Journal: Applied Mathematics and Computation

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Elsevier

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ISSN

0096-3003
1873-5649

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2013 - 2019172020 - 20253
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Publications 1 - 10 of 20
  • On a multidimensional half-discrete Hilbert-type inequality related to the hyperbolic cotangent function
    Item type: Journal Article
    Rassias, Michael Th.; Yang, Bicheng (2014)
    Applied Mathematics and Computation
  • On an n-dimensional mixed type additive and quadratic functional equation
    Item type: Journal Article
    Lee, Yang-Hi; Jung, Soon-Mo; Rassias, Michael Th. (2014)
    Applied Mathematics and Computation
  • Is cooperation sustained under increased mixing in evolutionary public goods games on networks?
    Item type: Journal Article
    Zhang, Wei; Brandes, Ulrik (2023)
    Applied Mathematics and Computation
    A qualitative difference between evolutionary public goods games in a single well-mixing population on the one hand and in neighborhoods of interaction networks on the other hand is the possibility of sustained cooperation within subpopulations. Compared with well-mixed populations, networks model local rather than global interactions by restricting them to social neighborhoods. In this work, we propose an evolutionary game model that is able to capture the effect of long-range links mixing local neighborhood and global group interactions in a finite networked population. We derived dynamical equations for the evolution of cooperation under weak selection by employing the mean-field and pair approximation approach. Using properties of Markov processes, we can approach a theoretical analysis of the effect of the density of mixing link. We find a rule governing the emergence and stabilization of cooperation, which shows that the positive or negative effect of mixing-link density for fixed group size depends on the global benefit in the public goods game. With mutations, we study the average abundance of cooperators and find that increasing mixing links promotes cooperation in strong dilemmas and hinders cooperation in weak dilemmas. These results are independent of whether strategy transfer is allowed via mixing links or not.
  • Transparent boundary conditions for a discontinuous Galerkin Trefftz method
    Item type: Journal Article
    Egger, Herbert; Kretzschmar, Fritz; Schnepp, Sascha M.; et al. (2015)
    Applied Mathematics and Computation
  • On half-discrete Hilbert's inequality
    Item type: Journal Article
    Rassias, Michael T.; Yang, Bicheng (2013)
    Applied Mathematics and Computation
  • On a functional equation of trigonometric type
    Item type: Journal Article
    Jung, Soon-Mo; Rassias, Michael T.; Mortici, Cristinel (2015)
    Applied Mathematics and Computation
  • Designing Gabor windows using convex optimization
    Item type: Journal Article
    Perraudin, Nathanaël; Holighaus, Nicki; Søndergaard, Peter L.; et al. (2018)
    Applied Mathematics and Computation
  • Overall error analysis for the training of deep neural networks via stochastic gradient descent with random initialisation
    Item type: Journal Article
    Jentzen, Arnulf; Welti, Timo (2023)
    Applied Mathematics and Computation
    In spite of the accomplishments of deep learning based algorithms in numerous applications and very broad corresponding research interest, at the moment there is still no rigorous understanding of the reasons why such algorithms produce useful results in certain situations. A thorough mathematical analysis of deep learning based algorithms seems to be crucial in order to improve our understanding and to make their implementation more effective and efficient. In this article we provide a mathematically rigorous full error analysis of deep learning based empirical risk minimisation with quadratic loss function in the probabilistically strong sense, where the underlying deep neural networks are trained using stochastic gradient descent with random initialisation. The convergence speed we obtain suffers under the curse of dimensionality. However, it is presumably close to optimal in the generality of the framework we consider and, to the best of our knowledge, we establish the first full error analysis in the scientific literature for a deep learning based algorithm in the probabilistically strong sense as well as the first full error analysis in the scientific literature for a deep learning based algorithm where stochastic gradient descent with random initialisation is the employed optimisation method.
  • A stencil-based implementation of Parareal in the C++ domain specific embedded language STELLA
    Item type: Conference Paper
    Arteaga, Andrea; Ruprecht, Daniel; Krause, Rolf (2015)
    Applied Mathematics and Computation
    In view of the rapid rise of the number of cores in modern supercomputers, time-parallel methods that introduce concurrency along the temporal axis are becoming increasingly popular. For the solution of time-dependent partial differential equations, these methods can add another direction for concurrency on top of spatial parallelization. The paper presents an implementation of the time-parallel Parareal method in a C++ domain specific language for stencil computations (STELLA). STELLA provides both an OpenMP and a CUDA backend for a shared memory parallelization, using the CPU or GPU inside a node for the spatial stencils. Here, we intertwine this node-wise spatial parallelism with the time-parallel Parareal. This is done by adding an MPI-based implementation of Parareal, which allows us to parallelize in time across nodes. The performance of Parareal with both backends is analyzed in terms of speedup, parallel efficiency and energy-to-solution for an advection–diffusion problem with a time-dependent diffusion coefficient.
  • Numerical approach to the Stokes problem with high contrasts in viscosity
    Item type: Journal Article
    Lobanov, I.S.; Popov, I.Y.; Popov, A.I.; et al. (2014)
    Applied Mathematics and Computation
Publications 1 - 10 of 20