Spectral multidomain technique with Local Fourier Basis

Israeli, M.; Vozovoi, L.; Averbuch, Amir

doi:10.1007/bf01060869

Cited by 40 publications

(35 citation statements)

References 3 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…This paper has presented a novel multi-GPU implementation of the Fourier spectral method using domain decomposition based on local Fourier basis [19]. The fundamental idea behind this work is the replacement of the global all-to-all communications introduced by the FFT (used to calculate spatial derivatives) by direct neighbour exchanges.…”

Section: Resultsmentioning

confidence: 99%

“…One way to overcome the global communication imposed by the Fourier spectral method is to use a local Fourier basis as proposed by Israeli, et al [19]. This allows the evaluation of derivatives to be splitted into multiple coupled subdomains, where the Fourier transforms for each subdomain are computed independently, followed by the exchange of data in an overlap or halo region.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Spectral Domain Decomposition Using Local Fourier Basis: Application to Ultrasound Simulation on a Cluster of GPUs

Jaroš

Vaverka

Treeby

2016

JSFI

View full text Add to dashboard Cite

The simulation of ultrasound wave propagation through biological tissue has a wide range of practical applications. However, large grid sizes are generally needed to capture the phenomena of interest. Here, a novel approach to reduce the computational complexity is presented. The model uses an accelerated k-space pseudospectral method which enables more than one hundred GPUs to be exploited to solve problems with more than 3 ×10 9 grid points. The classic communication bottleneck of Fourier spectral methods, all-to-all global data exchange, is overcome by the application of domain decomposition using local Fourier basis. Compared to global domain decomposition, for a grid size of 1536 × 1024 × 2048, this reduces the simulation time by a factor of 7.5 and the simulation cost by a factor of 3.8.

show abstract

Section: Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Spectral Domain Decomposition Using Local Fourier Basis: Application to Ultrasound Simulation on a Cluster of GPUs

Jaroš

Vaverka

Treeby

2016

JSFI

View full text Add to dashboard Cite

show abstract

“…In this case the implementation of Algorithm IV involves the interpolation step (from Chebyshev to Fourier nodes and vice versa) in order to combine u p and u h . An alternative approach for computation of the particular solution implemented in this paper, consists of expanding the source function into a finite trigonometric series along with a smoothing procedure near the boundaries (the LFB method of [5], [14]). The computational complexity of this method scales like…”

Section: The Analytic Case the Function Umentioning

confidence: 99%

“…In the first step, we construct the particular solution using the local Fourier basis (LFB) method of [5], [14]. The idea of this method is to project the source function in a smooth way on an extended domain.…”

mentioning

confidence: 99%

A Fast Poisson Solver of Arbitrary Order Accuracy in Rectangular Regions

Averbuch¹,

Israeli²,

Vozovoi³

1998

SIAM J. Sci. Comput.

Self Cite

View full text Add to dashboard Cite

Abstract. In this paper we propose a direct method for the solution of the Poisson equation in rectangular regions. It has an arbitrary order accuracy and low CPU requirements which makes it practical for treating large-scale problems.The method is based on a pseudospectral Fourier approximation and a polynomial subtraction technique. Fast convergence of the Fourier series is achieved by removing the discontinuities at the corner points using polynomial subtraction functions. These functions have the same discontinuities at the corner points as the sought solution. In addition to this, they satisfy the Laplace equation so that the subtraction procedure does not generate nonperiodic, nonhomogeneous terms.The solution of a boundary value problem is obtained in a series form in O(N log N ) floating point operations, where N 2 is the number of grid nodes. Evaluating the solution at all N 2 interior points requires O(N 2 log N ) operations.Key words. boundary value problem, Poisson equation, rectangular region, spectral method, corner discontinuities, polynomial subtraction AMS subject classifications. 35J05, 45L10, 65P05PII. S1064827595288589 1.Introduction. An important step in the development of fast numerical solvers for elliptic equations in complicated domains is an algorithm for the solution of boundary value problems for constant coefficient elliptic equations in rectangular regions. After solving a nonhomogeneous equation with some "convenient" boundary conditions, one must solve, in a correction step, the homogeneous problem with specified (Dirichlet or Neumann) boundary conditions. As a resulting boundary distribution may not satisfy the required compatibility conditions at the corner points, singularities may arise in the process of the solution, thus destroying the convergence rate even if the final solution is smooth.When the computational region is discretized on a N × N grid using some loworder (finite difference or finite element) scheme, the resulting system of linear algebraic equations is represented by a sparse matrix. Such a matrix can be inverted in O(N 2 ) (or O(N 2 log 2 N ) arithmetic operations using a "fast solver" [4]. However, since the method is of low order, the resolution N must be very large if a high accuracy is desired.Application of high-order (pseudo) spectral methods, based on global expansions into orthogonal polynomials, e.g., Chebyshev polynomials, to the solution of elliptic equations, results in a full matrix problem. The cost of inverting such a matrix using the best current algorithms is O(N 3 ) operations [2]. Besides, the accuracy decreases considerably as the dimension N 2 of the system grows due to accumulation of roundoff errors.When using the Chebyshev method, the fast computation of the expansion coefficients requires that the problem be discretized on a nonuniform grid. For timedependent problems, when elliptic equations arise due to a time-discretization pro-

show abstract

“…The connection functions are determined based on boundary information. This technique is similar to the superposition, filtering, and patching technique of Israeli et al [IVA93] but uses an exact representation of the connection functions and results in a fully coupled system. In two dimensions, for rectangular computational domains, each connection function can be decomposed into the sum of four separate connection functions.…”

Section: Introductionmentioning

confidence: 99%

A Cousin Formulation for Overlapped Domain Decomposition Applied to the Poisson Equation

Fabris

Zarantonello

Lecture Notes in Computational Science and Engineering

View full text Add to dashboard Cite

In this paper we present a domain decomposition method for the solution of a linear equation as a direct two-step procedure (in contrast to iterativebased Schwarz and Aitken-Schwarz procedures): first, local solutions to the nonhomogeneous equation are calculated on overlapping domains, and second, connection functions, solutions to the homogeneous equation, are calculated which satisfy the Cousin problem and correct for the initial solution mismatch at interior boundaries. The procedure is applied to Poisson's equation as a sample linear equation. The calculation of the connection functions is achieved through a classical orthogonal decomposition of the solution to Laplace's equation and can be achieved through progressive solutions in each direction for separable boundary conditions. The procedure is also applicable to more complicated domains with non-separable boundary conditions. A few examples will be given. The connection functions can be calculated to high precision and require minimal overlap of the subdomains.

show abstract

Spectral multidomain technique with Local Fourier Basis

Cited by 40 publications

References 3 publications

Spectral Domain Decomposition Using Local Fourier Basis: Application to Ultrasound Simulation on a Cluster of GPUs

Spectral Domain Decomposition Using Local Fourier Basis: Application to Ultrasound Simulation on a Cluster of GPUs

A Fast Poisson Solver of Arbitrary Order Accuracy in Rectangular Regions

A Cousin Formulation for Overlapped Domain Decomposition Applied to the Poisson Equation

Contact Info

Product

Resources

About