Very accurate posterior approximations based on finite mixtures of the hyperparameters conditionals

Consider a posterior density π(λ,φ) such that both and are known. We propose to approximate π(λ,φ) by , where is a finite mixture of the posterior conditionals . The weights and components of the mixture are chosen to minimize an approximate f-divergence between the approximate and the actual posterior. These approximate divergences are computed through an importance sampling idea using a simulated sample from the same finite mixture approximations. For the special case of the χ² or Harmonic divergences, once the minimum approximate divergences have been obtained, they can be plugged into total variation type inequalities to obtain precision limits for the corresponding approximations of posterior expectations of interest.When the algorithm can be used—namely, when both full conditionals and are known, it requires little computational, programming and diagnosing effort. Moreover, we present several examples which show that the approximations produced are extremely accurate, even when a small number of components are included in the mixture approximation.     

Adams methods for the efficient solution of stochastic differential equations with additive noise

The application of Adams methods for the numerical solution of stochastic differential equations is considered. Especially we discuss the path-wise (strong) solutions of stochastic differential equations with additive noise and their numerical computation. The special structure of these problems suggests the application of Adams methods, which are used for deterministic differential equations very successfully. Applications to circuit simulation are presented.     

Elasto-plasticity model in structural optimization of composite materials with periodic microstructures

The paper deals with a structural optimization of composite materials with periodic microstructures invoking an elasto-plasticity model with the von Mises yield criterion. Closest-point return mapping algorithms within the incremental finite element method are applied for the numerical solution of the problem. The latter iterative schemes are computationally effective, robust and stable, and have recently become the most popular means for numerical implementation of elasto-plastic models. The homogenized elasto-plastic equation is considered as an equality constraint in the structural optimization problem. Numerical experiments for the computation of the homogenized coefficients involving adaptive finite element discretizations of the three-dimensional periodicity microcell are presented.     

Rates of convergence of extremes for mixed exponential distributions

The limiting distributions of the extremes of mixed exponential distributions and the associated rates of convergence are derived. The practical values of the results are illustrated by a numerical study.     

Pseudospectra of linear matrix pencils by block diagonalization

We propose an algorithm that block-diagonalizes regular matrix pencils using well conditioned transformations. This algorithm is used for approximating the pseudospectra of matrix pencils. Several numerical experiments illustrate the behavior of the proposed algorithm.     

On the independence between risk profiles in the compound collective risk actuarial model

This paper examines a compound collective risk model in which the primary distribution comprised the Poisson–Lindley distribution with a λ parameter, and where the secondary distribution is an exponential one with a θ parameter. We consider the case of dependence between risk profiles (i.e., the parameters λ and θ), where the dependence is modelled by a Farlie–Gumbel–Morgenstern family. We analyze the consequences of the dependence on the Bayes premium. We conclude that the consequences of the dependence on the Bayes premium may vary considerably.     

An Alternative Algorithm for a Sliding Window ULV Decomposition

The ULV decomposition (ULVD) is an important member of a class of rank-revealing two-sided orthogonal decompositions used to approximate the singular value decomposition (SVD). The ULVD can be modified much faster than the SVD. When modifiying the ULVD, the accurate computation of the subspaces is required in certain time varying applications in signal processing. In this paper, we present an updating algorithm which is suitable for large scaled matrices of low rank and as effective as alternatives. The algorithm runs in O(n²) time.     

Weierstrass-like Methods with Corrections for the Inclusion of Polynomial Zeros

In this paper, we present iterative methods of Weierstress’ type for the simultaneous inclusion of all simple zeros of a polynomial. The main advantage of the proposed methods is the increase of the convergence rate attained by applying suitable correction terms. Using the concept of the R-order of convergence of mutually dependent sequences, we present the convergence analysis for the total-step and the single-step methods. Numerical examples are given.     

Measuring the network robustness by Monte Carlo estimation of shortest path length distribution

Two kinds of robustness measure for networks are introduced and applied to the road network systems in Japan. One is on the connectivity of randomly chosen pair of vertices, another is on the shortest path length between pair of connected vertices. We devise Monte Carlo methods for the computation of two measures.     

Error Analysis of a Derivative-Free Algorithm for Computing Zeros of Holomorphic Functions

We consider the quadrature method developed by Kravanja and Van Barel (Computing 63(1):69–91, 1999) for computing all the zeros of a holomorphic function that lie inside the unit circle. The algorithm uses only the function values and no (first or higher order) derivatives. Information about the location of the zeros is obtained from certain integrals along the unit circle. In numerical computations these are replaced by their trapezoidal rule approximations. We investigate the resulting quadrature error. Our error analysis shows that the zeros located inside the unit circle do not affect the accuracy of the computed approximations whereas the quadrature error related to the zeros located outside the unit circle tends to zero exponentially as the number of quadrature points tends to infinity.     

On the Convergence Acceleration of Slowly Convergent Sums Involving Oscillating Terms

In this paper, an acceleration technique based on a Kummer’s transformation method is developed for some slowly convergent series. The original series is decomposed into two parts; one part being rapidly convergent and the other part being slowly convergent. Then the series in the slowly convergent part is expressed as integrals of some auxiliary functions and subsequently they are written in terms of polynomials whose coefficients are given by the zeta functions. The given method is computationally oriented and does not involve much analytic effort. A numerical example is provided to illustrate the usage and the efficiency of the method.     

Implementations of a New Theorem for Computing Bounds for Positive Roots of Polynomials

Finding an upper bound for the positive roots of univariate polynomials is an important step of the continued fractions real root isolation algorithm. The revived interest in this algorithm has highlighted the need for better estimations of upper bounds of positive roots. In this paper we present a new theorem, based on a generalization of a theorem by D. Stefanescu, and describe several implementations of it – including Cauchy's and Kioustelidis' rules as well as two new rules recently developed by us. From the empirical results presented here we see that applying various implementations of our theorem – and taking the minimum of the computed values – greatly improves the estimation of the upper bound and hopefully that will affect the performance of the continued fractions real root isolation method.     

Data-sparse Approximation by Adaptive ℋ2-Matrices

A class of matrices (ℋ²-matrices) has recently been introduced for storing discretisations of elliptic problems and integral operators from the BEM. These matrices have the following properties: (i) They are sparse in the sense that only few data are needed for their representation. (ii) The matrix-vector multiplication is of linear complexity. (iii) In general, sums and products of these matrices are no longer in the same set, but after truncation to the ℋ²-matrix format these operations are again of quasi-linear complexity. We introduce the basic ideas of ℋ- and ℋ²-matrices and present an algorithm that adaptively computes approximations of general matrices in the latter format. Received April 17, 2002 Published online: July 26, 2002     

Accelerating the SVD Block-Jacobi Method

The paper discusses how to improve performance of the one-sided block-Jacobi algorithm for computing the singular value decomposition of rectangular matrices. In particular, it is shown how cosine-sine decomposition of orthogonal matrices can be used to accelerate the slowest part of the algorithm – updating the block-columns.     

Reduction of Smith Normal Form Transformation Matrices

Smith normal form computations are important in group theory, module theory and number theory. We consider the transformation matrices for the Smith normal form over the integers and give a presentation of arbitrary transformation matrices for this normal form. Our main contribution is an algorithm that replaces already computed transformation matrices by others with small entries. We combine methods from lattice basis reduction with a procedure to reduce the sum of the squared entries of both transformation matrices. This algorithm performs well even for matrices of large dimensions.     

Block triangular preconditioners for the discretized time-harmonic Maxwell equations in mixed form

In this paper, we consider the solution of the saddle point linear systems arising from the finite element discretization of the time-harmonic Maxwell equations in mixed form. Two types of block triangular Schur complement-free preconditioners used with Krylov subspace methods are proposed, involving the choice of the parameter. Furthermore, we give the optimal parameter in practice. Theoretical analysis shows that all eigenvalues of the preconditioned matrices are strongly clustered. Finally, numerical experiments that validate the analysis are presented.     

Derivative Free Inclusion Methods for Polynomial Zeros

Two iterative methods for the simultaneous inclusion of complex zeros of a polynomial are presented. Both methods are realized in circular interval arithmetic and do not use polynomial derivatives. The first method of the fourth order is composed as a combination of interval methods with the order of convergence two and three. The second method is constructed using double application of the inclusion method of Weierstrass’ type in serial mode. It is shown that its R-order of convergence is bounded below by the spectral radius of the corresponding matrix. Numerical examples illustrate the convergence rate of the presented methods     

A priori and a posteriori error analysis of an augmented mixed finite element method for incompressible fluid flows

In this paper we extend recent results on the a priori and a posteriori error analysis of an augmented mixed finite element method for the linear elasticity problem, to the case of incompressible fluid flows with symmetric stress tensor. Similarly as before, the present approach is based on the introduction of the Galerkin least-squares type terms arising from the constitutive and equilibrium equations, and from the relations defining the pressure in terms of the stress tensor and the rotation in terms of the displacement, all of them multiplied by stabilization parameters. We show that these parameters can be suitably chosen so that the resulting augmented variational formulation is defined by a strongly coercive bilinear form, whence the associated Galerkin scheme becomes well-posed for any choice of finite element subspaces. Next, we present a reliable and efficient residual-based a posteriori error estimator for the augmented mixed finite element scheme. Finally, several numerical results confirming the theoretical properties of this estimator, and illustrating the capability of the corresponding adaptive algorithm to localize the singularities and the large stress regions of the solution, are reported.     

Adaptive Recompression of -Matrices for BEM

The efficient treatment of dense matrices arising, e.g., from the finite element discretisation of integral operators requires special compression techniques. In this article, we use a hierarchical low-rank approximation, the so-called -matrix, that approximates the dense stiffness matrix in admissible blocks (corresponding to subdomains where the underlying kernel function is smooth) by low rank matrices. The low rank matrices are assembled by the ACA+ algorithm, an improved variant of the well-known ACA method. We present an algorithm that can determine a coarser block structure that minimises the storage requirements (enhanced compression) and speeds up the arithmetic (e.g., inversion) in the -matrix format. This coarse approximation is done adaptively and on-the-fly to a given accuracy such that the matrix is assembled with minimal storage requirements while keeping the desired approximation quality. The benefits of this new recompression technique are demonstrated by numerical examples.