Performance Of Numerical Algorithms Research Articles

Generating linear, semidefinite, and second-order cone optimization problems for numerical experiments

The numerical performance of algorithms can be studied using test sets or procedures that generate such problems. This paper proposes various methods for generating linear, semidefinite, and second-order cone optimization problems. Specifically, we are interested in problem instances requiring a known optimal solution, a known optimal partition, a specific interior solution, or all these together. In the proposed problem generators, different characteristics of optimization problems, including dimension, size, condition number, degeneracy, optimal partition, and sparsity, can be chosen to facilitate comprehensive computational experiments. We also develop procedures to generate instances with a maximally complementary optimal solution with a predetermined optimal partition to generate challenging semidefinite and second-order cone optimization problems. Generated instances enable us to evaluate efficient interior-point methods for conic optimization problems.

Accelerated augmented Lagrangian method for total variation minimization

The augmented Lagrangian method (TVAL3) (Li et al. in Comput Optim Appl 56(3):507–530, 2013), which combines an alternating direction technique with a nonmonotone line search to minimize the augmented Lagrangian function at each iteration, is a very efficient method for total variation image restoration. In this paper we present an accelerated augmented Lagrangian based on TVAL3 for total variation image restoration. It is widely accepted that the stepsize, is very crucial to the performance of numerical algorithms, especially for gradient-based methods. We design a new quadratic approximation model to generate an efficient approximately optimal stepsize, truncate it by the two well-known BB stepsizes and use the resulted approximately optimal stepsize to accelerate the augmented Lagrangian method. We establish the convergence of the proposed method under weaker condition. Numerical experiments show that the proposed method is very promising.

Solution method of optimisation problem based on a modified Armijo-type line search

In this paper, a new inexact line search rule is presented for solving a class of fundamental unconstrained optimisation problems. As a modified version of the standard Armijo line search rule, it incorporates the information of the second order derivative of objective function into determining a suitable step length based on the more accurate approximation of objective function. Owing to this line search, a greater reduction of objective function may be obtained at each iteration as well as with less cost of computation. Combined with a choice of descent search direction, a well defined algorithm is developed, where the line search rule allows the initial step length to be adjusted automatically such that the numerical performance of algorithm is improved. Under some mild assumptions, global convergence is established for the algorithm. Numerical results demonstrate that the proposed method is promising, especially in comparison with the existing methods.

Smoothing inexact Newton methods for NCP with various nonmonotone techniques

AbstractVarious iterative methods for solving nonlinear complementarity problems (NCP) are developed in recent years. In this paper we propose Jacobian smoothing inexact Newton methods for NCP with different nonmonotone strategies. The methods are based on semismooth equation reformulation of NCP by Fischer‐Burmeister function. Nonmonotone line‐search techniques are used for globalization procedure. Numerical performance of algorithms are compared. (© 2013 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Canonical transforms and the efficient integration of quantum mechanical wave equations

The integration of time-dependent quantum mechanical wave equations is a fundamental problem in computational physics and computational chemistry. The wave-function's energy spectrum as well as its momentum spectrum impose fundamental limits on the performance of numerical algorithms for the solution of wave equations. We demonstrate how canonical transforms may be applied to negotiate these limitations and to increase the performance of numerical algorithms by up to several orders of magnitude. Our approach includes the so-called Kramers-Henneberger transform as a special case and puts forward modifications toward an improved numerical efficiency.

Analysis of Boundary Element Methods

The workshop Analysis of Boundary Element Methods , organized by Martin Costabel (Rennes) and Ernst P. Stephan (Hannover). This meeting brought together 46 experts in numerical analysis. Boundary element methods (BEM) are well established numerical methods with a wide range of applications. There are still many challenging problems, the aim of this workshop was to coordinate the efforts to tackle these problems. A central theme of many talks was the error analysis. In the following we focus on some central aspects. 1. BEM for time dependent problems: Space-time boundary integral equations using integral representations with the fundamental solution of hyperbolic problems have become increasingly important in elastodynamics and electrodynamics. New results were presented concerning error analysis, stability and performance of numerical algorithms. Special attention was given to efficient quadrature of retarded potentials and also to the method of convolution quadrature for the computation of transient acoustic waves. 2.Stability at high frequencies: The numerical modeling of scattering problems at higher frequencies poses practical challenges due to the need to approximate highly oscillating functions. In addition, the analysis of many of the standard BEM loses usefulness in the high frequency range because the constants in the stability and error estimates either grow rapidly with the frequency of have unspecified dependency on the frequency. New boundary element methods have been presented whose stability deteriorates only slowly with frequency or are even unconditionally stable for all frequencies. 3. Fast algorithms: Multilevel approaches have produced several families of highly efficient algorithms. Several talks reported on the applications of the fast multipole method and on iterative techniques based on multigrid methods, H-matrices or on Schwarz preconditioners. 4. New concepts for BEM: Topics have been here: sparse p-version BEM with random loading, meshless methods for pseudodifferential equations, adaptive procedures, least squares FEM-BEM coupling, p-version applied to EFIE, BEM for eigenvalue problems. 5. Specific applications: Here the following have been addressed: Source identification and source reconstructions in acoustics with the help of inverse boundary elements, BEM for Maxwell's equations (currents and charges boundary integral equations, shape optimization of integrated lens antennas, electromagnetic scattering by a dielectric body), BEM for option pricing. The list of participants included several of the founders of the discipline of mathematical analysis of BEM, but also some PhD students. This mix of both established and younger researchers created a particularly stimulating research atmosphere.

NUMERICAL SOLUTION OF DISCRETE STABLE LINEAR MATRIX EQUATIONS ON MULTICOMPUTERS

We investigate the parallel performance of numerical algorithms for solving discrete Sylvester and Stein equations as they appear for instance in discrete-time control problems, filtering, and image restoration. The methods used here are the squared Smith iteration and the sign function method on a Cayley transformation of the original equation. For Stein equations with semidefinite right-hand side these methods are modified such that the Cholesky factor of the solution can be computed directly without forming the solution matrix explicitly. We report experimental results of these algorithms on distributed-memory multicomputers

Tracking ill-conditioning for the RLS-lattice algorithms

The numerical performance of lattice-based adaptive signal processing algorithms is shown to involve the conditioning of a 2/spl times/2 matrix whose off-diagonal elements contain reflection coefficients. Degraded algorithmic performance for the a posteriori recursive least squares lattice (RLSL) is shown to be attributed to the ill-conditioning of this matrix, Theoretical results are given which may be used to separate the conditioning of the underlying problem from issues concerning algorithmic stability. Although the results are not restricted to the all-pole case, for simplicity the authors make use of this well understood example since the condition number of the autocorrelation matrix will become arbitrarily close to singularity as the poles of an all-pole filter approach the unit circle. For a second-order prediction problem, four case studies of varying conditioning are provided which demonstrate the appropriateness of the theoretical bounds which analytically describe the sensitivity to perturbations in the residual update recursions. The paper illustrates the use of numerical linear algebra analysis techniques to better understand the numerical performance of algorithms in signal processing as well as emphasising that numerical performance is a function of the problem's conditioning as well as algorithmic stability.

Numerical solution of problems on unbounded domains. A review

While numerically solving a problem initially formulated on an unbounded domain, one typically truncates this domain, which necessitates setting the artificial boundary conditions (ABCs) at the newly formed external boundary. The issue of setting the ABCs appears most significant in many areas of scientific computing, for example, in problems originating from acoustics, electrodynamics, solid mechanics, and fluid dynamics. In particular, in computational fluid dynamics (where external problems represent a wide class of important formulations) the proper treatment of external boundaries may have a profound impact on the overall quality and performance of numerical algorithms and interpretation of the results. Most of the currently used techniques for setting the ABCs can basically be classified into two groups. The methods from the first group (global ABCs) usually provide high accuracy and robustness of the numerical procedure but often appear to be fairly cumbersome and (computationally) expensive. The methods from the second group (local ABCs) are, as a rule, algorithmically simple, numerically cheap, and geometrically universal; however, they usually lack accuracy of computations. In this paper we first present an extensive survey and provide a comparative assessment of different existing methods for constructing the ABCs. Then, we describe a new ABCs technique proposed in our recent work and review the corresponding results. This new technique enables one to construct the ABCs that largely combine the advantages relevant to the two aforementioned classes of existing methods. Our approach is based on application of the difference potentials method by Ryaben'kii. This approach allows one to obtain highly accurate ABCs in the form of certain (nonlocal) boundary operator equations. The operators involved are analogous to the pseudodifferential boundary projections first introduced by Calderon and then also studied by Seeley. In spite of the nonlocality, the new boundary conditions are geometrically universal, numerically inexpensive, and easy to implement along with the existing solvers.

The Use of Pivoting to Improve the Numerical Performance of Algorithms for Toeplitz Matrices

Classical $O( n^2 )$ Toeplitz solvers break down if any leading principal submatrices are singular, and as might be expected, the occurrence of nearly singular leading submatrices can cause a serious loss of accuracy in these solvers. In this paper, a pivoting scheme has been incorporated into the Toeplitz solver of Bareiss which allows near-singularities to be treated without significant loss of accuracy. As special cases, the pivoted Toeplitz solver also handles exact singularities and numerical singularities—the latter appear as near-singularities when finite-precision floating-point arithmetic is used. The Bareiss algorithm can also be used to compute the $LU$ factors of a Toeplitz matrix, and it is shown that a restricted version of the pivoted Toeplitz solver produces an $LU$ factorization of a row- and column-permuted version of the original Toeplitz matrix.Finally, it is shown how the inverter of Trench and Zohar can be derived from the multipliers produced by the Bareiss algorithm, and this conne...