Precise asymptotics – A general approach

The legendary 1947-paper by Hsu and Robbins, in which the authors introduced the concept of “complete convergence”, generated a series of papers culminating in the like-wise famous Baum–Katz 1965-theorem, which provided necessary and sufficient conditions for the convergence of the series $\sum_{n=1}^{\infty}n^{r/p-2}P (|S_{n}| \geqq \varepsilon n^{1/p})$ for suitable values of r and p, in which S _n denotes the n-th partial sum of an i.i.d. sequence. Heyde followed up the topic in his 1975-paper where he investigated the rate at which such sums tend to infinity as ε↘0 (for the case r=2 and p=1). The remaining cases have been taken care later under the heading “precise asymptotics”. An abundance of papers have since then appeared with various extensions and modifications of the i.i.d.-setting. The aim of the present paper is to show that the basis for the proof is essentially the same throughout, and to collect a number of examples. We close by mentioning that Klesov, in 1994, initiated work on rates in the sense that he determined the rate, as ε↘0, at which the discrepancy between such sums and their “Baum–Katz limit” converges to a nontrivial quantity for Heyde’s theorem. His result has recently been extended to the complete set of r- and p-values by the present authors.     

Differentiation of sets – The general case

In recent work by Khmaladze and Weil (2008) and by Einmahl and Khmaladze (2011), limit theorems were established for local empirical processes near the boundary of compact convex sets K   in R^d${\mathbb{R}}^d$. The limit processes were shown to live on the normal cylinder Σ of K, respectively on a class of set-valued derivatives in Σ. The latter result was based on the concept of differentiation of sets at the boundary ∂K of K, which was developed in Khmaladze (2007). Here, we extend the theory of set-valued derivatives to boundaries ∂F   of rather general closed sets F⊂R^d$F\subset {\mathbb{R}}^d$, making use of a local Steiner formula for closed sets, established in Hug, Last and Weil (2004).     

Numerical algorithm for solving the Stokes’ first problem for a heated generalized second grade fluid with fractional derivative

In this paper, based on the second-order compact approximation of first-order derivative, the numerical algorithm with second-order temporal accuracy and fourth-order spatial accuracy is developed to solve the Stokes’ first problem for a heated generalized second grade fluid with fractional derivative; the solvability, convergence, and stability of the numerical algorithm are analyzed in detail by algebraic theory and Fourier analysis, respectively; the numerical experiment support our theoretical analysis results.     

Creep of Single Crystals – Modelling and Numerical Aspects

A number of constitutive models, utilizing both microstructural and/or phenomenological considerations, have been developed for the simulation of the creep behaviour of nickel-base single crystal superalloys at elevated temperatures. In this work, emphasis is placed on the rate-dependent single crystal plasticity model [1]. A strategy for the identification of the material parameters of the model to fit the results from experiments has been implemented. The parameter fitting methodology rests upon a two-membered evolution strategy. In addition, a proposal is made for the extension of the Cailletaud model [1] by means of an evolution equation for a damage variable which enables the modelling of the tertiary creep stage. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Locally Embedded Strong Discontinuities at finite strains – Numerical Implementation

In this paper, a geometrically nonlinear finite element approximation for highly localized deformation in structures undergoing material failure in the form of strain softening, is developed. The basis for its numerical implementation in this class of problems is defined through the elaboration of Strong Discontinuity Approach-fundamentals. Proposed numerical model uses an Enhanced Assumed Strain Concept for the additive decomposition of the displacement gradient into a conforming and an enhanced part. The discontinuous component of the displacement field which is associated with the failure in the modeled structure is isolated in the enhanced part of the deformation gradient. In contrast to previous works, this part of the deformation mapping is condensed out at the material level, without the application of static condensation technique. The resulting set of constitutive equations is formally identical to that of standard plasticity and therefore, can be solved using the return-mapping algorithm. No assumptions regarding the interface law connecting the displacement discontinuity with the conjugate traction vector are made. As a result, the proposed numerical solution can be applied to a broad range of different mechanical problems including mode-I fracture in brittle materials or the analysis of shear bands. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Numerical analysis of quenching – Heat conduction in metallic materials

Metallic materials present a complex behavior during heat treatment processes. In a certain temperature range, change of temperature induces a phase transformation of metallic structure, which alters physical properties of the material. Indeed, measurements of specific heat and conductivity show strong temperature-dependence during processes such as quenching of steel. Several mathematical models, as solid mixtures and thermal–mechanical coupling, for problems of heat conduction in metallic materials, have been proposed. In this work, we take a simpler approach without thermal–mechanical coupling of deformation, by considering the nonlinear temperature-dependence of thermal parameters as the sole effect due to those complex behaviors. The above discussion of phase transformation of metallic materials serves only as a motivation for the strong temperature-dependence as material properties. In general, thermal properties of materials do depend on the temperature, and the present formulation of heat conduction problem may be served as a mathematical model when the temperature-dependence of material parameters becomes important. For this mathematical model we present the error estimate using the finite element method for the continuous-time case.     

PCR algorithm for parallel computing the solution of the general restricted linear equations

In this paper we present a parallel algorithm for parallel computing the solution of the general restricted linear equations Ax=b,x∈T, where T is a subspace of ? ⁿ and b∈AT. By this algorithm the solution x=A _T,S ⁽²⁾ b is obtained in n(log?₂ m+log?₂(n?s+1)+7)+log?₂ m+1 steps with P=mn processors when m≥2(n?1) and with P=2n(n?1) processors otherwise.     

Win–win match using a genetic algorithm

As the service industries grow, tasks are not directly assigned to the skills but the knowledge of the worker which is to be valued more in finding the best match. The problem becomes difficult mainly because the match has to be seen with the objectives of both sides. Assignment methods fail to respond to a multi-objective, multi-constraint problem with complicated match; whereas, metaheuristics is preferable based on computational simplicity. A conditional genetic algorithm is developed in this study to propose both global and composite match using different fitness functions. This algorithm kills the infeasibilities to achieve the maximum number of matches. The proposed algorithm is applied on an academic problem of multi-alternative candidates and multi-alternative tasks (m × n problem) in two stages. In the first stage, four different fitness functions are evaluated and in the second stage using one of the fitness functions global and composite matching have been compared. The achievements will contribute both to the academic and business world.     

Numerical discretization of a Darcy–Forchheimer model

We solve a steady Darcy–Forchheimer flow in a bounded region by means of piecewise constant velocities and nonconforming piecewise pressures. For the computation, we solve the nonlinearity by an alternating-directions algorithm and we decouple the computation of the velocity from that of the pressure by a gradient algorithm. We prove a priori error estimates of the scheme and convergence of the alternating-directions algorithm. The first author was supported by the J. Tinsley Oden Faculty Fellowship, ICES, The University of Texas at Austin.     

Coupled Multi-field and Multi-rate Problems – Numerical Solution and Stability Analysis

The numerical solution of coupled differential equation systems is usually done following a monolithic or a decoupled algorithm. In contrast to the holistic monolithic solvers, the decoupled solution strategies are based on breaking down the system into several subsystems. This results in different characteristics of these families of solvers, e. g., while the monolithic algorithms provide a relatively straight-forward solution framework, unlike their decoupled counterparts, they hinder software re-usability and customisation. This is a drawback for multi-field and multi-rate problems. The reason is that a multi-field problem comprises several subproblems corresponding to interacting subsystems. This suggests exploiting an individual solver for each subproblem. Moreover, for the efficient solution of a multi-rate problem, it makes sense to perform the temporal integration of each subproblem using a time-step size relative to its evolution rate. Nevertheless, decoupled solvers introduce additional errors to the solution and, thus, they must always be accompanied by a thorough stability analysis. Here, tailored solution schemes for the decoupled solution of multi-field and multi-rate problems are proposed. Moreover, the stability behaviour of the solutions obtained from these methods are studied. Numerical examples are solved and the reliability of the outcome of the stability analysis is investigated. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Enumeration of Pareto optimal multi-criteria spanning trees – a proof of the incorrectness of Zhou and Gen's proposed algorithm

The minimum spanning tree (MST) problem is a well-known optimization problem of major significance in operational research. In the multi-criteria MST (mc-MST) problem, the scalar edge weights of the MST problem are replaced by vectors, and the aim is to find the complete set of Pareto optimal minimum-weight spanning trees. This problem is NP-hard and so approximate methods must be used if one is to tackle it efficiently. In an article previously published in this journal, a genetic algorithm (GA) was put forward for the mc-MST. To evaluate the GA, the solution sets generated by it were compared with solution sets from a proposed (exponential time) algorithm for enumerating all Pareto optimal spanning trees. However, the proposed enumeration algorithm that was used is not correct for two reasons: (1) It does not guarantee that all Pareto optimal minimum-weight spanning trees are returned. (2) It does not guarantee that those trees that are returned are Pareto optimal. In this short paper we prove these two theorems.     

Adaptive Fourier series—a variation of greedy algorithm

We study decomposition of functions in the Hardy space \(H^2(\mathbb{D} )\) into linear combinations of the basic functions (modified Blaschke products) in the system

$\label{Walsh like} {B}_n(z)= \frac{\sqrt{1-|a_n|^2}}{1-\overline{a}_{n}z}\prod\limits_{k=1}^{n-1}\frac{z-a_k}{1-\overline{a}_{k}z}, \quad n=1,2,..., $

(1)

where the points a _n ’s in the unit disc \(\mathbb{D}\) are adaptively chosen in relation to the function to be decomposed. The chosen points a _n ’s do not necessarily satisfy the usually assumed hyperbolic non-separability condition

$\label{condition} \sum\limits_{k=1}^\infty (1-|a_k|)=\infty $

(2)

in the traditional studies of the system. Under the proposed procedure functions are decomposed into their intrinsic components of successively increasing non-negative analytic instantaneous frequencies, whilst fast convergence is resumed. The algorithm is considered as a variation and realization of greedy algorithm.

     

Numerical estimates for a Grötzsch ring constant

Numerical lower and upper estimates are obtained for the constant λ_n defined by λn =lim _a→0(modR _G,n (a)+loga) associated with the Grötzsch extremal ringR _G,n (a) in euclideann-space, for 3≤n≤22. Improved lower estimates in terms ofn are provided for λ_n, the modulus ofR _G,n (a) is compared with its counterpart in the plane, and bounds for modR _G,n (a) are obtained that are of the correct order asa tends either to 0 or to 1. The ratio of the latter bounds is bounded by constants depending only onn.     

Takagi’s decomposition of a symmetric unitary matrix as a finite algorithm

Takagi’s decomposition is an analog (for complex symmetric matrices and for unitary similarities replaced by unitary congruences) of the eigenvalue decomposition of Hermitian matrices. It is shown that, if a complex matrix is not only symmetric but is also unitary, then its Takagi decomposition can be found by quadratic radicals, that is, by means of a finite algorithm that involves arithmetic operations and quadratic radicals. A similar fact is valid for the eigenvalue decomposition of reflections, which are Hermitian unitary matrices.     

An improved algorithm for the $$L_2$$ – $$L_p$$ minimization problem

Mathematical Programming - In this paper we consider a class of non-Lipschitz and non-convex minimization problems which generalize the $$L_2$$ – $$L_p$$ minimization problem. We propose an...     

Exploiting special structure in a primal—dual path-following algorithm

A primal-dual path-following algorithm that applies directly to a linear program of the form, min{c ^t xAx = b, Hx u, x 0,x ⁿ }, is presented. This algorithm explicitly handles upper bounds, generalized upper bounds, variable upper bounds, and block diagonal structure. We also show how the structure of time-staged problems and network flow problems can be exploited, especially on a parallel computer. Finally, using our algorithm, we obtain a complexity bound of O( ds ² log(nk)) fortransportation problems withs origins,d destinations (s <d), andn arcs, wherek is the maximum absolute value of the input data.This research was supported in part by NSF Grants DMS-85-12277 and CDR-84-21402 and by ONR Contract N00014-87-K-0214.     

EigenBlock algorithm for change detection – An application of adaptive dictionary learning techniques

Change detection methods are very important in many areas such as medical imaging and remote sensing. In particular, identifying the changes in medical images taken at different times is of great relevance in clinical practice. The key of detecting changes in medical images is to detect disease-related changes while rejecting “unimportant” induced by noise, mis-alignment changes, and other common acquisition-related artifacts (such as inhomogeneity). In this paper we first summarize the existing methods for automatic change detection, and propose a new approach for detecting changes based on local dictionary learning techniques. In addition we aim to automatically ignore insignificant changes. Our new approach uses L₂ norm as similarity measure to learn the dictionary. We also apply knowledge of principal component analysis as a feature extraction tool, to eliminate the redundancy and hence to increase the computational efficiency. The performance of the algorithm is validated with synthetic and clinical images.     

Numerical solution of a variational problem with L∞ functionals

We consider the problem which consists in finding an optimal Lipschitz extension to the domain Ω of functions that verify the restriction u = g on ∂Ω. This work deals with the numerical approximations of the problem in dimension two. Using a discretization procedure based on finite differences method we obtain a large scale non smooth convex minimization problem, which is solved via Variable Metric Hibrid Proximal Point Method. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

PAT – a Reliable Path-Following Algorithm

This paper presents a new technique for the reliable computation of the -pseudospectrum defined by (A)={zC : _min(A–zI)} where _min is the smallest singular value. The proposed algorithm builds an orbit of adjacent equilateral triangles to capture the level curve (A)={zC : _min(A–zI)=} and uses a bisection procedure on specific triangle vertices to compute a numerical approximation to . The method is guaranteed to terminate, even in the presence of round-off errors.