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Setting the physical scale of dimensional reduction in causal dynamical triangulations

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arXiv 2018年

作者： Cooperman, Joshua H. Dorghabekov, Manuchehr Department of Physics and Astronomy Bucknell University LewisburgPA United States Physics Program Bard College Annandale-on-Hudson NY United States Applied Mathematics and Informatics Program American University of Central Asia Bishkek Kyrgyzstan

Within the causal dynamical triangulations approach to the quantization of gravity, striking evidence has emerged for the dynamical reduction of spacetime dimension on sufficiently small scales. Specifically, the spectral dimension decreases from the topological value of 4 towards a value near 2 as the scale being probed decreases. The physical scales over which this dimensional reduction occurs have not previously been ascertained. We present and implement a method to determine these scales in units of either the Planck length or the quantum spacetime geometry’s effective de Sitter length. We find that dynamical reduction of the spectral dimension occurs over physical scales of the order of 10 Planck lengths, which, for the numerical simulation considered below, corresponds to the order of 10−1 de Sitter lengths. Copyright © 2018, The Authors. All rights reserved.

关键词： Triangulation

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Sparse trace tests

arXiv 2022年

作者： Brysiewicz, Taylor Burr, Michael University of Notre Dame Department of Applied and Computational Mathematics and Statistics Notre DameIN46556 United States Clemson University School of Mathematical and Statistical Sciences 220 Parkway Drive ClemsonSC29634 United States

We establish how the coefficients of a sparse polynomial system influence the sum (or the trace) of its zeros. As an application, we develop numerical tests for verifying whether a set of solutions to a sparse system is complete. These algorithms extend the classical trace test in numerical algebraic geometry. Our results rely on both the analysis of the structure of sparse resultants as well as an extension of Esterov’s results on monodromy groups of sparse *** Codes 65H14, 14Q65, 14M25, 68W30 © 2022, CC BY.

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An improved discrete least-squares/reduced-basis method for parameterized elliptic PDEs

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arXiv 2017年

作者： Gunzburger, Max Schneier, Michael Webster, Clayton Zhang, Guannan Department of Scientific Computing Florida State University TallahasseeFL32306-4120 United States Computational and Applied Mathematics Division Oak Ridge National Laboratory Oak RidgeTN37831-6164 United States

It is shown that the computational efficiency of the discrete least-squares (DLS) approximation of solutions of stochastic elliptic PDEs is improved by incorporating a reduced-basis method into the DLS framework. In particular, we consider stochastic elliptic PDEs with an affine dependence on the random variable. The goal is to recover the entire solution map from the parameter space to the finite element space. To this end, first, a reduced-basis solution using a weak greedy algorithm is constructed, then a DLS approximation is determined by evaluating the reduced-basis approximation instead of the full finite element approximation. The main advantage of the new approach is that one only need apply the DLS operator to the coefficients of the reduced-basis expansion, resulting in huge savings in both the storage of the DLS coefficients and the online cost of evaluating the DLS approximation. In addition, the recently developed quasi-optimal polynomial space is also adopted in the new approach, resulting in superior convergence rates for a wider class of problems than previous analyzed. Numerical experiments are provided that illustrate the theoretical results. Copyright © 2017, The Authors. All rights reserved.

关键词： Least squares approximations

Local Discontinuous Galerkin Methods for Solving Convection-Diffusion and Cahn-Hilliard Equations on Surfaces

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arXiv 2024年

作者： Xu, Shixin Xu, Zhiliang Duke Kunshan University 8 Duke Ave Kunshan Shi Jiangsu Sheng Suzhou Shi215316 China Department of Applied and Computational Mathematics and Statistics University of Notre Dame Notre DameIN46556 United States

Local discontinuous Galerkin methods are developed for solving second order and fourth order time-dependent partial differential equations defined on static 2D manifolds. These schemes are second-order accurate with surfaces triangulized by planar triangles and careful design of numerical fluxes. The schemes are proven to be energy stable. Various numerical experiments are provided to validate the new schemes. © 2024, CC BY.

关键词： Galerkin methods

The dynamical behavior for a famous class of evolution equations with double exponential nonlinearities

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Journal of Ocean Engineering and Science 2022年

作者： Alharthi, Mohammed Shaaf Baleanu, Dumitru Ali, Khalid K. Nuruddeen, R.I. Muhammad, Lawal Aljohani, Abdulrahman F. Osman, M.S. Department of Mathematics and Statistics College of Science Taif University P.O. Box 11099 Taif 21944 Saudi Arabia Department of Mathematics Faculty of Arts and Sciences Çankaya University Ögretmenler Cad. Ankara 1406530 Turkey Institute of Space Sciences Magurele Bucharest Romania Mathematics Department Faculty of Science Al-Azhar University Nasr-City Cairo Egypt Department of Mathematics Faculty of Science Federal University Dutse Dutse Jigawa State Nigeria Department of Mathematics Faculty of Science Yusuf Maitama Sule University Kano Kano State Nigeria Computational & Analytical Mathematics and Their Applications Research Group Department of Applied Mathematics Faculty of Science University of Tabuk Tabuk 71491 Saudi Arabia Department of Mathematics Faculty of Science Cairo University Giza 12613 Egypt

An analytical investigation for a famous class of evolution equations with double exponential nonlinearities that has vast applications in many nonlinear sciences is presented. These equations include the Tzitzéica Equation (TE), Dodd-Bullough-Mikhailov Equation (DBME), Tzitzéica-Dodd-Bullough-Mikhailov equation (TDBME) and the Peyrard Bishop DNA Equation (PB-DNA-E). Furthermore, the Kudryashov method for constructing exponential function solutions has been employed to reveal various sets of traveling wave solutions with different geometrical structures to the identified models. We also give the graphical illustrations of certain solutions to further analyze the results. © 2022

关键词： Evolution equations Exponential nonlinearities Exponential solutions Kudryashov method

Artificial Immune Systems Approach for Surface Reconstruction of Shapes with Large Smooth Bumps 23rd

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Artificial Immune Systems Approach for Surface Reconstructi...

23rd International Conference on computational Science, ICCS 2023

作者： Gálvez, Akemi Fister, Iztok You, Lihua Fister, Iztok Iglesias, Andrés Department of Applied Mathematics and Computational Sciences University of Cantabria Santander39005 Spain Faculty of Pharmaceutical Sciences Toho University 2-2-1 Miyama Funabashi274-8510 Japan Faculty of Electrical Engineering and Computer Science University of Maribor Maribor Slovenia National Center for Computer Animation Faculty of Media and Communication Bournemouth University PooleBH12 5BB United Kingdom

ISBN: (纸本)9783031360268

Reverse engineering is one of the classical approaches for quailty assessment in industrial manufacturing. A key technology in reverse engineering is surface reconstruction, which aims at obtaining a digital model of a physical object from a cloud of 3D data points obtained by scanning the object. In this paper we address the surface reconstruction problem for surfaces that can exhibit large smooth bumps. To account for this type of features, our approach is based on using exponentials of polynomial functions in two variables as the approximating functions. In particular, we consider three different models, given by bivariate distributions obtained by combining a normal univariate distribution with a normal, Gamma, and Weibull distribution, respectively. The resulting surfaces depend on some parameters whose values have to be optimized. This yields a difficult nonlinear continuous optimization problem solved through an artificial immune systems approach based on the clonal selection theory. The performance of the method is discussed through its application to a benchmark comprised of three examples of point clouds. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023.

关键词： Weibull distribution

Nonuniform 3D finite difference elastic wave simulation on staggered grids

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arXiv 2020年

作者： Gao, Longfei Ghattas, Omar Keyes, David Oden Institute for Computational Engineering and Sciences The University of Texas at Austin AustinTX78712 United States Oden Institute for Computational Engineering and Sciences Jackson School of Geosciences Department of Mechanical Engineering The University of Texas at Austin AustinTX78712 United States Applied Mathematics and Computational Science Program Extreme Computing Research Center King Abdullah University of Science and Technology Thuwal23955-6900 Saudi Arabia

We present an approach to simulate the 3D isotropic elastic wave propagation using nonuniform finite difference discretization on staggered grids. Specifically, we consider simulation domains composed of layers of uniform grids with different grid spacings, separated by nonconforming interfaces. We demonstrate that this layer-wise finite difference discretization has the potential to significantly reduce the simulation cost, compared to its fully uniform counterpart. Stability of such a discretization is achieved by using specially designed difference operators, which are variants of the standard difference operators with adaptations near boundaries or interfaces, and penalty terms, which are appended to the discretized wave system to weakly impose boundary or interface conditions. Combined with specially designed interpolation operators, the discretized wave system is shown to preserve the energy conserving property of the continuous elastic wave equation, and a fortiori ensure the stability of the simulation. Numerical examples are presented to demonstrate the efficacy of the proposed simulation approach. © 2020, CC BY.

关键词： Elastic waves

Modeling of Measurement Error in Financial Returns Data

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arXiv 2024年

作者： Jasra, Ajay Maama, Mohamed Mijatović, Aleksandar School of Data Science The Chinese University of Hong Kong Shenzhen China Applied Mathematics and Computational Science Program Computer Electrical and Mathematical Sciences and Engineering Division King Abdullah University of Science and Technology Thuwal23955-6900 Saudi Arabia Department of Statistics University of Warwick United Kingdom

In this paper we consider the modeling of measurement error for fund returns data. In particular, given access to a time-series of discretely observed log-returns and the associated maximum over the observation period, we develop a stochastic model which models the true log-returns and maximum via a Lévy process and the data as a measurement error there-of. The main technical difficulty of trying to infer this model, for instance Bayesian parameter estimation, is that the joint transition density of the return and maximum is seldom known, nor can it be simulated exactly. Based upon the novel stick breaking representation of [12] we provide an approximation of the model. We develop a Markov chain Monte Carlo (MCMC) algorithm to sample from the Bayesian posterior of the approximated posterior and then extend this to a multilevel MCMC method which can reduce the computational cost to approximate posterior expectations, relative to ordinary MCMC. We implement our methodology on several applications including for real data. Copyright © 2024, The Authors. All rights reserved.

关键词： Markov chains

Improved error estimates for a modified exponential Euler method for the semilinear stochastic heat equation with rough initial data

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arXiv 2023年

作者： Gui, Xinping Li, Buyang Wang, Jilu Beijing Computational Science Research Center Beijing100193 China Department of Applied Mathematics The Hong Kong Polytechnic University Hung Hom Hong Kong School of Science Harbin Institute of Technology Shenzhen518055 China

A class of stochastic Besov spaces BpL2(Ω;(Equation presented α(O)), 1 ⩽ p ⩽ ∞ and α ∈ [−2, 2], is introduced to characterize the regularity of the noise in the semilinear stochastic heat equation du − ∆udt = f(u)dt + dW(t), under the following conditions for some α ∈ (0, 1]: (Equation presented). The conditions above are shown to be satisfied by both trace-class noises (with α = 1) and one-dimensional space-time white noises (with α = 1/2). The latter would fail to satisfy the conditions with α = 1/2 if the stochastic Besov norm (Equation presented) is replaced by the classical Sobolev norm k · kL2(Ω;(Equation presented α(O)), and this often causes reduction of the convergence order in the numerical analysis of the semilinear stochastic heat equation. In this article, the convergence of a modified exponential Euler method, with a spectral method for spatial discretization, is proved to have order α in both time and space for possibly nonsmooth initial data in L4(Ω;(Equation presented β(O)) with β > −1, by utilizing the real interpolation properties of the stochastic Besov spaces and a class of locally refined stepsizes to resolve the singularity of the solution at t = 0. Copyright © 2023, The Authors. All rights reserved.

关键词： Additive noise