A Jacobi–Davidson method for two‐real‐parameter nonlinear eigenvalue problems arising from delay‐differential equations

The critical delays of a delay‐differential equation can be computed by solving a nonlinear two‐parameter eigenvalue problem. The solution of this two‐parameter problem can be translated to solving a quadratic eigenvalue problem of squared dimension. We present a structure preserving QR‐type method for solving such quadratic eigenvalue problem that only computes real‐valued critical delays; that is, complex critical delays, which have no physical meaning, are discarded. For large‐scale problems, we propose new correction equations for a Newton‐type or Jacobi–Davidson style method, which also forces real‐valued critical delays. We present three different equations: one real‐valued equation using a direct linear system solver, one complex valued equation using a direct linear system solver, and one Jacobi–Davidson style correction equation that is suitable for an iterative linear system solver. We show numerical examples for large‐scale problems arising from PDEs. Copyright © 2012 John Wiley & Sons, Ltd.     

Compact representation of the full Broyden class of quasi‐Newton updates

In this paper, we present the compact representation for matrices belonging to the Broyden class of quasi‐Newton updates, where each update may be either rank one or rank two. This work extends previous results solely for the restricted Broyden class of rank‐two updates. In this article, it is not assumed that the same Broyden update is used in each iteration; rather, different members of the Broyden class may be used in each iteration. Numerical experiments suggest that a practical implementation of the compact representation is able to accurately represent matrices belonging to the Broyden class of updates. Furthermore, we demonstrate how to compute the compact representation for the inverse of these matrices and a practical algorithm for solving linear systems with members of the Broyden class of updates. We demonstrate through numerical experiments that the proposed linear solver is able to efficiently solve linear systems with members of the Broyden class of matrices with high accuracy. As an immediate consequence of this work, it is now possible to efficiently compute the eigenvalues of any limited‐memory member of the Broyden class of matrices, allowing for the computation of condition numbers and the ability to perform sensitivity analysis.     

A geometric‐based algebraic multigrid method for higher‐order finite element equations in two‐dimensional linear elasticity

In this paper, we will discuss the geometric‐based algebraic multigrid (AMG) method for two‐dimensional linear elasticity problems discretized using quadratic and cubic elements. First, a two‐level method is proposed by analyzing the relationship between the linear finite element space and higher‐order finite element space. And then a geometric‐based AMG method is obtained with the existing solver used as a solver on the first coarse level. The resulting AMG method is applied to some typical elasticity problems including the plane strain problem with jumps in Young's modulus. The results of various numerical experiments show that the proposed AMG method is much more robust and efficient than a classical AMG solver that is applied directly to the high‐order systems alone. Moreover, we present the corresponding theoretical analysis for the convergence of the proposed AMG algorithms. These theoretical results are also confirmed by some numerical tests. Copyright © 2008 John Wiley & Sons, Ltd.     

An optimal burn‐in policy based on a degradation model

Burn‐in tests help manufacturers detect defective items and remove them before being sold to customers. In a competitive marketplace, cost is a major consideration and not employing a burn‐in test may result in higher and needless expenses. With this in mind, we consider degradation‐based burn‐in tests in which the degradation path follows a Wiener process and weak items are identified when the process crosses a piecewise linear function. We also study linear functions as a special case of such a piecewise linear barrier. Within this setup, we apply a cost model to determine the optimal burn‐in test. Finally, we discuss an illustrative example using GaAs laser degradation data and present an optimal burn‐in test for it.     

One‐level Newton–Krylov–Schwarz algorithm for unsteady non‐linear radiation diffusion problem

In this paper, we present a parallel Newton–Krylov–Schwarz (NKS)‐based non‐linearly implicit algorithm for the numerical solution of the unsteady non‐linear multimaterial radiation diffusion problem in two‐dimensional space. A robust solver technology is required for handling the high non‐linearity and large jumps in material coefficients typically associated with simulations of radiation diffusion phenomena. We show numerically that NKS converges well even with rather large inflow flux boundary conditions. We observe that the approach is non‐linearly scalable, but not linearly scalable in terms of iteration numbers. However, CPU time is more important than the iteration numbers, and our numerical experiments show that the algorithm is CPU‐time‐scalable even without a coarse space given that the mesh is fine enough. This makes the algorithm potentially more attractive than multilevel methods, especially on unstructured grids, where course grids are often not easy to construct. Copyright © 2004 John Wiley & Sons, Ltd.     

First‐Order system least squares for generalized‐Newtonian coupled Stokes‐Darcy flow

The coupled problem for a generalized Newtonian Stokes flow in one domain and a generalized Newtonian Darcy flow in a porous medium is studied in this work. Both flows are treated as a first‐order system in a stress‐velocity formulation for the Stokes problem and a volumetric flux‐hydraulic potential formulation for the Darcy problem. The coupling along an interface is done using the well‐known Beavers–Joseph–Saffman interface condition. A least squares finite element method is used for the numerical approximation of the solution. It is shown that under some assumptions on the viscosity the error is bounded from above and below by the least squares functional. An adaptive refinement strategy is examined in several numerical examples where boundary singularities are present. Due to the nonlinearity of the problem a Gauss–Newton method is used to iteratively solve the problem. It is shown that the linear variational problems arising in the Gauss–Newton method are well posed. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 1150–1173, 2015     

A two‐grid method for mixed finite‐element solution of reaction‐diffusion equations

We present a scheme for solving two‐dimensional, nonlinear reaction‐diffusion equations, using a mixed finite‐element method. To linearize the mixed‐method equations, we use a two grid scheme that relegates all the Newton‐like iterations to a grid Δ_H much coarser than the original one Δ_h, with no loss in order of accuracy so long as the mesh sizes obey . The use of a multigrid‐based solver for the indefinite linear systems that arise at each coarse‐grid iteration, as well as for the similar system that arises on the fine grid, allows for even greater efficiency. © 1999 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 15: 317–332, 1999     

Newton‐Krylov iterative matrix representative spectrum

The concept of a representative spectrum is introduced in the context of Newton‐Krylov methods. This concept allows a better understanding of convergence rate accelerating techniques for Krylov‐subspace iterative methods in the context of CFD applications of the Newton‐Krylov approach to iteratively solve sets of non‐linear equations. The dependence of the representative spectrum on several parameters such as mesh, Mach number or discretization techniques is studied and analyzed. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006     

Mixed antiplane boundary‐value problem for a piecewise‐homogeneous elastic body with a semi‐infinite interfacial crack

Antiplane stress state of a piecewise‐homogeneous elastic body with a semi‐infinite crack along the interface is considered. The longitudinal displacements along one of the crack edges on a finite interval, adjacent to the crack tip, are known. Shear stresses are applied to the body along the crack edges and at infinity. The problem reduces to a Riemann–Hilbert boundary‐value matrix problem with a piecewise‐constant coefficient for a complex potential in the class of symmetric functions. The complex potential is found explicitly using a Gaussian hypergeometric function. The stress state of the body close to the singular points is investigated. The stress intensity factors are determined. Copyright © 2016 John Wiley & Sons, Ltd.     

Rank‐tolerance graph classes

In this article we introduce certain classes of graphs that generalize ?‐tolerance chain graphs. In a rank‐tolerance representation of a graph, each vertex is assigned two parameters: a rank, which represents the size of that vertex, and a tolerance which represents an allowed extent of conflict with other vertices. Two vertices are adjacent if and only if their joint rank exceeds (or equals) their joint tolerance. This article is concerned with investigating the graph classes that arise from a variety of functions, such as min, max, sum, and prod (product), that may be used as the coupling functions ? and ρ to define the joint tolerance and the joint rank. Our goal is to obtain basic properties of the graph classes from basic properties of the coupling functions. We prove a skew symmetry result that when either ? or ρ is continuous and weakly increasing, the (?,ρ)‐representable graphs equal the complements of the (ρ,?)‐representable graphs. In the case where either ? or ρ is Archimedean or dual Archimedean, the class contains all threshold graphs. We also show that, for min, max, sum, prod (product) and, in fact, for any piecewise polynomial ?, there are infinitely many split graphs which fail to be representable. In the reflexive case (where ? = ρ), we show that if ? is nondecreasing, weakly increasing and associative, the class obtained is precisely the threshold graphs. This extends a result of Jacobson, McMorris, and Mulder [10] for the function min to a much wider class, including max, sum, and prod. We also give results for homogeneous functions, powers of sums, and linear combinations of min and max. © 2006 Wiley Periodicals, Inc. J Graph Theory     

A nonconventional Eulerian‐Lagrangian single‐node collocation method with Hermite polynomials for unsteady‐state advection‐diffusion equations

We developed a nonconventional Eulerian‐Lagrangian single‐node collocation method (ELSCM) with piecewise‐cubic Hermite polynomials as basis functions for the numerical simulation to unsteady‐state advection‐diffusion transport partial differential equations. This method greatly reduces the number of unknowns in the conventional collocation method, and generates accurate numerical solutions even if very large time steps are taken. The method is relatively easy to formulate. Numerical experiments are presented to show the strong potential of this method. © 2003 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 19: 271–283, 2003.     

Analysis of a 3‐node mixed‐shear‐projected triangular element method for Reissner–Mindlin plates

This article analyses an existing 3‐node hybrid triangular element, called MiSP3, for Reissner–Mindlin plates which behaves robustly in numerical benchmark tests (Ayad, Dhatt, and Batoz, Int J Numer Method Eng 42 (1998), 1149–1179). Based on Hellinger‐Reissner variational principle and the mixed shear interpolation/projection technique of MITC family, the MiSP3 element uses continuous piecewise linear polynomials for the approximations of displacements and a piecewise‐independent equilibrium mode for the approximations of bending moments/shear stresses. Due to local elimination of the parameters of moments/stresses, the element is almost of the same computational cost as the conforming linear triangular displacement element. We derive uniform stability and convergence results with respect to the plate thickness. The main tools of our analysis are the self‐equilibrium relation of the moments/stresses approximations, the properties of the mixed shear interpolation and the discrete Helmholtz decomposition of the shear stress approximation. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 241–258, 2017     

Solution of Complementarity Problems via Piecewise Linearization

My master thesis concerns the solution linear complementarity problems (LCP). The Lemke algorithm, the most commonly used algorithm for solving a LCP until this day, was compared with the piecewise Newton method (PLN algorithm). The piecewise Newton method is an algorithm to solve a piecewise linear system on the basis of damped Newton methods. The linear complementarity problem is formulated as a piecewise linear system for the applicability of the PLN algorithm. Then, different application examples will be presented, solved with the PLN algorithm. As a result of the findings (of my master thesis) it can be assumed that – under the condition of coherent orientation – the PLN-algorithm requires fewer iterations to solve a linear complementarity problem than the Lemke algorithm. The coherent orientation for piecewise linear problems corresponds for linear complementarity problems to the P-matrix-property. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Mesh‐independent convergence of the modified inexact Newton method for a second order non‐linear problem

In this paper, we consider an inexact Newton method applied to a second order non‐linear problem with higher order non‐linearities. We provide conditions under which the method has a mesh‐independent rate of convergence. To do this, we are required, first, to set up the problem on a scale of Hilbert spaces and second, to devise a special iterative technique which converges in a higher than first order Sobolev norm. We show that the linear (Jacobian) system solved in Newton's method can be replaced with one iterative step provided that the initial non‐linear iterate is accurate enough. The closeness criteria can be taken independent of the mesh size. Finally, the results of numerical experiments are given to support the theory. Published in 2005 by John Wiley & Sons, Ltd.     

Dual‐primal mixed finite elements for elliptic problems

In this article, a novel dual‐primal mixed formulation for second‐order elliptic problems is proposed and analyzed. The Poisson model problem is considered for simplicity. The method is a Petrov—Galerkin mixed formulation, which arises from the one‐element formulation of the problem and uses trial functions less regular than the test functions. Thus, the trial functions need not be continuous while the test functions must satisfy some regularity constraint. Existence and uniqueness of the solution are proved by using the abstract theory of Nicolaides for generalized saddle‐point problems. The Helmholtz Decomposition Principle is used to prove the inf‐sup conditions in both the continuous and the discrete cases. We propose a family of finite elements valid for any order, which employs piecewise polynomials and Raviart—Thomas elements. We show how the method, with this particular choice of the approximation spaces, is linked to the superposition principle, which holds for linear problems and to the standard primal and dual formulations, addressing how this can be employed for the solution of the final linear system. © 2001 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 17: 137–151, 2001     

A quasi‐static boundary value problem in multi‐surface elastoplasticity: part 2—numerical solution

Multi‐yield elastoplasticity models a material with more than one plastic state and hence allows for refined approximation of irreversible deformations. Aspects of the mathematical modelling and a proof of unique existence of weak solutions can be found in part I of this paper (Math. Models Methods Appl. Sci. 2004). In this part II we establish a canonical time–space discretization of the evolution problem and present various algorithms for the solving really discrete problems. Based on a global Newton–Raphson solver, we carefully study and solve elementwise inner iterations. Numerical examples illustrate the model and its flexibility to allow for refined hysteresis curves. Copyright © 2005 John Wiley & Sons, Ltd.     

Saddle representations of positively homogeneous functions by linear functions

We say that a positively homogeneous function admits a saddle representation by linear functions iff it admits both an inf-sup-representation and a sup-inf-representation with the same two-index family of linear functions. In the paper we show that each continuous positively homogeneous function can be associated with a two-index family of linear functions which provides its saddle representation. We also establish characteristic properties of those two-index families of linear functions which provides saddle representations of functions belonging to the subspace of Lipschitz continuous positively homogeneous functions as well as the subspaces of difference sublinear and piecewise linear functions.     

Complexiton solutions to the Hirota‐Satsuma‐Ito equation

The Hirota bilinear method is a powerful tool for solving nonlinear evolution equations. Together with the linear superposition principle, it can be used to find a special class of explicit solutions that correspond to complex eigenvalues of associated characteristic problems. These solutions are known as complexiton solutions or simply complexitons. In this article, we study complexiton solutions of the the Hirota‐Satsuma‐Ito equation which is a (2 + 1)‐dimensional extension of the Hirota‐Satsuma shallow water wave equation known to describe propagation of unidirectional shallow water waves. We first construct hyperbolic function solutions and consequently derive the so‐called complexitons via the Hirota bilinear method and the linear superposition principle. In particular, we find nonsingular complexiton solutions to the Hirota‐Satsuma‐Ito equation. Finally, we give some illustrative examples and a few concluding remarks.     

Fully Piecewise Linear Vector Optimization Problems

Journal of Optimization Theory and Applications - We distinguish two kinds of piecewise linear functions and provide an interesting representation for a piecewise linear function between two normed...     

On the representation formula for well‐ordered elastic composites: a convergence of measure approach

The aim of this paper is to derive an integral representation formula for the effective elasticity tensor for a two‐component well‐ordered composite of elastic materials without using a third reference medium and without assuming the completeness of the eigenspace of the operator ? defined in (2.16) in (J. Mech. Phys. Solids 1984; 32 (1):41–62). As shown in (J. Mech. Phys. Solids 1984; 32 (1):41–62) and (Math. Meth. Appl. Sci. 2006; 29 (6):655–664), this integral representation formula implies a relation which links the effective elastic moduli to the N‐point correlation functions of the microstructure. Such relation not only facilitates a powerful scheme for systematic incorporation of microstructural information into bounds on the effective elastic moduli but also provides a theoretical foundation for de‐homogenization. The analysis presented in this paper can be generalized to an n‐component composite of elastic materials. The relations developed here can be applied to the de‐homogenization for a special class of linear viscoelastic composites. The results will be presented in another paper. Copyright © 2006 John Wiley & Sons, Ltd.