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Computational efficiency of fractional diffusion using adaptive time step memory [article]

Brian P. Sprouse, Christopher L. MacDonald, Gabriel A. Silva
2010 arXiv   pre-print
Here we present an adaptive time method for smooth functions that is computationally efficient and results in smaller errors during numerical simulations.  ...  In finite difference methods this has been approximated using the 'short memory effect' where it is assumed that previous events prior to some certain time point are insignificant and thus not calculated  ...  Igor Podlubny for helpful discussions during the course of this work.  ... 
arXiv:1004.5128v1 fatcat:j3awgxk2cnevxpv7izpyfsxgsm

Efficient computation of the Grünwald–Letnikov fractional diffusion derivative using adaptive time step memory

Christopher L. MacDonald, Nirupama Bhattacharya, Brian P. Sprouse, Gabriel A. Silva
2015 Journal of Computational Physics  
Here we present an adaptive time step memory method for smooth functions applied to the Grunwald-Letnikov fractional diffusion derivative.  ...  Computing numerical solutions to fractional differential equations can be computationally intensive due to the effect of non-local derivatives in which all previous time points contribute to the current  ...  Igor Podlubny for helpful discussions during the course of this work. This work was supported by NIH NS054736 and ARO 63795EGII.  ... 
doi:10.1016/j.jcp.2015.04.048 fatcat:g2wwtlgjqbddldua5b6qju36ky

An adaptive memory method for accurate and efficient computation of the Caputo fractional derivative [article]

Daegeun Yoon, Donghyun You
2017 arXiv   pre-print
The sub-diffusion process of a time-fractional diffusion equation and the creeping response of a fractional viscoelastic model are simulated to demonstrate the accuracy as well as the computational efficiency  ...  The present adaptive memory method significantly reduces the requirement for computational memory for storing function values at past time points and also significantly improves the accuracy by calculating  ...  Using the analytic solution of a time-fractional diffusion equation, the accuracy and computational benefits of each method are compared quantitatively.  ... 
arXiv:1711.10071v1 fatcat:urhvjdkh7bax5crvtu4auuh24q

High throughput software for direct numerical simulations of compressible two-phase flows

Babak Hejazialhosseini, Diego Rossinelli, Christian Conti, Petros Koumoutsakos
2012 2012 International Conference for High Performance Computing, Networking, Storage and Analysis  
The software enables 3D simulations of shockbubble interaction including, for the first time, effects of diffusion and surface tension, by efficiently employing two hundred billion computational elements  ...  We suggest that the present work may serve as a performance upper bound, regarding achievable GFLOP/s, for two-phase flow solvers using adaptive mesh refinement.  ...  ACKNOWLEDGMENT The authors would like to thank Mauro Bianco of the Swiss National Supercomputing Center (CSCS) and Dr.  ... 
doi:10.1109/sc.2012.66 dblp:conf/sc/HejazialhosseiniRCK12 fatcat:gazke6q2gjbatpksmsplfspkt4

Efficient numerical solution of the time fractional diffusion equation by mapping from its Brownian counterpart

Peter W. Stokes, Bronson Philippa, Wayne Read, Ronald D. White
2015 Journal of Computational Physics  
The solution of a Caputo time fractional diffusion equation of order 0<α<1 is expressed in terms of the solution of a corresponding integer order diffusion equation.  ...  The mapping is applied successfully to the least-squares fitting of a fractional advection diffusion model for the current in a time-of-flight experiment, resulting in a computational speed up in the range  ...  Figure 5 . 5 Comparison of computation time versus number of time steps for least squares fitting performed using the explicit finite difference method  ... 
doi:10.1016/j.jcp.2014.11.023 fatcat:fifog4meaffhhkhwyptgipw3au

Stability and Complexity Analyses of Finite Difference Algorithms for the Time-Fractional Diffusion Equation [article]

Nirupama Bhattacharya, Gabriel A. Silva
2020 arXiv   pre-print
We also analyze the time complexity of the algorithms, and describe an alternate adaptive time step approach that utilizes a linked list implementation, which yields better algorithmic efficiency.  ...  In previous work we implemented the full 2D time-fractional diffusion equation as a Forward Time Central Space finite difference equation by using the Grünwald-Letnikov definition of the fractional derivative  ...  We used adaptive step adaptive step parameter a and η values of 20. Figure 4 . 1 : 41 Comparing error of adaptive memory and linked list algorithms, using the full implementation as the reference.  ... 
arXiv:2007.08660v1 fatcat:e5lu2rjaybalbncappnazhaioq

Dynamics of the Human Structural Connectome Underlying Working Memory Training

K. Caeyenberghs, C. Metzler-Baddeley, S. Foley, D. K. Jones
2016 Journal of Neuroscience  
indexed by the [quantitative] longitudinal relaxation rate, R 1 ) than previously used diffusion MRI metrics (fractional anisotropy or fiber-tracking recovered streamlines).  ...  Adaptive working memory training led to improvement on (non)trained working memory tasks and generalization to tasks of reasoning and inhibition.  ...  We now describe the processing steps starting from analyses of the neuroimaging data to computation of the topological metrics of the graph.  ... 
doi:10.1523/jneurosci.1973-15.2016 pmid:27053212 pmcid:PMC4821915 fatcat:vpgsih74fvcvxeckbf3bbhnwge

New Resolution Strategy for Multi-scale Reaction Waves using Time Operator Splitting and Space Adaptive Multiresolution: Application to Human Ischemic Stroke

Max Duarte, Marc Massot, Stéphane Descombes, Christian Tenaud, Thierry Dumont, Violaine Louvet, Frédérique Laurent
2011 ESAIM: Proceedings and Surveys  
for shared memory architectures was implemented, in order to reduce computing costs related to "detailed chemistry" features of the model.  ...  Based on recent theoretical studies of numerical analysis, such a strategy leads to a splitting time step which is not restricted neither by the fastest scales in the source term nor by restrictive diffusive  ...  , then the numerical strategy can be used exactly as it is provided with a dynamical self-adapted splitting time step for more general unsteady solutions of reaction-diffusion and convection-reaction-diffusion  ... 
doi:10.1051/proc/201134006 fatcat:iegcgotdlfhujneg6hmaov2m2a

Task-based adaptive multiresolution for time-space multi-scale reaction-diffusion systems on multi-core architectures

Stéphane Descombes, Max Duarte, Thierry Dumont, Thomas Guillet, Violaine Louvet, Marc Massot
2017 SMAI Journal of Computational Mathematics  
A new solver featuring time-space adaptation and error control has been recently introduced to tackle the numerical solution of stiff reaction-diffusion systems.  ...  Based on operator splitting, finite volume adaptive multiresolution and high order time integrators with specific stability properties for each operator, this strategy yields high computational efficiency  ...  Part of this work was conducted by the Exascale Computing Research laboratory, thanks to the support of CEA, GENCI, Intel, UVSQ.  ... 
doi:10.5802/smai-jcm.19 fatcat:udd3cgm2zzdadmvuaunysfmhda

Numerical simulations of 2D fractional subdiffusion problems

Hermann Brunner, Leevan Ling, Masahiro Yamamoto
2010 Journal of Computational Physics  
In this work, the 2D fractional subdiffusion problems are solved by an algorithm that couples an adaptive time stepping and adaptive spatial basis selection approach.  ...  Currently, most algorithms are designed for 1D problems due to the memory effect in fractional derivatives.  ...  Note that using small τ > 0 requires more (but still much faster than using fixed small time step) computational time.  ... 
doi:10.1016/j.jcp.2010.05.015 fatcat:q2bdpa7gd5glpbhdletui5vzse

Adaptation of the in situ adaptive tabulation (ISAT) procedure for efficient computation of surface reactions

Sandip Mazumder
2005 Computers and Chemical Engineering  
Computationally efficient implementation of combustion chemistry using in situ adaptive tabulation.  ...  Combustion Theory and Modeling, 1, 41-63] for the efficient computation of homogeneous reactions in chemically reacting flows, is adapted for the efficient computation of heterogeneous surface reactions  ...  Research has revealed that if the chemistry is complex, a large fraction of the overall computational time is spent in computing surface reaction fluxes, i.e., in solving the reaction-diffusion balance  ... 
doi:10.1016/j.compchemeng.2005.08.008 fatcat:nqgtndborfa5vlzmowisbwwsm4

Parallel Adaptive Simulation of Coupled Incompressible Viscous Flow and Advective-Diffusive Transport Using Stabilized FEM Formulation

André L. Rossa, Alvaro L.G.A. Coutinho
2012 CLEI Electronic Journal  
In this work we study coupled incompressible viscous flow and advective-diffusive trans- port of a scalar. Both the Navier-Stokes and transport equations are solved using an Eulerian approach.  ...  The implementation is held using the libMEsh finite element library which provides support for parallel adaptive mesh refinement and coarsening.  ...  We set the exponent of the nonlinear diffusion operator (19) as β = 1 and the time step is set to ∆t = 0.025.  ... 
doi:10.19153/cleiej.15.3.1 fatcat:qeydz2znabcijlcvayo373scmu

A fast, low-memory, and stable algorithm for implementing multicomponent transport in direct numerical simulations [article]

Aaron J. Fillo, Jason Schlup, Guillaume Beardsell, Guillaume Blanquart, Kyle E. Niemeyer
2019 arXiv   pre-print
In this study we propose a fast,efficient, low-memory algorithm and use that to evaluate the role of multicomponent mass diffusion in reacting-flow simulations.  ...  A semi-implicit algorithm decreases the computational expense of inverting the full multicomponent ordinary diffusion array while maintaining accuracy and fidelity.  ...  A time-step size of ∆t =10 −5 s was used for B. Figure 2 : 2 Two-dimensional schematic of the three-dimensional flame configuration. Adapted from Burali et al. and Schlup and Blanquart [5, 6] .  ... 
arXiv:1808.05463v3 fatcat:xkxxdznrxfbiddu2co5xu75m7e

radamesh: cosmological radiative transfer for Adaptive Mesh Refinement simulations

Sebastiano Cantalupo, Cristiano Porciani
2010 Monthly notices of the Royal Astronomical Society  
This results in an increased accuracy level and, at the same time, a huge gain in computational speed with respect to a "classical" Monte Carlo RT, especially when combined with an Adaptive Mesh Refinement  ...  The chemical state of six species (HI, HII, HeI, HeII, HeIII, e) and gas temperatures are computed with a time-dependent, non-equilibrium chemistry solver.  ...  This ensures an efficient memory usage and reduces computational overhead during ray-tracing.  ... 
doi:10.1111/j.1365-2966.2010.17799.x fatcat:f4f7k46t6jg5hg5fatrfqh6moi

Adaptive Time-Space Algorithms for the Simulation of Multi-scale Reaction Waves [chapter]

Max Duarte, Marc Massot, Stéphane Descombes, Thierry Dumont
2011 Springer Proceedings in Mathematics  
We aim at time-space accuracy control of the solution and splitting time steps purely dictated by the physics of the phenomenon and not by stability constraints associated with mesh size or source time  ...  The main goal is to perform computationally efficient simulations of the dynamics of multi-scale phenomena under study, considering large simulation domains with conventional computing resources.  ...  step due to stability requirements or the unreasonable memory and computing time required by implicit methods.  ... 
doi:10.1007/978-3-642-20671-9_40 fatcat:f5zvmn2iwnc7xabm6zwrvatocm
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