoustonTX77030 United States

We formulate truncated singular value decompositions of entire convolutional neural networks. We demonstrate the computed left and right singular vectors are useful in identifying which images the convolutional neural network is likely to perform poorly on. To create this diagnostic tool, we define two metrics: the Right Projection Ratio and the Left Projection Ratio. The Right (Left) Projection Ratio evaluates the fidelity of the projection of an image (label) onto the computed right (left) singular vectors. We observe that both ratios are able to identify the presence of class imbalance for an image classification problem. Additionally, the Right Projection Ratio, which only requires unlabeled data, is found to be correlated to the model’s performance when applied to image segmentation. This suggests the Right Projection Ratio could be a useful metric to estimate how likely the model is to perform well on a sample. © 2025, CC BY.

关键词： Convolutional neural networks