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PHoM is a software package in C++ for finding all isolated solutions of polynomial systems using a polyhedral homotopy continuation method. Among three modules constituting the package, the first module StartSystem constructs a family of polyhedral-linear homotopy functions, based on the polyhedral homotopy theory, from input data for a given system of polynomial equations f(x)=0. The second module...
This paper considers the problem of on-line scheduling a list of independent jobs in which each job has an arbitrary release time on m parallel identical machines. A tight bound is given for List Scheduling(LS) algorithm and a better algorithm is given for m2.
The integrands in M. Schrders integral representation for the price of Asian options are highly oscillatory yielding cancellations of many digits in the integration. Furthermore, obtaining multiple precision function values is costly. Numerical integration is nevertheless possible by using Hermitian quadrature formulas, techniques of automatic differentiation and explicit error bounds combined with...
In this paper, we give a class of Alternating Segment CrankNicolson (ASC-N) method for the convection-diffusion equation. The method is unconditionally stable and has the advantages of parallel computing. The numerical examples show that the accuracy of the method is better than that of the existing method in [4].
In [8], a class of (data-sparse) hierarchical (-) matrices is introduced that can be used to efficiently assemble and store stiffness matrices arising in boundary element applications. In this paper, we develop and analyse modifications in the construction of an -matrix that will allow an efficient application to problems involving adaptive mesh refinement. In particular, we present a new clustering...
We study a system of coupled reaction-diffusion equations. The equations have diffusion parameters of different magnitudes associated with them. Near each boundary, their solution exhibit two overlapping layers. A central difference scheme on layer-adapted piecewise uniform meshes is used to solve the system numerically. We show that the scheme is almost second-order convergent, uniformly in both...
In this paper, we present a robust distributive smoother in a multigrid method for the system of poroelasticity equations. Within the distributive framework, we deal with a decoupled system, that can be smoothed with basic iterative methods like an equation-wise red-black Jacobi point relaxation. The properties of the distributive relaxation are optimized with the help of Fourier smoothing analysis...
The problem of factoring a linear partial differential operator is studied. An algorithm is designed which allows one to factor an operator when its symbol is separable, and if in addition the operator has enough right factors then it is completely reducible. Since finding the space of solutions of a completely reducible operator reduces to the same for its right factors, we apply this approach and...
For derivative calculations, debugging, and interactive control one may need to reverse the execution of a computer program for given inputs. If any increase of the time needed for the reversal is unacceptable, the availability of enough auxiliary processors provides the possibility to reverse the computer program with minimal temporal complexity and a surprisingly small spatial complexity using parallel...
In this paper, we propose a new stabilized three-field formulation applied to the advection-diffusion equation. Using finite elements with SUPG stabilization in the interior of the subdomains our approach enables us to use almost arbitrary discrete function spaces. They need not to satisfy the inf-sup conditions of the standard three-field formulation. We can prove the stability of the scheme and...
In this note we refine strategies of the so called dual-weighted-residual (DWR) approach to a posteriori error control for FE-schemes. We derive rigorous error bounds, especially we control the approximation process of the (unknown) dual solution entering the proposed estimate.
In this paper, we analyze and enhance previously achieved results related to the generalized Grover search algorithm in terms of arbitrary initial pure state, arbitrary unitary transformation, arbitrary phase rotations, and arbitrary number of marked items. This allows us to construct an unsorted database search algorithm which can be included inside a quantum computing system. Because of its constructive...
In preceding papers [8], [11], [12], [6], a class of matrices (-matrices) has been developed which are data-sparse and allow to approximate integral and more general nonlocal operators with almost linear complexity. In the present paper, a weaker admissibility condition is described which leads to a coarser partitioning of the hierarchical -matrix format. A coarser format yields smaller constants...
We discuss a possibility to construct high-order numerical algorithms on uniform or mildly graded grids for solving linear Volterra integral equations of the second kind with weakly singular or other nonsmooth kernels. We first regularize the solution of integral equation by introducing a suitable new independent variable and then solve the transformed equation by a piecewise polynomial collocation...
. The presented image registration method uses a regularized gradient flow to correlate the intensities in two images. Thereby, an energy functional is successively minimized by descending along its regularized gradient. The gradient flow formulation makes use of a robust multi-scale regularization, an efficient multi-grid solver and an effective time-step control. The data processing is arranged...
. The paper presents a class of numerical methods to compute the stationary distribution of Markov chains (MCs) with large and structured state spaces. A popular way of dealing with large state spaces in Markovian modeling and analysis is to employ Kronecker-based representations for the generator matrix and to exploit this matrix structure in numerical analysis methods. This paper presents various...
. We derive limit values of high-order derivatives of the Cauchy integrals, which are extensions of the Plemelj-Sokhotskyi formula. We then use them to develop the Taylor expansion of the logarithmic potentials at the normal direction. Based on the Taylor expansion and numerical integration methods for weekly singular functions using grid points, we design fast algorithms for computing the logarithmic...
. The reliability of polyhedral homotopy continuation methods for solving a polynomial system becomes increasingly important as the dimension of the polynomial system increases. High powers of the homotopy continuation parameter t and ill-conditioned Jacobian matrices encountered in tracing of homotopy paths affect the numerical stability. We present modified homotopy functions with a new homotopy...
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