A New Framework of GPU-Accelerated Spectral Solvers: Collocation and Glerkin Methods for Systems of Coupled Elliptic Equations

Chen, Feng

doi:10.1007/s10915-014-9868-3

Cited by 6 publications

(3 citation statements)

References 21 publications

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“…which may not be coercive in general. Thus if f = (f 1 , f 2 ) ∈ (L 2 (Ω)) 2 , we cannot use Lax-Milgram lemma directly to prove the existence of a solution to the problem: Find u ∈ V such that…”

Section: Existence and Uniquenessmentioning

confidence: 99%

“…The aim of this paper is to obtain an existence and uniqueness of the solutions on any bounded Lipschitz domain, in addition to carry out a new meshless (WEB-S) numerical method for the approximate solutions of the system. On an arbitrary domain finite element approximation of these equations is a highly sought after approach to obtain their numerical solution;In particular WEB-FEM combines the computational advantages of B-Splines and standard mesh-based finite elements.Further it attains the degree and smoothness to be chosen flexibly without substantially increasing the size of problem.Off late spectral Galerkin method [2], Lattice-Boltzmann schemes [3] etc have been used for numerical approximations.In [4] Boglaev have used method of upper and lower solutions , and construct monotone sequence for difference scheme to approximate the solution of coupled system and Xiu et.al [5] used stochastic Galerkin and stochastic collocation method in conjunction with the gPC expansions. In view of the computational advantages WEBS-FEA is one of highly desired approach to solve the coupled elliptic system.In the current literature no work is reported on WEBS-FEA of the general coupled elliptic problem.…”

Section: Introductionmentioning

confidence: 99%

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Weighted extended B-spline finite element analysis of a coupled system of general elliptic equations

Chakraborty

Kumar

2018

Int J Adv Eng Sci Appl Math

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In this study we establish the existence and uniqueness of the solution of a coupled system of general elliptic equations with anisotropic diffusion , non-uniform advection and variably influencing reaction terms on Lipschitz continuous domain Ω ⊂ R m (m≥1) with a Dirichlet boundary. Later we consider the finite element (FE) approximation of the coupled equations in a meshless framework based on weighted extended B-Spine functions (WEBS).The a priori error estimates corresponding to the finite element analysis are derived to establish the convergence of the corresponding FE scheme and the numerical methodology has been tested on few examples.

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Section: Existence and Uniquenessmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Weighted extended B-spline finite element analysis of a coupled system of general elliptic equations

Chakraborty

Kumar

2018

Int J Adv Eng Sci Appl Math

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“…Yang et al [20] simulated plunging black-hole binaries in the BSSN formalism using a finite-difference CUDA port of the numerical relativity code AMSS-NCKU [21,22] in concert with an AMR-like algorithm of their own design. Chen [23] used a GPU-accelerated approach to solve sample coupled elliptic equations using the spectral collocation and spectral Galerkin methods. The Teukolsky master equations [24] describing perturbations to the Kerr spacetime have been solved using the Cell processor SDK [25], OpenCL [26,27], and CUDA [28].…”

Section: Introductionmentioning

confidence: 99%

GPU-accelerated simulations of isolated black holes

Lewis

Pfeiffer

2018

Class. Quantum Grav.

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We present a port of the numerical relativity code SpEC which is capable of running on NVIDIA GPUs. Since this code must be maintained in parallel with SpEC itself, a primary design consideration is to perform as few explicit code changes as possible. We therefore rely on a hierarchy of automated porting strategies. At the highest level we use TLoops, a C++ library of our design, to automatically emit CUDA code equivalent to tensorial expressions written into C++ source using a syntax similar to analytic calculation. Next, we trace out and cache explicit matrix representations of the numerous linear transformations in the SpEC code, which allows these to be performed on the GPU using pre-existing matrix-multiplication libraries. We port the few remaining important modules by hand. In this paper we detail the specifics of our port, and present benchmarks of it simulating isolated black hole spacetimes on several generations of NVIDIA GPU. arXiv:1804.09101v1 [gr-qc] 24 Apr 2018 GPU-Accelerated Simulations of Isolated Black Holes 2 IntroductionNumerical relativity (NR), the direct numerical integration of the Einstein field equations, is now a mature subfield of computational physics, with stable binary black hole evolutions possible since 2005 [1-5]. Binary black hole NR is of central importance for gravitational waveform modeling. For instance, the SpEC waveform catalogues [6,7] were used in the development of EOB-waveform models [8][9][10]. Waveform models calibrated to numerical relativity were used to analyse the resent BBH gravitational wave events detected by LIGO and Virgo [11,12]. Furthermore, NR binary black hole simulations were used to assess systematic errors of parameter estimation of these GW events [13][14][15].Detailed knowledge of expected waveforms, themselves coming ultimately from NR simulations, are required by these detectors to maximize sensitivities, to interpret observation, and to make tests of general relativity [16]. Ground based detectors' relative insensitivity to e.g. eccentric binaries is, conversely, in part due to a lack of productionquality simulations in the eccentric region of parameter space [17], a situation which also impairs comparisons with analytic theory.The intricacy of the Einstein equations presents two challenges to NR. First, interesting simulations are expensive, with wallclock times measured in weeks or months. Second, codes able to perform such simulations are quite intricate from a software engineering perspective. Applying them to new regions of the binary black hole parameter space -let alone to new spacetimes -can require months of effort by small groups of experts. These issues are difficult to address simultaneously, since improving runtime tends to complicate code, and vice versa.Twenty years ago, the simplest solution would have been to simply wait for hardware to improve. Unfortunately CPU clock frequencies have been essentially static for some time now, with new high-performance computers instead employing increasingly massive levels of parallelism. But...

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An efficient space-splitting method for simulating brain neurons by neuronal synchronization to control epileptic activity

Moayeri

Rasanan

Latifi

et al. 2020

Engineering with Computers

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A New Framework of GPU-Accelerated Spectral Solvers: Collocation and Glerkin Methods for Systems of Coupled Elliptic Equations

Cited by 6 publications

References 21 publications

Weighted extended B-spline finite element analysis of a coupled system of general elliptic equations

Weighted extended B-spline finite element analysis of a coupled system of general elliptic equations

GPU-accelerated simulations of isolated black holes

An efficient space-splitting method for simulating brain neurons by neuronal synchronization to control epileptic activity

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