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Scalable diffusion posterior sampling in infinite-dimensional inverse problems

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arXiv 2024年

作者： Schneider, Fabian Duong, Duc-Lam Lassas, Matti de Hoop, Maarten V. Helin, Tapio School of Engineering Science Lappeenranta-Lahti University of Technology Lappeenranta Finland Vienna University of Technology Austria Department of Mathematics and Statistics University of Helsinki Helsinki Finland Department of Computational and Applied Mathematics Rice University Houston United States

Score-based diffusion models (SDMs) have emerged as a powerful tool for sampling from the posterior distribution in Bayesian inverse problems. However, existing methods often require multiple evaluations of the forward mapping to generate a single sample, resulting in significant computational costs for large-scale inverse problems. To address this issue, we propose a scalable diffusion posterior sampling (SDPS) method to bypass forward mapping evaluations during sampling by shifting computational effort to an offline training phase, where a task-dependent score is learned based on the forward mapping. Crucially, the conditional posterior score is then derived from this trained score using affine transformations, ensuring no conditional score approximation is needed. The approach is shown to generalize to infinite-dimensional diffusion models and is validated through rigorous convergence analysis and high-dimensional CT imaging *** Codes 62F15, 65N21, 68Q32, 60Hxx, 60Jxx, 68T07, 92C55, © 2024, CC BY.

关键词： Affine transforms

An assessment of the potential impact of the Omicron variant of SARS-CoV-2 in Aotearoa New Zealand

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Infectious Disease Modelling 2022年第2期7卷 94-105页

作者： Giorgia Vattiato Oliver Maclaren Audrey Lustig Rachelle N.Binny Shaun C.Hendy Michael J.Plank School of Mathematics and Statistics University of CanterburyChristchurchNew Zealand Department of Physics University of AucklandAucklandNew Zealand Manaaki Whenua LincolnNew Zealand Department of Engineering Science University of AucklandAucklandNew Zealand

New Zealand delayed the introduction of the Omicron variant of SARS-CoV-2 into the community by the continued use of strict border controls through to January *** allowed time for vaccination rates to increase and the roll out of third doses of the vaccine(boosters)to *** also meant more data on the characteristics of Omicron became available prior to the first cases of community *** we present a mathematical model of an Omicron epidemic,incorporating the effects of the booster roll out and waning of vaccine-induced immunity,and based on estimates of vaccine effectiveness and disease severity from international *** model considers differing levels of immunity against infection,severe illness and death,and ignores waning of infection-induced *** model was used to provide an assessment of the potential impact of an Omicron wave in the New Zealand population,which helped inform government preparedness and *** the time the modelling was carried out,the date of introduction of Omicron into the New Zealand community was *** therefore simulated outbreaks with different start dates,as well as investigating different levels of booster *** found that an outbreak starting on 1 February or 1 March led to a lower health burden than an outbreak starting on 1 January because of increased booster coverage,particularly in older age *** also found that outbreaks starting later in the year led to worse health outcomes than an outbreak starting on 1 *** is because waning immunity in older groups started to outweigh the increased protection from higher booster coverage in younger *** an outbreak starting on 1 February and with high booster uptake,the number of occupied hospital beds in the model peaked between 800 and 3,300 depending on assumed transmission *** conclude that combining an accelerated booster programme with public health measures to flatten the curve are key to avoid overwhelming the hea

关键词： COVID-19 Epidemiological modelling Immunity Infectious diseases Public health Vaccination

A PROBABILISTIC PARKING PROCESS AND LABELED IDLA

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arXiv 2025年

作者： Harris, Pamela E. Holleben, Thiago Mori, J. Carlos Martínez Priestley, Amanda Sullivan, Keith Wagenius, Per Department of Mathematical Sciences University of Wisconsin Milwaukee United States Department of Mathematics & Statistics Dalhousie University Canada H. Milton Stewart School of Industrial and Systems Engineering Georgia Institute of Technology United States Department of Computer Science The University of Texas Austin United States Department of Mathematics and Statistics University of Vermont United States

We introduce and study a new probabilistic variant of the classical parking protocol of Konheim and Weiss [29], which is closely related to Internal Diffusion Limited Aggregation, or IDLA, introduced in 1991 by Diaconis and Fulton [15]. In particular, we show that if one runs our parking protocol starting with a parking function whose outcome permutation (in the sense of the classical parking process of Konheim and Weiss) is the identity permutation, then we can compute the exact probability that all of the cars park. Furthermore, we compute the expected time it takes for the protocol to complete assuming all of the cars park, and prove that the parking process is negatively correlated. We also study statistics of uniformly random weakly increasing parking functions, a subset of parking functions whose outcome is the identity permutation. We give the distribution of the last entry, along with the probability that a specific set of cars is lucky, and the expected number of lucky *** Codes Primary: 60C05, Secondary: 60K35, 60J05 Copyright © 2025, The Authors. All rights reserved.

关键词： Probability distributions

EXPONENTIAL STABILITY FOR AN INFINITE MEMORY WAVE EQUATION WITH FRICTIONAL DAMPING AND LOGARITHMIC NONLINEAR TERMS

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arXiv 2025年

作者： Peng, Qingqing Liu, Yikan School of Mathematics and Statistics Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Science and Technology Wuhan430074 China Department of Mathematics Kyoto University Kitashirakawa-Oiwakecho Sakyo-ku Kyoto606-8502 Japan

This article is concerned with the energy decay of an infinite memory wave equation with a logarithmic nonlinear term and a frictional damping term. The problem is formulated in a bounded domain in d (d ≥ 3) with a smooth boundary, on which we prescribe a mixed boundary condition of the Dirichlet and the acoustic types. We establish an exponential decay result for the energy with a general material density ρ(x) under certain assumptions on the involved coefficients. The proof is based on a contradiction argument, the multiplier method and some microlocal analysis techniques. In addition, if ρ(x) takes a special form, our result even holds without the damping effect, that is, the infinite memory effect alone is strong enough to guarantee the exponential stability of the *** Codes 35L20, 35L70, 35B40 Copyright © 2025, The Authors. All rights reserved.

关键词： Damping

Diffeomorphic Latent Neural Operators for Data-Efficient Learning of Solutions to Partial Differential Equations

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arXiv 2024年

作者： Ahmad, Zan Chen, Shiyi Yin, Minglang Kumar, Avisha Charon, Nicolas Trayanova, Natalia Maggioni, Mauro Johns Hopkins University Department of Applied Mathematics and Statistics United States Johns Hopkins University Department of Biomedical Engineering United States Johns Hopkins University Department of Mechanical Engineering United States Johns Hopkins University Department of Electrical and Computer Engineering United States University of Houston Department of Mathematics United States

A computed approximation of the solution operator to a system of partial differential equations (PDEs) is needed in various areas of science and engineering. Neural operators have been shown to be quite effective at predicting these solution generators after training on high-fidelity ground truth data (e.g. numerical simulations). However, in order to generalize well to unseen spatial domains, neural operators must be trained on an extensive amount of geometrically varying data samples that may not be feasible to acquire or simulate in certain contexts (e.g., patient-specific medical data, large-scale computationally intensive simulations.) We propose that in order to learn a PDE solution operator that can generalize across multiple domains without needing to sample enough data expressive enough for all possible geometries, we can train instead a latent neural operator on just a few ground truth solution fields diffeomorphically mapped from different geometric/spatial domains to a fixed reference configuration. Furthermore, the form of the solutions is dependent on the choice of mapping to and from the reference domain. We emphasize that preserving properties of the differential operator when constructing these mappings can significantly reduce the data requirement for achieving an accurate model due to the regularity of the solution fields that the latent neural operator is training on. We provide motivating numerical experimentation that demonstrates an extreme case of this consideration by exploiting the conformal invariance of the Laplacian. © 2024, CC BY-NC-SA.

关键词： Mathematical operators

RESULTS ON THE GENERALIZED NUMERICAL RANGES IN MAX ALGEBRA

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arXiv 2024年

作者： Aboutalebi, Narges Haj Fallat, Shaun Peperko, Aljoša Taghizadeh, Davod Zahraei, Mohsen Department of Mathematics Shahrood Branch Islamic Azad University Shahrood Iran Department of Mathematics and Statistics University of Regina SK Canada Faculty of Mechanical Engineering University of Ljubljana Aškerčeva 6 LjubljanaSI-1000 Slovenia Institute of Mathematics Physics and Mechanical Jadranska 19 Ljubljana1000 Slovenia Department of Mathematics Ahvaz Branch Islamic Azad University Ahvaz Iran

Let n and k be two positive integers with k ≤ n and C an n × n matrix with nonnegative entries. In this paper, the rank-k numerical range in the max algebra setting is introduced and studied. The related notions of the max joint k-numerical range and the max joint C-numerical range of an entry-wise nonnegative matrix and an m-tuple of nonnegative matrices are also introduced. Some interesting algebraic properties of these concepts are *** Codes Primary 39B82, Secondary 44B20, 46C05 Copyright © 2024, The Authors. All rights reserved.

关键词： Matrix algebra

An Intelligent Secure Adversarial Examples Detection Scheme in Heterogeneous Complex Environments

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Computers, Materials & Continua 2023年第9期76卷 3859-3876页

作者： Weizheng Wang Xiangqi Wang Xianmin Pan Xingxing Gong Jian Liang Pradip Kumar Sharma Osama Alfarraj Wael Said College of Information Science and Engineering HunanWomen’s UniversityChangsha410138China School of Mathematics and Statistics Hunan First Normal UniversityChangsha410138China School of Computer&Communication Engineering Changsha University of Science&TechnologyChangsha410114China Department of Computing Science University of AberdeenAberdeenAB243FXUK Department of Computer Science Community CollegeKing Saud UniversityRiyadh11437Saudi Arabia Department of Computer Science Faculty of Computers and InformaticsZagazig UniversityZagazig44511Egypt

Image-denoising techniques are widely used to defend against Adversarial Examples(AEs).However,denoising alone cannot completely eliminate adversarial *** remaining perturbations tend to amplify as they propagate through deeper layers of the network,leading to ***,image denoising compromises the classification accuracy of original *** address these challenges in AE defense through image denoising,this paper proposes a novel AE detection *** proposed technique combines multiple traditional image-denoising algorithms and Convolutional Neural Network(CNN)network *** used detector model integrates the classification results of different models as the input to the detector and calculates the final output of the detector based on a machine-learning voting *** analyzing the discrepancy between predictions made by the model on original examples and denoised examples,AEs are detected *** technique reduces computational overhead without modifying the model structure or parameters,effectively avoiding the error amplification caused by *** proposed approach demonstrates excellent detection performance against mainstream AE *** results show outstanding detection performance in well-known AE attacks,including Fast Gradient Sign Method(FGSM),Basic Iteration Method(BIM),DeepFool,and Carlini&Wagner(C&W),achieving a 94%success rate in FGSM detection,while only reducing the accuracy of clean examples by 4%.

关键词： Deep neural networks adversarial example image denoising adversarial example detection machine learning adversarial attack

Improving Gamma Imaging in Proton Therapy by Sanitizing Compton Camera Simulated Patient Data using Neural Networks through the BRIDE Pipeline

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Improving Gamma Imaging in Proton Therapy by Sanitizing Comp...

2024 IEEE International Conference on Big Data, BigData 2024

作者： Chen, Michael O. Hodge, Julian Jin, Peter L. Protz, Ella Wong, Elizabeth Obe, Ruth Shakeri, Ehsan Cham, Mostafa Gobbert, Matthias K. Barajas, Carlos A. Sharma, Vijay R. Mossahebi, Sina Ren, Lei Peterson, Stephen W. Polf, Jerimy C. Dartmouth College Departments of Mathematics United States U. of Maryland Dept. of Mathematics and Statistics United States James M. Bennett High School SalisburyMD United States Florida Atlantic University Department of Mathematics and Sciences United States Brookdale Community College Department of Mathematics United States U. of Houston Dept. of Computer Science and of Software Engineering Clear Lake United States U. of Maryland Dept. of Information Systems United States U. of Maryland School of Medicine Dept. of Radiation Oncology United States U. of Cape Town Dept. of Physics South Africa M3D Inc. United States

ISBN: (纸本)9798350362480

Precision medicine in cancer treatment increasingly relies on advanced radiotherapies, such as proton beam radiotherapy, to enhance efficacy of the treatment. When the proton beam in this treatment interacts with patient matter, the excited nuclei may emit prompt gamma ray interactions that can be captured by a Compton camera. The image reconstruction from this captured data faces the issue of mischaracterizing the sequences of incoming scattering events, leading to excessive background noise. To address this problem, several machine learning models such as Feedfoward Neural Networks (FNN) and Recurrent Neural Networks (RNN) were developed in PyTorch to properly characterize the scattering sequences on simulated datasets, including newly-created patient medium data, which were generated by using a pipeline comprised of the GEANT4 and Monte-Carlo Detector Effects (MCDE) softwares. These models were implemented using the novel 'Big-data REU Integrated Development and Experimentation' (BRIDE) platform, a modular pipeline that streamlines preprocessing, feature engineering, and model development and evaluation on parallelized GPU processors. Hyperparameter studies were done on the novel patient data as well as on water phantom datasets used during previous research. Patient data was more difficult than water phantom data to classify for both FNN and RNN models. FNN models had higher accuracy on patient medium data but lower accuracy on water phantom data when compared to RNN models. Previous results on several different datasets were reproduced on BRIDE and multiple new models achieved greater performance than in previous research. © 2024 IEEE.

关键词： Phantoms

ENERGY DECAY OF NONLOCAL VISCOELASTIC EQUATIONS WITH NONLINEAR DAMPING AND POLYNOMIAL NONLINEARITY

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arXiv 2025年

This paper is concerned with the energy decay of a viscoelastic variable coefficient wave equation with nonlocality in time as well as nonlinear damping and polynomial nonlinear terms. Using the Lyapunov method, we establish a polynomial energy decay for the solution under relatively weak assumptions regarding the kernel of the nonlocal term. More specifically, we improve the decay rate of the energy by additionally imposing a certain convexity assumption on the kernel. Several examples are provided to confirm such improvements to faster polynomial or even exponential *** Codes 35L20, 35L70, 35B40 Copyright © 2025, The Authors. All rights reserved.

关键词： Damping