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arXiv 2022年

作者： Lee, Donghwan Huang, Xinmeng Hassani, Hamed Dobriban, Edgar Graduate Group in Applied Mathematics and Computational Science Univ. of Pennsylvania United States Department of Electrical and Systems Engineering Univ. of Pennsylvania United States Department of Statistics and Data Science Univ. of Pennsylvania United States

The prediction accuracy of machine learning methods is steadily increasing, but the calibration of their uncertainty predictions poses a significant challenge. Numerous works focus on obtaining well-calibrated predictive models, but less is known about reliably assessing model calibration. This limits our ability to know when algorithms for improving calibration have a real effect, and when their improvements are merely artifacts due to random noise in finite datasets. In this work, we consider detecting mis-calibration of predictive models using a finite validation dataset as a hypothesis testing problem. The null hypothesis is that the predictive model is calibrated, while the alternative hypothesis is that the deviation from calibration is sufficiently large. We find that detecting mis-calibration is only possible when the conditional probabilities of the classes are sufficiently smooth functions of the predictions. When the conditional class probabilities are Hölder continuous, we propose T-Cal, a minimax optimal test for calibration based on a debiased plug-in estimator of the 2-Expected Calibration Error (ECE). We further propose adaptive T-Cal, a version that is adaptive to unknown smoothness. We verify our theoretical findings with a broad range of experiments, including with several popular deep neural net architectures and several standard post-hoc calibration methods. T-Cal is a practical general-purpose tool, which-combined with classical tests for discrete-valued predictors-can be used to test the calibration of virtually any probabilistic classification method. T-Cal is available at https://***/dh7401/T-Cal. Copyright © 2022, The Authors. All rights reserved.

关键词： Forecasting

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Soliton interaction with external forcing within the Korteweg–de Vries equation

arXiv

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arXiv 2018年

作者： Ermakov, Andrei Stepanyants, Yury School of Agricultural Computational and Environmental Sciences University of Southern Queensland QLD4350 Australia School of Agricultural Computational and Environmental Sciences University of Southern Queensland QLD4350 Australia Department of Applied Mathematics Nizhny Novgorod State Technical University Nizhny Novgorod603950 Russia

We revise the solutions of the forced Korteweg–de Vries equation describing a resonant interaction of a solitary wave with external pulse-type perturbations. In contrast to previous works where only the limiting cases of a very narrow forcing in comparison with the initial soliton or a very narrow soliton in comparison with the width of external perturbation were studied, we consider here an arbitrary relationship between the widths of soliton and external perturbation of a relatively small amplitude. In many particular cases, exact solutions of the forced Korteweg–de Vries equation can be obtained for the specific forcings of arbitrary amplitude. We use the earlier developed asymptotic method to derive an approximate set of equations up to the second-order on a small parameter characterising the amplitude of external force. The analysis of exact solutions of the derived equations is presented and illustrated graphically. It is shown that the theoretical outcomes obtained by asymptotic method are in a good agreement with the results of direct numerical modelling within the framework of forced Korteweg–de Vries equation. Copyright © 2018, The Authors. All rights reserved.

关键词： Solitons

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Fixation dynamics on hypergraphs

arXiv

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arXiv 2023年

作者： Liu, Ruodan Masuda, Naoki Department of Mathematics State University of New York at Buffalo BuffaloNY United States Computational and Data-Enabled Sciences and Engineering Program State University of New York at Buffalo BuffaloNY United States

Hypergraphs have been a useful tool for analyzing population dynamics such as opinion formation and the public goods game occurring in overlapping groups of individuals. In the present study, we propose and analyze evolutionary dynamics on hypergraphs, in which each node takes one of the two types of different but constant fitness values. For the corresponding dynamics on conventional networks, under the birth-death process and uniform initial conditions, most networks are known to be amplifiers of natural selection;amplifiers by definition enhance the difference in the strength of the two competing types in terms of the probability that the mutant type fixates in the population. In contrast, we provide strong computational evidence that a majority of hypergraphs are suppressors of selection under the same conditions by combining theoretical and numerical analyses. We also show that this suppressing effect is not explained by one-mode projection, which is a standard method for expressing hypergraph data as a conventional network. Our results suggest that the modeling framework for structured populations in addition to the specific network structure is an important determinant of evolutionary dynamics, paving a way to studying fixation dynamics on higher-order networks including hypergraphs. Copyright © 2023, The Authors. All rights reserved.

关键词： Dynamics

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Hybrid Direct and Iterative Solver with Library of Multi-criteria Optimal Orderings for h Adaptive Finite Element Method Computations

引用

Procedia Computer Science 2016年 80卷 865-874页

作者： Hassan AbouEisha Konrad Jopek Bartłomiej Medygrał Mikhail Moshkov Szymon Nosek Anna Paszyńska Maciej Paszyński Keshav Pingali Computer Science King Abdullah University of Science and Technology Thuwal Saudi Arabia Department of Computer Science AGH University of Science and Technology Kraków Poland Applied Mathematics and Computational Science King Abdullah University of Science and Technology Thuwal Saudi Arabia Faculty of Physics Astronomy and Applied Computer Science Jagiellonian University Kraków Poland Institute for Computational and Engineering Sciences The University of Texas at Austin USA

In this paper we present a multi-criteria optimization of element partition trees and resulting orderings for multi-frontal solver algorithms executed for two dimensional h adaptive finite element method. In particular, the problem of optimal ordering of elimination of rows in the sparse matrices resulting from adaptive finite element method computations is reduced to the problem of finding of optimal element partition trees. Given a two dimensional h refined mesh, we find all optimal element partition trees by using the dynamic programming approach. An element partition tree defines a prescribed order of elimination of degrees of freedom over the mesh. We utilize three different metrics to estimate the quality of the element partition tree. As the first criterion we consider the number of floating point operations(FLOPs) performed by the multi-frontal solver. As the second criterion we consider the number of memory transfers (MEMOPS) performed by the multi-frontal solver algorithm. As the third criterion we consider memory usage (NONZEROS) of the multi-frontal direct solver. We show the optimization results for FLOPs vs MEMOPS as well as for the execution time estimated as FLOPs+100*MEMOPS vs NONZEROS. We obtain Pareto fronts with multiple optimal trees, for each mesh, and for each refinement level. We generate a library of optimal elimination trees for small grids with local singularities. We also propose an algorithm that for a given large mesh with identified local sub-grids, each one with local singularity. We compute Schur complements over the sub-grids using the optimal trees from the library, and we submit the sequence of Schur complements into the iterative solver ILUPCG.

关键词： finite element method h adaptivity multi-frontal direct solver element partition tree ordering algorithms ILUPCG

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An informed operator based genetic algorithm for tuning the reaction rate parameters of chemical kinetics mechanisms

引用

6th Annual Genetic and Evolutionary Computation Conference, GECCO 2004

作者： Elliott, Lionel Ingham, Derek B. Kyne, Adrian G. Mera, Nicolae S. Pourkashanian, Mohamed Wilson, Christopher W. Department of Applied Mathematics University of Leeds LeedsLS2 9JT United Kingdom Centre for Computational Fluid Dynamics Energy and Resources ResearchInstitute University of Leeds LeedsLS2 9JT United Kingdom Department of Mechanical Engineering University of Sheffield Mappin Street SheffieldS1 3JD United Kingdom

ISBN: (纸本)3540223436

A reduced model technique based on a reduced number of numerical simulations at a subset of operating conditions for a perfectly stirred reactor is developed in order to increase the rate of convergence of a genetic algorithm (GA) used for determining new reaction rate parameters of chemical kinetics mechanisms. The genetic algorithm employed uses perfectly stirred reactor, laminar premixed flame and ignition delay time data in the inversion process in order to produce efficient reaction mechanisms that are valid for a wide range of combustion processes and various operating conditions. © Springer-Verlag Berlin Heidelberg 2004.

关键词： Genetic algorithms

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Flexible Categorization Using Formal Concept Analysis and Dempster-Shafer Theory

SSRN

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SSRN 2023年

作者： Manoorkar, Krishna Boersma, Marcel Panettiere, Mattia Palmigiano, Alessandra Tzimoulis, Apostolos Wijnberg, Nachoem School of Business and Economics Vrije Universiteit Amsterdam Netherlands Department of Mathematics and Applied Mathematics University of Johannesburg South Africa College of Business and Economics University of Johannesburg South Africa Faculty of Economics and Business University of Amsterdam Netherlands Computational Science Lab University of Amsterdam Amsterdam Netherlands KPMG Amstelveen Netherlands

Categorization of business processes is an important part of auditing. Large amounts of transactional data in auditing can be represented as transactions between financial accounts using weighted bipartite graphs. We view such bipartite graphs as many-valued formal contexts, which we use to obtain explainable categorization of these business processes in terms of financial accounts involved in a business process by using methods in formal concept analysis. We use Dempster-Shafer mass functions to represent agendas showing different interest in different set of financial accounts. We also model some possible deliberation scenarios between agents with different interrogative agendas to reach an aggregated agenda and categorization. The framework developed in this paper provides a formal ground to obtain and study explainable categorizations from the data represented as bipartite graphs according to the agendas of different agents in an organization (e.g.~an audit firm), and interaction between these through deliberation. We use this framework to describe a machine-leaning meta algorithm for outlier detection and classification which can provide local and global explanations of its result and demonstrate it through an outlier detection algorithm. © 2023, The Authors. All rights reserved.

关键词： Formal concept analysis

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A General Positivity-Preserving Algorithm for Implicit High-Order Finite Volume Schemes Solving the Euler and Navier-Stokes Equations

SSRN

引用

SSRN 2023年

作者： Huang, Qian-Min Ren, Yu-Xin Wang, Qian Department of Engineering Mechanics Tsinghua University Beijing100084 China CAEP Software Center for High Performance Numerical Simulation Beijing100088 China Institute of Applied Physics and Computational Mathematics Beijing100088 China Beijing Computational Science Research Center Beijing100193 China

This paper presents a general positivity-preserving algorithm for implicit high-order finite volume schemes solving Euler and Navier-Stokes equations. Previous positivity-preserving algorithms are mainly based on mathematical analyses, being highly dependent on the existence of low-order positivity-preserving numerical schemes for specific governing equations. This dependency poses serious restrictions on extending these algorithms to temporally implicit schemes, since it is difficult to know if a low-order implicit scheme is positivity-preserving. The present positivity-preserving algorithm is based on an asymptotic analysis of the solutions near local vacuum minimum points. The asymptotic analysis shows that the solutions decay exponentially with time to maintain non-negative density and pressure at a local vacuum minimum point. In its neighborhood, the exponential evolution leads to a modification of the linear evolution process, which can be modelled by a direct correction of the linear residual to ensure positivity. This correction however destroys the conservation of the numerical scheme. Therefore, a second correction procedure is proposed to recover conservation. The proposed positivity-preserving algorithm is considerably less restrictive than existing algorithms. It does not rely on the existence of low-order positivity-preserving baseline schemes and the convex decomposition of volume integrals of flow quantities. It does not need to reduce the time step size for maintaining the stability either. Furthermore, it can be implemented iteratively in the implicit dual time-stepping schemes to preserve positivity of the intermediate and converged states of the sub-iterations. It is proved that the present positivity-preserving algorithm is accuracy-preserving. Numerical results demonstrate that the proposed algorithm preserve the positive density and pressure in all test cases, and considerably improve the robustness of the numerical schemes. © 2023, The Author

关键词： Finite volume method

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Proximal methods for structured nonsmooth optimization over Riemannian submanifolds

arXiv

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arXiv 2024年

作者： Li, Qia Zhang, Na Yan, Hanwei Feng, Junyu School of Computer Science and Engineering Guangdong Province Key Laboratory of Computational Science Sun Yat-sen University Guangzhou China Department of Applied Mathematics College of Mathematics and Informatics South China Agricultural University Guangzhou China School of Computer Science and Engineering Sun Yat-sen University Guangzhou China

In this paper, we consider a class of structured nonsmooth optimization problems over an embedded submanifold of a Euclidean space, where the first part of the objective is the sum of a difference-of-convex (DC) function and a smooth function, while the remaining part is the square of a weakly convex function over a smooth function. This model problem has many important applications in machine learning and scientific computing, for example, the sparse generalized eigenvalue problem. We propose a manifold proximal-gradient-subgradient algorithm (MPGSA) and show that under mild conditions any accumulation point of the solution sequence generated by it is a critical point of the underlying problem. By assuming the Kurdyka-Lojasiewicz property of an auxiliary function, we further establish the convergence of the full sequence generated by MPGSA under some suitable conditions. When the second component of the DC function involved is the maximum of finite continuously differentiable convex functions, we also propose an enhanced MPGSA with guaranteed subsequential convergence to a lifted B-stationary points of the optimization problem. Finally, some preliminary numerical experiments are conducted to illustrate the efficiency of the proposed *** Codes 90C26, 90C30, 65K05 Copyright © 2024, The Authors. All rights reserved.

关键词： Discriminant analysis

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Color-avoiding percolation in edge-colored Erdős–Rényi graphs

arXiv

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arXiv 2022年

作者： Ráth, Balázs Varga, Kitti Fekete, Panna Tímea Molontay, Roland Afréd Rényi Institute of Mathematics Hungary Department of Stochastics Institute of Mathematics Budapest University of Technology and Economics Hungary ELKH-BME Stochastics Research Group Chile Department of Computer Science and Information Theory Faculty of Electrical Engineering and Informatics Budapest University of Technology and Economics Hungary ELKH-ELTE Egerváry Research Group Hungary Department of Applied Analysis and Computational Mathematics Eötvös Loránd University Hungary

We study a variant of the color-avoiding percolation model introduced by Krause et al., namely we investigate the color-avoiding bond percolation setup on (not necessarily properly) edge-colored Erd̋ os–Rényi random graphs. We say that two vertices are color-avoiding connected in an edge-colored graph if after the removal of the edges of any color, they are in the same component in the remaining graph. The color-avoiding connected components of an edge-colored graph are maximal sets of vertices such that any two of them are color-avoiding connected. We consider the fraction of vertices contained in color-avoiding connected components of a given size as well as the fraction of vertices contained in the giant color-avoiding connected component. Under some mild assumptions on the color-densities, we prove that these quantities converge and the limits can be expressed in terms of probabilities associated to edge-colored branching process trees. We provide explicit formulas for the limit of the normalized size of the giant color-avoiding connected component, and in the two-colored case we also provide explicit formulas for the limit of the fraction of vertices contained in color-avoiding connected components of a given size. Copyright © 2022, The Authors. All rights reserved.

关键词： Color

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Learning from 2D: machine learning of 3D effective properties of heterogeneous materials based on 2D microstructure sections

arXiv

引用

arXiv 2023年

作者： Hu, Guangyu Latypov, Marat I. Department of Materials Science and Engineering University of Arizona TucsonAZ85721 United States Graduate Interdisciplinary Program in Applied Mathematics University of Arizona TucsonAZ85721 United States

Microstructure–property relationships are key to effective design of structural materials for advanced applications. Advances in computational methods enabled modeling microstructure-sensitive properties using 3D models (e.g., finite elements) based on microstructure representative volumes. 3D microstructure data required as input to these models are typically obtained from either 3D characterization experiments or digital reconstruction based on statistics from 2D microstructure images. In this work, we present machine learning (ML) approaches to modeling effective properties of heterogeneous materials directly from 2D microstructure sections. To this end, we consider statistical learning models based on spatial correlations and convolutional neural networks as two distinct ML strategies. In both strategies, models are trained on a dataset of synthetically generated 3D microstructures and their properties obtained from micromechanical 3D simulations. Upon training, the models predict properties from 2D microstructure sections. The advantage of the presented models is that they only need 2D sections, whose experimental acquisition is more accessible compared to 3D characterization. Furthermore, the present models do not require digital reconstruction of 3D microstructures. © 2023, CC BY.

关键词： Microstructure

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