A direct linear systems solver for pipe networks

In this paper, an efficient method for the solution of matrix equations resulting from the flow of water in a pipe network is presented. The method is found to be rapidly convergent, and computationally efficient, and the results obtained compare favourably with those found in the literature.     

A sixth order fast direct helmholtz equation solver

An O(h⁶) accurate difference approximation to solutions of the Helmholtz equation is derived. The discrete equations are solved using a reduction procedure and Fourier analysis. Its computational performance is compared with a fourth order similar method over a set of linear and mildly nonlinear elliptic boundary value problems.     

A structured low-rank wavelet solver for the Ornstein-Zernike integral equation

In this article, we present a new structured wavelet algorithm to solve the Ornstein-Zernike integral equation for simple liquids. This algorithm is based on the discrete wavelet transform of radial distribution functions and different low-rank approximations of the obtained convolution matrices. The fundamental properties of wavelet bases such as the interpolation properties and orthogonality are employed to improve the convergence and speed of the algorithm. In order to solve the integral equation we have applied a combined scheme in which the coarse part of the solution is calculated by the use of wavelets and Newton-Raphson algorithm, while the fine part is solved by the direct iteration. Tests have indicated that the proposed procedure is more effective than the conventional method based on hybrid algorithms.     

A parallel direct solver for the self-adaptive hp Finite Element Method

In this paper we present a new parallel multi-frontal direct solver, dedicated for the hp Finite Element Method (hp-FEM). The self-adaptive hp-FEM generates in a fully automatic mode, a sequence of hp-meshes delivering exponential convergence of the error with respect to the number of degrees of freedom (d.o.f.) as well as the CPU time, by performing a sequence of hp refinements starting from an arbitrary initial mesh. The solver constructs an initial elimination tree for an arbitrary initial mesh, and expands the elimination tree each time the mesh is refined. This allows us to keep track of the order of elimination for the solver. The solver also minimizes the memory usage, by de-allocating partial LU factorizations computed during the elimination stage of the solver, and recomputes them for the backward substitution stage, by utilizing only about 10% of the computational time necessary for the original computations. The solver has been tested on 3D Direct Current (DC) borehole resistivity measurement simulations problems. We measure the execution time and memory usage of the solver over a large regular mesh with 1.5 million degrees of freedom as well as on the highly non-regular mesh, generated by the self-adaptive hp$hp$-FEM, with finite elements of various sizes and polynomial orders of approximation varying from p=1$p=1$ to p=9$p=9$. From the presented experiments it follows that the parallel solver scales well up to the maximum number of utilized processors. The limit for the solver scalability is the maximum sequential part of the algorithm: the computations of the partial LU factorizations over the longest path, coming from the root of the elimination tree down to the deepest leaf.     

swSuperLU: A highly scalable sparse direct solver on Sunway manycore architecture

The Journal of Supercomputing - Sparse LU factorization is essential for scientific and engineering simulations. In this work, we present swSuperLU, a highly scalable sparse direct solver on Sunway...     

Fill-ins number reducing direct solver designed for FIT-type matrix

This paper investigates the particular problem of matrices appearing during the modeling of Integrated Circuits with Finite Integration Technique (FIT) method. We present the key points of FIT approach followed by an illustration of the structure and the properties of the FIT-type matrix. A novel algorithm SMark is proposed, which focuses on fill-ins number reduction. The main idea of SMark is the concept of a dual architecture—symbolic and numeric factorization.     

A hybrid method for the solution of crack problems based on the cauchy integral formula and the Riemann-Hilbert problem

Using the values of a complex potential on a closed contour surrounding a crack (derived either by an experimental or by a numerical method), this potential (as well as the corresponding stress intensity factors) is determined inside the contour by using the classical theory of the Riemann-Hilbert boundary value problem together with the well-known Cauchy integral formula. The case of an array of periodic collinear cracks is considered in detail and the corresponding numerical results (based on the trapezoidal quadrature role) are displayed.     

A cyclic block-tridiagonal solver

A simple algorithm for solving a cyclic block-tridiagonal system of equations is presented. Introducing a special form of a new variable, the system is split into two block-tridiagonal systems, which can be solved by known methods. Implementation details of the algorithm are discussed and numerical examples of diagonal and random generated systems are presented.     

A recurrence formula for the direct solution of singular integral equations

An iterative method for the solution of singular integral equations is given in this paper by developing a recurrence formula. Discretizing the above formula, by using appropriate quadrature rules, the solution of the singular integral equation is given in an extremely simple form. The number of numerical operations required for such a solution is considerably reduced, when compared to the number of operations required for a classical type of solution. Illustrative examples are given, indicating the efficiency of the method. It is shown that the number of operations in this procedure is only half the number of the operations for a typical numerical method. The convergence of the method is studied in the space of Hölder continuous functions. In the particular case of plane elasticity more efficient bounds are given. In the same case it is proved that the procedure is equivalent to the Schwarz's alternating method and convergence is assured [18].     

On the natural interpolation formula for cauchy type singular integral equations of the first kind

A Cauchy type singular integral equation of the first kind can be numerically solved either directly, through the use of a Gaussian numerical integration rule, or by reduction to an equivalent Fredholm integral equation of the second kind, where the Nyström method is applicable. In this note it is proved that under appropriate but reasonable conditions the expressions of the unknown function of the integral equation, resulting from the natural interpolation formulae of the direct method, as well as of the Nyström method, are identical along the whole integration interval.     

A direct solver with reutilization of LU factorizations for h-adaptive finite element grids with point singularities

This paper describes a direct solver algorithm for a sequence of finite element meshes that are $h$-refined towards one or several point singularities. For such a sequence of grids, the solver delivers linear computational cost $O\left(N\right)$ in terms of CPU time and memory with respect to the number of unknowns $N$. The linear computational cost is achieved by utilizing the recursive structure provided by the sequence of $h$-adaptive grids with a special construction of the elimination tree that allows for reutilization of previously computed partial LU (or Cholesky) factorizations over the entire unrefined part of the computational mesh. The reutilization technique reduces the computational cost of the entire sequence of $h$-refined grids from $O\left(N^2\right)$ down to $O\left(N\right)$. Theoretical estimates are illustrated with numerical results on two- and three-dimensional model problems exhibiting one or several point singularities.     

A parallel linear system solver

A new factorisation method for the solution of a linear system is proposed. The method is similar to an LU type factorisation of the coefficient matrix A where the factors are interlocking matrix quadrants and can be applied on a single-instruction stream parallel machine.     

从第一积分构造Lagrange函数的直接方法

提出力学系统Lagrange函数和第一积分之间存在一种新关联,在此基础上给出变分法逆问题的一种新的直接解法.证明系统Lagrange函数可以由带修正因子的第一积分构成,导出修正因子应满足的偏微分方程,运用此解法构建不同系统的Lagrange函数和函数族,并讨论新解法的特点.     

A cost accounting problem solver and generator

Well-structured problems in many fields can be solved by the computer. By a procedure similar to finding the inverse of a function, the computer can also be used to generate problems with specified characteristics. This paper gives an example of the complementarity of problem solving and problem generation in the discipline of cost accounting     

On the numerical solution of a certain class of non-linear cauchy problems arising from integral equation theory

Summary A numerical algorithm for solving a system of non-linear partial differential integral equations is presented. These equations result from applying the method of invariant imbedding to the solution of a certain class ofFredholm integral equations of the second kind [2]. The algorithm is compared with a more standard method of solution and some numerical results presented.     

A FEM-based direct method for material reconstruction inverse problem in soft tissue elastography

A novel finite element method (FEM) based direct method is developed for the material reconstruction inverse problem in soft tissue elastography. The solution is obtained by minimising an objective function, defined as the sum of the square of the residual norms at all nodes, where the nodal residual norm is defined as a linear function of elasticity parameters of the associated elements. The measured deformation is enforced directly and satisfying the equilibrium at every node is utilised as the optimisation objective. As a result, the soft tissue elastography can be obtained directly by solving the resulting set of linear equations.     

On the improvement of a scalable sparse direct solver for unsymmetrical linear equations

This paper focuses on the application level improvements in a sparse direct solver specifically used for large-scale unsymmetrical linear equations resulting from unstructured mesh discretization of coupled elliptic/hyperbolic PDEs. Existing sparse direct solvers are designed for distributed server systems taking advantage of both distributed memory and processing units. We conducted extensive numerical experiments with three state-of-the-art direct linear solvers that can work on distributed-memory parallel architectures; namely, MUMPS (MUMPS solver website, http://graal.ens-lyon.fr/MUMPS), WSMP (Technical Report TR RC-21886, IBM, Watson Research Center, Yorktown Heights, 2000), and SUPERLU_DIST (ACM Trans Math Softw 29(2):110–140, 2003). The performance of these solvers was analyzed in detail, using advanced analysis tools such as Tuning and Analysis Utilities (TAU) and Performance Application Programming Interface (PAPI). The performance is evaluated with respect to robustness, speed, scalability, and efficiency in CPU and memory usage. We have determined application level issues that we believe they can improve the performance of a distributed-shared memory hybrid variant of this solver, which is proposed as an alternative solver [SuperLU_MCDT (Many-Core Distributed)] in this paper. The new solver utilizing the MPI/OpenMP hybrid programming is specifically tuned to handle large unsymmetrical systems arising in reservoir simulations so that higher performance and better scalability can be achieved for a large distributed computing system with many nodes of multicore processors. Two main tasks are accomplished during this study: (i) comparisons of public domain solver algorithms; existing state-of-the-art direct sparse linear system solvers are investigated and their performance and weaknesses based on test cases are analyzed, (ii) improvement of direct sparse solver algorithm (SuperLU_MCDT) for many-core distributed systems is achieved. We provided results of numerical tests that were run on up to 16,384 cores, and used many sets of test matrices for reservoir simulations with unstructured meshes. The numerical results showed that SuperLU_MCDT can outperform SuperLU_DIST 3.3 in terms of both speed and robustness.     

A branch-and-prune solver for distance constraints

Given some geometric elements such as points and lines in R/sup 3/, subject to a set of pairwise distance constraints, the problem tackled in this paper is that of finding all possible configurations of these elements that satisfy the constraints. Many problems in robotics (such as the position analysis of serial and parallel manipulators) and CAD/CAM (such as the interactive placement of objects) can be formulated in this way. The strategy herein proposed consists of looking for some of the a priori unknown distances, whose derivation permits solving the problem rather trivially. Finding these distances relies on a branch-and-prune technique, which iteratively eliminates from the space of distances entire regions which cannot contain any solution. This elimination is accomplished by applying redundant necessary conditions derived from the theory of distance geometry. The experimental results qualify this approach as a promising one.     

A cost-optimal parallel tridiagonal system solver

We first show how to transform the solution of an n × n tridiagonal system into suffix computations of continued fractions. Then a parallel substitution scheme is introduced to compute the suffix values. The derived parallel algorithm allows the tridiagonal system to be solved in O(log n) time on an unshuffle network with Θ(n /log n) processors. It is cost-optimal in the sense that processor number times execution time is minimized. Our solver is conceptually simple and easy for implementation.