检索结果-内蒙古大学图书馆

GENERALIZING RIEMANN CURVATURE TO REGGE METRICS

学校读者我要写书评

暂无评论

arXiv 2023年

作者： Gopalakrishnan, Jay Neunteufel, Michael Schöberl, Joachim Wardetzky, Max Portland State University PO Box 751 PortlandOR97201 United States Institute for Analysis and Scientific Computing TU Wien Wiedner Hauptstr. 8-10 Wien1040 Austria Institute of Numerical and Applied Mathematics University of Göttingen Lotzestr. 16-18 Göttingen37083 Germany

In this paper we propose a generalization of the Riemann curvature tensor on manifolds (of dimension two or higher) endowed with a Regge metric. Specifically, while all components of the metric tensor are assumed to be smooth within elements of a triangulation of the manifold, they need not be smooth across element interfaces, where only continuity of the tangential components are assumed. While linear derivatives of the metric can be generalized as Schwartz distributions, similarly generalizing the classical Riemann curvature tensor, a nonlinear second-order derivative of the metric, requires more care. We propose a generalization combining the classical angle defect and jumps of the second fundamental form across element interfaces, and rigorously prove correctness of this generalization. Specifically, if a piecewise smooth metric approximates a globally smooth metric, our generalized Riemann curvature tensor approximates the classical Riemann curvature tensor arising from a globally smooth metric. Moreover, we show that if the metric approximation converges at some rate in a piecewise norm that scales like the L2-norm, then the curvature approximation converges in the H-2-norm at the same rate, under additional assumptions. By appropriate contractions of the generalized Riemann curvature tensor, this work also provides generalizations of scalar curvature, the Ricci curvature tensor, and the Einstein tensor in any *** Codes 65N30 (Primary) 53A70, 83C27 (Secondary) Copyright © 2023, The Authors. All rights reserved.

关键词： Tensors

Asymptotic Consistency for Nonconvex Risk-Averse Stochastic Optimization with Infinite Dimensional Decision Spaces

学校读者我要写书评

暂无评论

arXiv 2022年

作者： Milz, Johannes Surowiec, Thomas M. H. Milton Stewart School of Industrial and Systems Engineering Georgia Institute of Technology AtlantaGA30332 United States Department of Numerical Analysis and Scientific Computing Simula Research Laboratory Oslo0164 Norway

Optimal values and solutions of empirical approximations of stochastic optimization problems can be viewed as statistical estimators of their true values. From this perspective, it is important to understand the asymptotic behavior of these estimators as the sample size goes to infinity. This area of study has a long tradition in stochastic programming. However, the literature is lacking consistency analysis for problems in which the decision variables are taken from an infinite dimensional space, which arise in optimal control, scientific machine learning, and statistical estimation. By exploiting the typical problem structures found in these applications that give rise to hidden norm compactness properties for solution sets, we prove consistency results for nonconvex risk-averse stochastic optimization problems formulated in infinite dimensional space. The proof is based on several crucial results from the theory of variational convergence. The theoretical results are demonstrated for several important problem classes arising in the *** Codes 90C15, 90C06, 62F12, 35Q93, 49M41, 49J52 © 2022, CC BY-SA.

关键词： Stochastic programming

Exploration of core concepts required for mid- and domain-level ontology development to facilitate explainable-AI-readiness of data and models 4

学校读者我要写书评

暂无评论

Exploration of core concepts required for mid- and domain-le...

4th International Workshop on Data meets Ontologies in Explainable AI, DAO-XAI 2024

作者： Horsch, Martin T. Chiacchiera, Silvia Todorov, Ilian T. Correia, Ana Teresa Dey, Aditya Konchakova, Natalia A. Scholze, Sebastian Stephan, Simon Tøndel, Kristin Sarkar, Arkopaul Karray, M. Hedi Al Machot, Fadi Schembera, Björn Norwegian University of Life Sciences Department of Data Science Postboks 5003 Ås1432 Norway UKRI Science and Technology Facilities Council Scientific Computing Department DaresburyWA4 4AD United Kingdom ATB Institut für Angewandte Systemtechnik Bremen GmbH Wiener Str. 1 Bremen28359 Germany Helmholtz-Zentrum Hereon Institute of Surface Science Max-Planck-Str. 1 Geesthacht21502 Germany RPTU Kaiserslautern Laboratory of Engineering Thermodynamics Kaiserslautern67663 Germany University of Technology Tarbes Occitanie Pyrénées Production Engineering Lab. 47 av. d’Azereix Tarbes France University of Stuttgart Institute of Applied Analysis and Numerical Simulation Stuttgart70569 Germany

This position paper reports on the initial discussions within the Knowledge Graph Alliance’s working group on explainable-AI-ready data and metadata principles, which was created in March 2024. At present, we are taking initial steps toward capturing core concepts related to explanation, grounding, reliance, and trust;the scope also extends to potential dual notions such as explainability, verifiability/reproducibility, reliability, and trustworthiness. These initial steps consist in reviewing core concepts as they are discussed in the literature and exploring what could be practically useful definitions of these most central concepts. One of the conclusions is that the metadata standards will need to be suitable for documenting three kinds of grounding: Grounding of knowledge, grounding of reliance, and grounding of trust. Pre-existing metadata standards at the mid and domain level are presently undergoing a redesign in order to become more modular, computationally tractable, intelligible to humans, and adjustable, which will be needed as we continue our work toward actionable recommendations. The development of this system of lite (OWL 2 EL) ontologies, called MSO-EM: Ontologies for modelling, simulation, optimization (MSO) and epistemic metadata (EM), is carried out on a public repository. © 2024 Copyright for this paper by its authors.

关键词： Applied ontology epistemic metadata explainable-AI-ready (XAIR) ontology redesign reliability reproducibility

A fast and oblivious matrix compression algorithm for Volterra integral operators

学校读者我要写书评

暂无评论

arXiv 2021年

作者： Dölz, Jürgen Egger, Herbert Shashkov, Vsevolod Institute for Numerical Simulation University of Bonn Friedrich-Hirzebruch-Allee 7 Bonn53115 Germany Numerical Analysis and Scientific Computing Department of Mathematics TU Darmstadt Dolivostr. 15 Darmstadt64293 Germany

The numerical solution of dynamical systems with memory requires the efficient evaluation of Volterra integral operators in an evolutionary manner. After appropriate discretisation, the basic problem can be represented as a matrix-vector product with a lower diagonal but densely populated matrix. For typical applications, like fractional diffusion or large scale dynamical systems with delay, the memory cost for storing the matrix approximations and complete history of the data then becomes prohibitive for an accurate numerical approximation. For Volterra-integral operators of convolution type, the fast and oblivious convolution quadrature method of Schädle, Lopez-Fernandez, and Lubich resolves this issue and allows to compute the discretized evaluation with N time steps in O(N log N) complexity and only requires O(log N) active memory to store a compressed version of the complete history of the data. We will show that this algorithm can be interpreted as an H-matrix approximation of the underlying integral operator. A further improvement can thus be achieved, in principle, by resorting to H2matrix compression techniques. Following this idea, we formulate a variant of the H2-matrix vector product for discretized Volterra integral operators that can be performed in an evolutionary and oblivious manner and requires only O(N) operations and O(log N) active memory. In addition to the acceleration, more general asymptotically smooth kernels can be treated and the algorithm does not require a-priori knowledge of the number of time steps. The efficiency of the proposed method is demonstrated by application to some typical test problems. Copyright © 2021, The Authors. All rights reserved.

关键词： Matrix algebra

Convergence of Anisotropic Consensus-Based Optimization in Mean-Field Law

学校读者我要写书评

暂无评论

arXiv 2021年

作者： Fornasier, Massimo Klock, Timo Riedl, Konstantin Technical University of Munich Department of Mathematics Munich Germany Munich Data Science Institute Munich Germany Simula Research Laboratory Department of Numerical Analysis and Scientific Computing Oslo Norway

In this paper we study anisotropic consensus-based optimization (CBO), a population-based metaheuristic derivative-free optimization method capable of globally minimizing nonconvex and nonsmooth functions in high dimensions. CBO is based on stochastic swarm intelligence, and inspired by consensus dynamics and opinion formation. Compared to other metaheuristic algorithms like Particle Swarm Optimization, CBO is of a simpler nature and therefore more amenable to theoretical analysis. By adapting a recently established proof technique, we show that anisotropic CBO converges globally with a dimension-independent rate for a rich class of objective functions under minimal assumptions on the initialization of the method. Moreover, the proof technique reveals that CBO performs a convexification of the optimization problem as the number of particles goes to infinity, thus providing an insight into the internal CBO mechanisms responsible for the success of the method. To motivate anisotropic CBO from a practical perspective, we further test the method on a complicated high-dimensional benchmark problem, which is well understood in the machine learning *** Codes 65K10, 90C26, 90C56, 35Q90, 35Q84 Copyright © 2021, The Authors. All rights reserved.

关键词： Anisotropy

analysis OF CURVATURE APPROXIMATIONS VIA COVARIANT CURL AND INCOMPATIBILITY FOR REGGE METRICS

学校读者我要写书评

暂无评论

arXiv 2022年

作者： Gopalakrishnan, Jay Neunteufel, Michael Schöberl, Joachim Wardetzky, Max Portland State University PO Box 751 PortlandOR97207 United States Institute for Analysis and Scientific Computing Tu Wien Wiedner Hauptstr. 8-10 Wien1040 Austria Institute of Numerical and Applied Mathematics University of Göttingen Lotzestr. 16-18 Göttingen37083 Germany

The metric tensor of a Riemannian manifold can be approximated using Regge finite elements and such approximations can be used to compute approximations to the Gauss curvature and the Levi-Civita connection of the manifold. It is shown that certain Regge approximations yield curvature and connection approximations that converge at a higher rate than previously known. The analysis is based on covariant (distributional) curl and incompatibility operators which can be applied to piecewise smooth matrix fields whose tangential-tangential component is continuous across element interfaces. Using the properties of the canonical interpolant of the Regge space, we obtain superconvergence of approximations of these covariant operators. numerical experiments further illustrate the results from the error *** Codes 65N30 (Primary) 53A70, 83C27 (Secondary) Copyright © 2022, The Authors. All rights reserved.

关键词： Geometry

A Sensitivity analysis of Cellular Automata and Heterogeneous Topology Networks: Partially-Local Cellular Automata and Homogeneous Homogeneous Random Boolean Networks

学校读者我要写书评

暂无评论

arXiv 2024年

作者： Glover, Tom Eivind Jahren, Ruben Martinuzzi, Francesco Lind, Pedro Gonçalves Nichele, Stefano Department of Computer Science OsloMet - Oslo Metropolitan University Oslo Norway Department of Computer Science and Communication Østfold University College Halden Norway Simula Research Laboratory Numerical Analysis and Scientific Computing Oslo0164 Norway Center for Scalable Data Analytics and Artificial Intelligence Leipzig University Leipzig Germany Institute for Earth System Science & Remote Sensing Leipzig University Leipzig Germany Remote Sensing Centre for Earth System Research Leipzig University Leipzig Germany

Elementary Cellular Automata (ECA) are a well-studied computational universe that is, despite its simple configurations, capable of impressive computational variety. Harvesting this computation in a useful way has historically shown itself to be difficult, but if combined with reservoir computing (RC), this becomes much more feasible. Furthermore, RC and ECA enable energy-efficient AI, making the combination a promising concept for Edge AI. In this work, we contrast ECA to substrates of Partially-Local CA (PLCA) and Homogeneous Homogeneous Random Boolean Networks (HHRBN). They are, in comparison, the topological heterogeneous counterparts of ECA. This represents a step from ECA towards more biological-plausible substrates. We analyse these substrates by testing on an RC benchmark (5-bit memory), using Temporal Derrida plots to estimate the sensitivity and assess the defect collapse rate. We find that, counterintuitively, disordered topology does not necessarily mean disordered computation. There are countering computational"forces" of topology imperfections leading to a higher collapse rate (order) and yet, if accounted for, an increased sensitivity to the initial condition. These observations together suggest a shrinking critical range. © 2024, CC BY.

关键词： Sensitivity analysis

Springer Proceedings in Mathematics and Statistics

学校读者我要写书评

暂无评论

Preface

Springer Proceedings in Mathematics and Statistics 2023年 432卷 vii-viii页

作者： Franck, Emmanuel Michel-Dansac, Victor Navoret, Laurent Fuhrmann, Jürgen Université de Strasbourg CNRS Inria IRMA Strasbourg France Weierstrass Institute for Applied Analysis and Stochastics Numerical Mathematics and Scientific Computing Berlin Germany

OPTIMIZATION OF HOPF BIFURCATION POINTS

学校读者我要写书评

暂无评论

arXiv 2022年

作者： Boullé, Nicolas Farrell, Patrick E. Rognes, Marie E. Isaac Newton Institute for Mathematical Sciences University of Cambridge CB3 0EH United Kingdom Mathematical Institute University of Oxford OxfordOX2 6GG United Kingdom Department of Scientific Computing and Numerical Analysis Simula Research Laboratory Oslo Norway Department of Mathematics University of Bergen Bergen Norway

We introduce a numerical technique for controlling the location and stability properties of Hopf bifurcations in dynamical systems. The algorithm consists of solving an optimization problem constrained by an extended system of nonlinear partial differential equations that characterizes Hopf bifurcation points. The flexibility and robustness of the method allows us to advance or delay a Hopf bifurcation to a target value of the bifurcation parameter, as well as controlling the oscillation frequency with respect to a parameter of the system or the shape of the domain on which solutions are defined. numerical applications are presented in systems arising from biology and fluid dynamics, such as the FitzHugh-Nagumo model, Ginzburg-Landau equation, Rayleigh-Bénard convection problem, and Navier-Stokes equations, where the control of the location and oscillation frequency of periodic solutions is of high *** Codes 65P30, 65P40, 37M20, 65K10, 49M41 Copyright © 2022, The Authors. All rights reserved.

关键词： Hopf bifurcation