Nested Dissection Method Research Articles

Parallel QR Factorization of Block-Tridiagonal Matrices

In this work, we deal with the QR factorization of block-tridiagonal matrices, where the blocks are dense and rectangular. This work is motivated by a novel method for computing geodesics over Riemannian man-ifolds. If blocks are reduced sequentially along the diagonal, only limited parallelism is available. We propose a matrix permutation approach based on the Nested Dissection method which improves parallelism at the cost of additional computations and storage. We provide a detailed analysis of the approach showing that this extra cost is bounded. Finally, we present an implementation for shared memory systems relying on task parallelism and the use of a runtime system. Experimental results support the conclusions of our analysis and show that the proposed approach leads to good performance and scalability.

A parallel shared-memory implementation of a high-order accurate solution technique for variable coefficient Helmholtz problems

The recently developed Hierarchical Poincaré–Steklov (HPS) method is a high-order discretization technique that comes with a direct solver. Results from previous papers demonstrate the method’s ability to solve Helmholtz problems to high accuracy without the so-called pollution effect. While the asymptotic scaling of the direct solver’s computational cost is the same as the nested dissection method, serial implementations of the solution technique are not practical for large scale numerical simulations. This manuscript presents the first parallel implementation of the HPS method. Specifically, we introduce an approach for a shared memory implementation of the solution technique utilizing parallel linear algebra. This approach is the foundation for future large scale simulations on supercomputers and clusters with large memory nodes. Performance results on a desktop computer (resembling a large memory node) are presented.

Экспериментальное сравнение алгоритмов в параллельном методе вложенных сечений

Пирова Анна Юрьевна, ассистент, кафедра математического обеспечения и суперкомпьютерных технологий, Институт информационных технологий, математики и механики, Нижегородский государственный университет им. Н.И. Лобачевского (Нижний Новгород, Российская Федерация) Кудрявцев Никита Юрьевич, магистрант, кафедра математического обеспечения и суперкомпьютерных технологий, Институт информационных технологий, математики и механики, Нижегородский государственный университет им. Н.И. Лобачевского (Нижний Новгород, Российская Федерация) Мееров Иосиф Борисович, к.т.н., доцент, зам. заведующего кафедрой, кафедра математического обеспечения и суперкомпьютерных технологий, Институт информацион- ных технологий, математики и механики, Нижегородский государственный университет им. Н.И. Лобачевского (Нижний Новгород, Российская Федерация). Yu. Pirova, N.Yu. Kudriavtsev, I.B. Meyerov Lobachevsky State University of Nizhni Novgorod (23 Gagarin Avenue, Nizhni Novgorod, 603022 Russia) E-mail: anna.pirova@itmm.unn.ru, n.yu.kudriavtsev@gmail.com, iosif.meyerov@itmm.unn.ru

Matrix sparsification and nested dissection over arbitrary fields

The generalized nested dissection method, developed by Lipton et al. [1979], is a seminal method for solving a linear system Ax = b where A is a symmetric positive definite matrix. The method runs extremely fast whenever A is a well-separable matrix (such as matrices whose underlying support is planar or avoids a fixed minor). In this work, we extend the nested dissection method to apply to any nonsingular well-separable matrix over any field. The running times we obtain essentially match those of the nested dissection method. An important tool is a novel method for matrix sparsification that preserves determinants and minors, and that guarantees that constant powers of the sparsified matrix remain sparse.

Оптимизация расчётов в пакете OpenFOAM на GPU

The paper presents preliminary research on improving performance of CFD simulations in OpenFOAM via offloading parts of computations (specifically, solution of linear systems) to a graphics accelerator (GPU). We present a short review of OpenFOAM package and describe porting conjugate gradient method to the GPU architecture using CUDA programming model. Porting the basic algorithm is straightforward, however care should be taken to avoid unnecessary copying over PCI-Express bus. Efficient preconditioning on the GPU is then discussed. We use approximate inverse preconditioning, which can be implemented with good parallelism on the GPU. To amortize the cost of preparing the preconditioner, we allow reuse of preconditioners on the GPU and compute them on the CPU in a helper thread asynchronously. We mention several optimization opportunities: reordering the preconditioner to upper-left triangular form so that CUDA blocks multiplying by denser parts of preconditiner factors are scheduled first; using single-precision storage for the preconditioner to save memory bandwidth; reordering the mesh with nested dissection method from Metis library and using mixed-precision iteration for the conjugate gradient method. Preliminary performance testing results show performance improvement starting from 64000-cell meshes and reaching 2x for a 1-million cell mesh for a non-parallel run. As future work we mention support for parallel runs with MPI, research of other solvers such as multigrid, BiCGStab and IDR, and choosing drop tolerance automatically for the AINV preconditioner.

Parallel Direct Solver for the Finite Integration Technique in Electrokinetic Problems

The finite integration technique allows the simulation of real-world electromagnetic field problems with complex geometries. It provides a discrete reformulation of Maxwell's equations in their integral form suitable for numerical computing. The resulting matrix equations of the discretized fields can be used for efficient numerical simulations on modern computers and can be exploited to use a parallel computing. In fact, by reordering the unknowns by the nested dissection method, it is possible to directly construct the lower triangular matrix of the Cholesky factorization with many processors without assembling the matrix system. In this paper, a parallel algorithm is proposed for the direct solution of large sparse linear systems with the finite integration technique. This direct solver has the advantage of handling singularities in the matrix of linear systems. The computational effort for these linear systems, often encountered in numerical simulation of electromagnetic phenomena by finite integration technique, is very significant in terms of run-time and memory requirements. Many numerical tests have been carried out to evaluate the performance of the parallel direct solver.

Parallel direct solver for the finite integration technique in electromagnetics

PurposeThe purpose of this paper is to develop a solving method based on the parallelization of the calculus of the finite integration technique on many processors and discuss its efficiency following the number of processors.Design/methodology/approachThe finite integration technique is used in different engineering domains as well as to compute the electromagnetic phenomena. This technique is efficient and generates a linear system with regular structure, this system can be implemented and solved in parallel with a direct solver. In fact, in reordering the unknowns by the nested dissection method, the lower triangular matrix of the Cholesky factorization can be constructed with many processors without assembling the matrix system.FindingsThis paper deals with the parallelization of the finite integration technique which is performed by data partitioning and leads to a high‐performance.Originality/valueThe paper presents a parallel implementation of the finite integration technique associated to a direct solver which is practical and efficient.

Building graph separators with the recursive rotation algorithm for the nested dissection method

A recursive rotation algorithm is built and investigated. The algorithm is a possible version of the nested dissection algorithm. The Liu algorithm builds a matrix graph separator by means of rotation of an elimination tree, which reduces the height of the latter. In this case, the nodes of the matrix graph are previously reordered by one of the well-known Cuthill-McKee algorithms, the reverse Cuthill-McKee algorithm, and the King algorithm. Then this procedure is recursively repeated. The recursive rotation algorithm is compared with the multilevel and spectral methods of graph separation for 2D finite-element grids.

Stress–Strain Analysis of Thin-Walled Shells with Massive Ribs

A multifrontal method is proposed for solving large-scale finite-element problems with a symmetric stiffness matrix. This approach is based on node ordering rather than element ordering as in the traditional frontal or multifrontal methods. This allows us to employ the efficient well-known ordering techniques such as the minimum-degree algorithm or the nested dissection method. A thin-walled cylindrical shell with massive ribs is considered as an example. The efficiency of the method proposed is illustrated against the classical skyline and aggregation multilevel conjugate-gradient iterative solvers

Weighted-Averaging Finite-Element Method for 2D Elastic Wave Equations in the Frequency Domain

We present a weighted-averaging frequency-domain finite-element method for an accurate and efficient 2D elastic wave modeling technique. Our method introduces three kinds of supplementary element sets in addition to a basic element set that is used in the standard finite-element method. By constructing global stiffness and mass matrices for four kinds of element sets and then averaging them with weighting coefficients, we obtain a new global stiffness and mass matrix. With optimal weighting coefficients determined by a Marquardt–Levenberg method to minimize grid dispersion and grid anisotropy, we can reduce the number of nodal points per shear wavelength from 33.3 (using the standard finite-element method) and 20 (using the eclectic method) to 5, with the errors of group velocities no larger than 1%. By reducing the number of grid points per wavelength, we achieve a 97% and 75% reduction of computer memory required to store the complex impedance matrix for a band-type matrix solver and a nested dissection method, respectively, compared with those of the eclectic method. Our method gives approximate solutions compatible with exact solutions for an infinite homogeneous, a semi-infinite homogeneous (Lamb9s problem), and a horizontal two-layer model with fewer grid points than the standard and the eclectic method. A major advantage of the weighted-averaging finite-element method for the elastic wave equation is that it provides solutions very close to correct solutions for Lamb9s problem economically, unlike most of the displacement approaches. In addition, our scheme makes the complex impedance matrix symmetric, which satisfies reciprocity. Seismic forward modeling techniques that satisfy reciprocity are of critical importance in seismic imaging and inversion because we can economically calculate a Jacobian matrix using the reciprocity. Successful simulation of a large-size model shows that our method can be used for the simulation of wave propagation in the geological model needed in the reverse-time migration or seismic inversion.

The Nested Substructures Method for Solving Large Finite-Element Systems as Applied to Thin-Walled Shells with High Ribs

A direct method is proposed to solve large systems of linear algebraic equations that arise in using the finite-element method. The method suggests to subdivide the initial structure into nonoverlapping, deeply nested substructures. The subdivision is performed automatically by the nested dissection method. Initially, the structure is subdivided into isolated finite elements. The solution process is step-by-step assembly of subsystems with simultaneous elimination of unknowns for completely assembled ones. The efficiency of the method is demonstrated by comparing it with the incomplete Cholesky conjugate-gradient method and the traditional envelope method as applied to a circular cylindrical shell with high ribs

New algorithm for the Ising problem: partition function for finite lattice graphs.

We present a new efficient method to find the Ising problem partition function for finite lattice graphs embeddable on an arbitrary orientable surface, with integral coupling constants bounded in the absolute value by a polynomial of the size of the lattice graph. The algorithm has been implemented for toroidal lattices using modular arithmetic and the generalized nested dissection method. The implementation has substantially better performance than any other as far as we know.

Fast and efficient solution of path algebra problems

This paper extends the author's parallel nested dissection algorithm (Pan and Reif, Technical Report 88-18, Computer Science Department, SUNY Albany, 1988) originally devised for solving sparse linear systems. We present a class of new applications of the nested dissection method, this time to path algebra computations (in both cases of the single source path problems and of the all pair path problems), where the path algebra problem is defined by a symmetric matrix A whose associated undirected graph G has a known family of separators of small size s( n) (in many cases of interest, s( n) = O(√ n).) The assumption that G has known separators is reasonable in a large variety of practical dynamic situations, where G is fixed and the entries of the matrix A associated with the edges of G may vary with the input. We substantially improve the known algorithms for path algebra problems of this general class: Previous estimates Path problem Sequential time Parallel time Proccessors Precomputed separators Single source O(n 2) O(n) O(I(n) log n) n n 3/(I(n) log n) No Yes All pairs O(n 3) O(n) O(I(n) log n) n 2 n 3/(I(n) log n) No Yes New Algrithms Path problem Sequential time Parallel time Processors Precomputed separators Single source O(n 1.5) O(( log n) √n) O(I(n) log 2 n) n/ log n n 1.5/(I(n) log n) No Yes All pairs O(n 2 log n) O(( log n) √n) O(I(n) log 2 n) n 1.5 n 2/I(n) log n) No Yes Here we assume that G is given with its O(√ n)-separator family. The latter assumption can be lifted for the sequential time estimates and for O((log n) √ n) parallel time estimates for planar graphs, because the evaluation of an O(√)-separator family can be done in O( n) sequential time (Lipton and Tarjan, SIAM J. Appl. Math. 36, No. 2, (1979), 177–189) or, on PRAM, in O((log n) √ n) parallel time with (√ n)/log n processors (Gazit and Miller, manuscript, Computer Sci. Dept., University of Southern California, 1986), with small overhead constants in both cases. I( n) denotes parallel time of computing the sum of n values, I( n)= O(log n) for any EREW PRAM, I( n) = O(1) on a randomized CRCW PRAM. Furthermore using the randomized algorithm of (Gazit and Miller, in “Proceedings, 28th Annu. IEEE Symp. FOCS, 1987,” pp. 238–248), we may precompute separators of a planar graph using O(log 2 n) time, n + ƒ 1+ϵ processors for a positive ϵ where ƒ is the number of faces; this is less than the cost of the subsequent path computation. Moreover, we preserve the above processor bounds but further decrease the parallel time by a factor of log n (via a modification of our new algorithms based on pipelining) in the important case of computing the minimum cost paths in a planar graph. Further applications lead, in particular, to computing a maxflow and a mincut in an undirected planar network using O( I( n) log 2 n) parallel steps, n 1.5 I(n) log n processors, versus the known bounds, O(log 2 n) and n 4, of (Johnson, J. ACM 34, No. 4 (1987), 950–967).

Efficient solutions of hierarchical systems of linear equations

Several methods have been developed in the past for the efficient solution of sparse systems of linear equations with Gaussian elimination. In [5] the generalized nested dissection method is introduced. This method finds an ordering of the rows and columns of the coefficient matrix that produces small fill-in. [6] exploits the hierarchical structure imposed on the coefficient matrix by nested dissection to circumvent the necessity for using general sparse matrix algorithms. He decomposes the matrix into submatrices, solves them and reassembles the solution. We take the approach of [6] one step further: In many engineering applications such as finite element problems and circuit analysis problems many of the submatrices are identical. We exploit this fact to drastically reduce the space requirements for solving systems of linear equations that are defined succinctly using hierarchical definitions. Furthermore, if only a few components of the solution vector are asked for, we can by the same token achieve drastic savings of computing time. We get our results by extending a method for hierarchical graph processing presented in [4] to matrix computation. Our approach generalizes special solutions given in [9, 10].

Parallel nested dissection for path algebra computations

This paper extends the authors' parallel nested dissection algorithm of [13] originally devised for solving sparse linear systems. We present a class of new applications of the nested dissection method, this time to path algebra computations (in both cases of single source and all pair paths), where the path algebra problem is defined by a symmetric matrix A whose associated graph G with n vertices is planar. We substantially improve the known algorithms for path algebra problems of that general class; this has further applications to maximum flow and minimum cut problems in an undirected planar network and to the feasibility testing of a multicommodity flow in a planar network.