Approximation of integral operators by Green quadrature and nested cross approximation

Börm, Steffen; Christophersen, Sven

doi:10.1007/s00211-015-0757-y

Cited by 24 publications

(37 citation statements)

References 38 publications

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“…The definition of η-admissibility specifies when two subsetsX ⊂ X andỸ ⊂ Y are well separated in the sense of the error estimates cited above. There are also methods which combine ideas of the ACA with degenerated expansions, for instance HCA [11] or the Green hybrid method [9]. Searching for the optimal partition of I ×J in a sense that equation (21) holds for most blocks with minimal rank r is prohibitively expensive.…”

Section: Boundary Elementmentioning

confidence: 99%

Vlasov--Poisson System Tackled by Particle Simulation Utilizing Boundary Element Methods

Keßler¹,

Rjasanow²,

Weißer³

2020

SIAM J. Sci. Comput.

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This paper presents a grid-free simulation algorithm for the fully three-dimensional Vlasov-Poisson system for collisionless electron plasmas. We employ a standard particle method for the numerical approximation of the distribution function. Whereas the advection of the particles is grid-free by its very nature, the computation of the acceleration involves the solution of the non-local Poisson equation. To circumvent a volume mesh, we utilise the Fast Boundary Element Method, which reduces the three-dimensional Poisson equation to a system of linear equations on its twodimensional boundary. This gives rise to fully populated matrices which are approximated by the H 2 -technique, reducing the computational time from quadratic to linear complexity. The approximation scheme based on interpolation has shown to be robust and flexible, allowing a straightforward generalisation to vector-valued functions. In particular, the Coulomb forces acting on the particles are computed in linear complexity. In first numerical tests, we validate our approach with the help of classical non-linear plasma phenomena. Furthermore, we show that our method is able to simulate electron plasmas in complex three-dimensional domains with mixed boundary conditions in linear complexity.

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Section: Boundary Elementmentioning

confidence: 99%

Vlasov--Poisson System Tackled by Particle Simulation Utilizing Boundary Element Methods

Keßler¹,

Rjasanow²,

Weißer³

2020

SIAM J. Sci. Comput.

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“…Hierarchical matrices (H-matrices) are widely used to accommodate the large covariance matrices dimension by applying a low-rank approximation to the off-diagonal matrix [16]. Different data approximation techniques based on H-matrices have been proposed in literature such as Hierarchically Off-Diagonal Low-Rank (HODLR) [17], Hierarchically Semi-Separable (HSS) [18], H 2 -matrices [19], [20], and Block/Tile Low-Rank (BLR/TLR) [21], [22].…”

Section: Related Workmentioning

confidence: 99%

Geostatistical Modeling and Prediction Using Mixed Precision Tile Cholesky Factorization

Abdulah

Ltaief

Sun

et al. 2019

2019 IEEE 26th International Conference on High Performance Computing, Data, and Analytics (HiPC)

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Geostatistics represents one of the most challenging classes of scientific applications due to the desire to incorporate an ever increasing number of geospatial locations to accurately model and predict environmental phenomena. For example, the evaluation of the Gaussian log-likelihood function, which constitutes the main computational phase, involves solving systems of linear equations with a large dense symmetric and positive definite covariance matrix. Cholesky, the standard algorithm, requires O(n 3 ) floating point operators and has an O(n 2 ) memory footprint, where n is the number of geographical locations. Here, we present a mixed-precision tile algorithm to accelerate the Cholesky factorization during the log-likelihood function evaluation. Under an appropriate ordering, it operates with double-precision arithmetic on tiles around the diagonal, while reducing to single-precision arithmetic for tiles sufficiently far off. This translates into an improvement of the performance without any deterioration of the numerical accuracy of the application. We rely on the StarPU dynamic runtime system to schedule the tasks and to overlap them with data movement. To assess the performance and the accuracy of the proposed mixedprecision algorithm, we use synthetic and real datasets on various shared and distributed-memory systems possibly equipped with hardware accelerators. We compare our mixed-precision Cholesky factorization against the double-precision reference implementation as well as an independent block approximation method. We obtain an average of 1.6X performance speedup on massively parallel architectures, while maintaining the accuracy necessary for modeling and prediction.

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“…If the block tree is constructed by standard algorithms [15], stiffness matrices corresponding to the discretization of a partial differential operator are hierarchical matrices of local rank zero, while integral operators can be approximated by hierarchical matrices of low rank [2,8,9,7].…”

Section: Basic Arithmetic Operationsmentioning

confidence: 99%

Hierarchical matrix arithmetic with accumulated updates

Börm

2019

Comput. Visual Sci.

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Hierarchical matrices can be used to construct efficient preconditioners for partial differential and integral equations by taking advantage of lowrank structures in triangular factorizations and inverses of the corresponding stiffness matrices.The setup phase of these preconditioners relies heavily on low-rank updates that are responsible for a large part of the algorithm's total run-time, particularly for matrices resulting from three-dimensional problems.This article presents a new algorithm that significantly reduces the number of low-rank updates and can shorten the setup time by 50 percent or more.

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Approximation of integral operators by Green quadrature and nested cross approximation

Cited by 24 publications

References 38 publications

Vlasov--Poisson System Tackled by Particle Simulation Utilizing Boundary Element Methods

Vlasov--Poisson System Tackled by Particle Simulation Utilizing Boundary Element Methods

Geostatistical Modeling and Prediction Using Mixed Precision Tile Cholesky Factorization

Hierarchical matrix arithmetic with accumulated updates

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