232 research outputs found
Very fast simulation of nonlinear water waves in very large numerical wave tanks on affordable graphics cards
Spectral tensor-train decomposition
The accurate approximation of high-dimensional functions is an essential task
in uncertainty quantification and many other fields. We propose a new function
approximation scheme based on a spectral extension of the tensor-train (TT)
decomposition. We first define a functional version of the TT decomposition and
analyze its properties. We obtain results on the convergence of the
decomposition, revealing links between the regularity of the function, the
dimension of the input space, and the TT ranks. We also show that the
regularity of the target function is preserved by the univariate functions
(i.e., the "cores") comprising the functional TT decomposition. This result
motivates an approximation scheme employing polynomial approximations of the
cores. For functions with appropriate regularity, the resulting
\textit{spectral tensor-train decomposition} combines the favorable
dimension-scaling of the TT decomposition with the spectral convergence rate of
polynomial approximations, yielding efficient and accurate surrogates for
high-dimensional functions. To construct these decompositions, we use the
sampling algorithm \texttt{TT-DMRG-cross} to obtain the TT decomposition of
tensors resulting from suitable discretizations of the target function. We
assess the performance of the method on a range of numerical examples: a
modifed set of Genz functions with dimension up to , and functions with
mixed Fourier modes or with local features. We observe significant improvements
in performance over an anisotropic adaptive Smolyak approach. The method is
also used to approximate the solution of an elliptic PDE with random input
data. The open source software and examples presented in this work are
available online.Comment: 33 pages, 19 figure
Efficient p-multigrid spectral element model for water waves and marine offshore structures
In marine offshore engineering, cost-efficient simulation of unsteady water
waves and their nonlinear interaction with bodies are important to address a
broad range of engineering applications at increasing fidelity and scale. We
consider a fully nonlinear potential flow (FNPF) model discretized using a
Galerkin spectral element method to serve as a basis for handling both wave
propagation and wave-body interaction with high computational efficiency within
a single modellingapproach. We design and propose an efficientO(n)-scalable
computational procedure based on geometric p-multigrid for solving the Laplace
problem in the numerical scheme. The fluid volume and the geometric features of
complex bodies is represented accurately using high-order polynomial basis
functions and unstructured meshes with curvilinear prism elements. The new
p-multigrid spectralelement model can take advantage of the high-order
polynomial basis and thereby avoid generating a hierarchy of geometric meshes
with changing number of elements as required in geometric h-multigrid
approaches. We provide numerical benchmarks for the algorithmic and numerical
efficiency of the iterative geometric p-multigrid solver. Results of numerical
experiments are presented for wave propagation and for wave-body interaction in
an advanced case for focusing design waves interacting with a FPSO. Our study
shows, that the use of iterative geometric p-multigrid methods for theLaplace
problem can significantly improve run-time efficiency of FNPF simulators.Comment: Submitted to an international journal for peer revie
Computational Hydrodynamics: How Portable and Scalable Are Heterogeneous Programming Paradigms?
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