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This paper deals with a fully implicit time discretization scheme with variable time-step for a nonlinear system modelling phase transition and mechanical deformations in shape memory alloys. The model is studied in the non-stationary case and accounts for local microscopic interactions between the phases introducing the gradients of the phase parameters. The resulting initial-boundary value problem has already been studied by the author who proved existence, uniqueness and continuous dependence on data for a suitable weak solution along some regularity results. A careful and detailed investigation of the variable time-step discretization is the goal of this paper. Thus, we deduce some estimates for the discretization error. These estimates depend only on data, impose no constraints between consecutive time-steps and show an optimal order of convergence. Finally, we prove another regularity result for the solution under stronger regularity assumptions on data. [ABSTRACT FROM AUTHOR]

The notion of extended-well-posedness has been introduced by Zolezzi for scalar minimization problems and has been further generalized to vector minimization problems by Huang. In this paper, we study the extended well-posedness properties of vector minimization problems in which the objective function is C-quasiconvex. To achieve this task, we first study some stability properties of such problems. [ABSTRACT FROM AUTHOR]

This paper is concerned with the analysis of the discontinuous Galerkin finite element method applied to the space semidiscretization of nonlinear nonstationary convection-diffusion problems. The attention is paid to the derivation of the method and the investigation of error estimates for schemes with a symmetric as well as nonsymmetric discretization of diffusion terms and with the interior and boundary penalty. The theoretical results are illustrated by numerical experiments. [ABSTRACT FROM AUTHOR]

This paper is an extension of previously published work by Bernard O. Koopman to the general case of any number of tasks and any effect function. A systematic method for the numerical calculation of the maximum effect and its corresponding optimum distribution is presented in a form which is also suitable for solution on an automatic digital computer. For the special case of additive returns which saturate, a special graphical or numerical method for any number of tasks is presented with an example worked out in detail A similar method for multiplicative effects is sketched. [ABSTRACT FROM AUTHOR]

In this paper, a new conjugate gradient formula βNewk is given to compute the search directions for unconstrained optimization problems. General convergence results for the proposed formula with some line searches such as the exact line search, the Grippo-Lucidi line search and the Wolfe-Powell line search are discussed. Under the above line searches and some assumptions, the global convergence properties of the given methods are discussed. The given formula βNewk ⩾ 0, and the search directions dk which are generated by the given βNewk under the strong Wolfe-Powell line search satisfy the sufficient descent condition. Preliminary numerical results show that the proposed methods are efficient. [ABSTRACT FROM AUTHOR]

Research on multigrid methods for optimization problems is reviewed. Optimization problems considered include shape design, parameter optimization, anti optimal control problems governed by partial differential equations of elliptic, parabolic, and hyperbolic type. [ABSTRACT FROM AUTHOR]

A hyperplane search tree is a binary tree used to store a set S of n d-dimensional data points. In a random hyperplane search tree for S, the root represents a hyperplane defined by d data points drawn uniformly at random from S. The remaining data points are split by the hyperplane, and the definition is used recursively on each subset. We assume that the data are points in general position in Rd. We show that, uniformly over all such data sets S, the expected height of the hyperplane tree is not worse than that of the k-d tree or the ordinary one-dimensional random binary search tree, and that, for any fixed d ≥ 3, the expected height improves over that of the standard random binary search tree by an asymptotic factor strictly greater than one. [ABSTRACT FROM AUTHOR]

Iterative methods are proposed for certain augmented systems of linear equations that arise in interior methods for general nonlinear optimization, Interior methods define a sequence of KKT equations that represent the symmetrized (but indefinite) equations associated with Newton's method for a point satisfying the perturbed optimality conditions. These equations involve both the primal and dual variables and become increasingly ill-conditioned as the optimization proceeds. In this context, an iterative linear solver must not only handle the ill-conditioning but also detect the occurrence of KKT matrices with the wrong matrix inertia. A one-parameter family of equivalent linear equations is formulated that includes the KKT system as a special case. The discussion focuses on a particular system from this family, known as the "doubly augmented system," that is positive definite with respect to both the primal and dual variables. This property means that a standard preconditioned conjugate-gradient method involving both primal and dual variables will either terminate successfully or detect if the KKT matrix has the wrong inertia. Constraint preconditioning is a well-known technique for preconditioning the conjugate-gradient method on augmented systems. A family of constraint preconditioners is proposed that provably eliminates the inherent ill-conditioning in the augmented system. A considerable benefit of combining constraint preconditioning with the doubly augmented system is that the preconditioner need not be applied exactly. Two particular "active-set" constraint preconditioners are formulated that involve only a subset of the rows of the augmented system and thereby may be applied with considerably less work. Finally, some numerical experiments illustrate the numerical performance of the proposed preconditioners and highlight some theoretical properties of the preconditioned matrices. [ABSTRACT FROM AUTHOR]

A mixed optimal model approximation is presented to obtain reduced-order models for truly fast descriptor systems. By a projection from truly fast descriptor systems to discrete-time systems, a mixed optimal model approximation for truly fast descriptor systems is transformed to a mixed optimal model approximation of the corresponding discrete-time systems. The structure of the fast descriptor systems is preserved in the model approximation procedure. The expression of the error and its gradient are given explicitly in terms of the solutions of certain Lyapunov equations. A numerical example is provided to illustrate the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]