97 research outputs found
Hierarchical interpolative factorization for elliptic operators: integral equations
This paper introduces the hierarchical interpolative factorization for
integral equations (HIF-IE) associated with elliptic problems in two and three
dimensions. This factorization takes the form of an approximate generalized LU
decomposition that permits the efficient application of the discretized
operator and its inverse. HIF-IE is based on the recursive skeletonization
algorithm but incorporates a novel combination of two key features: (1) a
matrix factorization framework for sparsifying structured dense matrices and
(2) a recursive dimensional reduction strategy to decrease the cost. Thus,
higher-dimensional problems are effectively mapped to one dimension, and we
conjecture that constructing, applying, and inverting the factorization all
have linear or quasilinear complexity. Numerical experiments support this claim
and further demonstrate the performance of our algorithm as a generalized fast
multipole method, direct solver, and preconditioner. HIF-IE is compatible with
geometric adaptivity and can handle both boundary and volume problems. MATLAB
codes are freely available.Comment: 39 pages, 14 figures, 13 tables; to appear, Comm. Pure Appl. Mat
White Dwarf Mergers on Adaptive Meshes I. Methodology and Code Verification
The Type Ia supernova progenitor problem is one of the most perplexing and
exciting problems in astrophysics, requiring detailed numerical modeling to
complement observations of these explosions. One possible progenitor that has
merited recent theoretical attention is the white dwarf merger scenario, which
has the potential to naturally explain many of the observed characteristics of
Type Ia supernovae. To date there have been relatively few self-consistent
simulations of merging white dwarf systems using mesh-based hydrodynamics. This
is the first paper in a series describing simulations of these systems using a
hydrodynamics code with adaptive mesh refinement. In this paper we describe our
numerical methodology and discuss our implementation in the compressible
hydrodynamics code CASTRO, which solves the Euler equations, and the Poisson
equation for self-gravity, and couples the gravitational and rotation forces to
the hydrodynamics. Standard techniques for coupling gravitation and rotation
forces to the hydrodynamics do not adequately conserve the total energy of the
system for our problem, but recent advances in the literature allow progress
and we discuss our implementation here. We present a set of test problems
demonstrating the extent to which our software sufficiently models a system
where large amounts of mass are advected on the computational domain over long
timescales. Future papers in this series will describe our treatment of the
initial conditions of these systems and will examine the early phases of the
merger to determine its viability for triggering a thermonuclear detonation.Comment: Accepted for publication in the Astrophysical Journa
Hierarchical interpolative factorization for elliptic operators: differential equations
This paper introduces the hierarchical interpolative factorization for
elliptic partial differential equations (HIF-DE) in two (2D) and three
dimensions (3D). This factorization takes the form of an approximate
generalized LU/LDL decomposition that facilitates the efficient inversion of
the discretized operator. HIF-DE is based on the multifrontal method but uses
skeletonization on the separator fronts to sparsify the dense frontal matrices
and thus reduce the cost. We conjecture that this strategy yields linear
complexity in 2D and quasilinear complexity in 3D. Estimated linear complexity
in 3D can be achieved by skeletonizing the compressed fronts themselves, which
amounts geometrically to a recursive dimensional reduction scheme. Numerical
experiments support our claims and further demonstrate the performance of our
algorithm as a fast direct solver and preconditioner. MATLAB codes are freely
available.Comment: 37 pages, 13 figures, 12 tables; to appear, Comm. Pure Appl. Math.
arXiv admin note: substantial text overlap with arXiv:1307.266
Parallel-MLFMA solution of CFIE discretized with tens of millions of unknowns
We consider the solution of large scattering problems in electromagnetics involving three-dimensional arbitrary geometries with closed surfaces. The problems are formulated accurately with the combined-field integral equation and the resulting dense matrix equations are solved iteratively by employing the multilevel fast multipole algorithm (MLFMA). With an efficient parallelization of MLFMA on relatively inexpensive computing platforms using distributed-memory architectures, we easily solve large-scale problems that are discretized with tens of millions of unknowns. Accuracy of the solutions is demonstrated on scattering problems involving spheres of various sizes, including a sphere of radius 110 λ discretized with 41,883,638 unknowns, which is the largest integral-equation problem ever solved, to the best of our knowledge. In addition to canonical problems, we also present the solution of real-life problems involving complicated targets with large dimensions
GPU Acceleration of a Non-Standard Finite Element Mesh Truncation Technique for Electromagnetics
The emergence of General Purpose Graphics Processing Units (GPGPUs) provides new opportunities to accelerate applications involving a large number of regular computations. However, properly leveraging the computational resources of graphical processors is a very challenging task. In this paper, we use this kind of device to parallelize FE-IIEE (Finite Element-Iterative Integral Equation Evaluation), a non-standard finite element mesh truncation technique introduced by two of the authors. This application is computationally very demanding due to the amount, size and complexity of the data involved in the procedure. Besides, an efficient implementation becomes even more difficult if the parallelization has to maintain the complex workflow of the original code. The proposed implementation using CUDA applies different optimization techniques to improve performance. These include leveraging the fastest memories of the GPU and increasing the granularity of the computations to reduce the impact of memory access. We have applied our parallel algorithm to two real radiation and scattering problems demonstrating speedups higher than 140 on a state-of-the-art GPU.This work was supported in part by the Spanish Government under Grant TEC2016-80386-P, Grant TIN2017-82972-R,
and Grant ESP2015-68245-C4-1-P, and in part by the Valencian Regional Government under Grant PROMETEO/2019/109
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