Parallel solution of elasticity problems using overlapping aggregations

The finite element (FE) solution of geotechnical elasticity problems leads to the solution of a large system of linear equations. For solving the system, we use the preconditioned conjugate gradient (PCG) method with two-level additive Schwarz preconditioner. The preconditioning is realised in parallel. A coarse space is usually constructed using an aggregation technique. If the finite element spaces for coarse and fine problems on structural grids are fully compatible, relations between elements of matrices of the coarse and fine problems can be derived. By generalization of these formulae, we obtain an overlapping aggregation technique for the construction of a coarse space with smoothed basis functions. The numerical tests are presented at the end of the paper.     

Parallel solution of certain Toeplitz least-squares problems

We describe a systolic algorithm for solving a Toeplitz least-squares problem of special form. Such problems arise, for example, when Volterra convolution equations of the first kind are solved by regularization. The systolic algorithm is based on a sequential algorithm of Eldén, but we show how the storage requirements of Eldén's algorithm can be reduced from O(n²) to O(n). The sequential algorithm takes time O(n²); the systolic algorithm takes time O(n) using a linear systolic array of O(n) cells. We also show how large problems may be decomposed and solved on a small systolic array.     

Parallel solution of variational inequality problems with nonlinear source terms

Current address: Department of Mathematics, University of California at Los Angeles, CA 90095-1555 and Computing Centre, Academia Sinica, People's Republic of China. The work of this author was supported by the Alexander von Humboldt Foundation. Using the finite-element method to discretize the variationalinequality problems related to some free boundary problems withnonlinear source terms, we then solve the resulting algebraicsystems by the combination of the quasi-Newton method and thedomain decomposition techniques. The numerical solution processesare completely parallelizable. Both the convergence of the finite-elementapproximation and of the parallel iterative processes are proved.     

A Lagrange multiplier method for the finite element solution of elliptic interface problems using non-matching meshes

Summary. In this paper we propose a Lagrange multiplier method for the finite element solution of multi-domain elliptic partial differential equations using non-matching meshes. The interface Lagrange multiplier is chosen with the purpose of avoiding the cumbersome integration of products of functions on unrelated meshes (e.g, we will consider global polynomials as multiplier). The ideas are illustrated using Poissons equation as a model, and the proposed method is shown to be stable and optimally convergent. Numerical experiments demonstrating the theoretical results are also presented.Mathematics Subject Classification (2000): 65N30, 65N12     

Numerical aspects in the SGBEM solution of softening cohesive interface problems

Some numerical aspects of a new symmetric Galerkin boundary integral formulation presented in [A. Salvadori, A symmetric boundary integral formulation for cohesive interface problems, Comput. Mech. 32 (2003) 381–391] connected with a like arc-length technique for softening cohesive interface problems introduced in [F. Freddi, Cohesive interface analysis via boundary integral equations, Ph.D. thesis, University of Bologna, Italy, 2004] are here investigated. Further, if the problem is invariant with respect to a finite group of congruences of the Euclidean space , we propose to reduce the computational cost of the symmetric Galerkin boundary element method (SGBEM) matrix evaluation and linear system solution using suitable restriction matrices strictly related to a system of irreducible matrix representation of . Some numerical simulations are presented and discussed.     

Small perturbations of an interface for elastostatic problems

We consider the Lamé system for an elastic medium consisting of an inclusion embedded in a homogeneous background medium. Based on the field expansion method and layer potential techniques, we rigorously derived the asymptotic expansion of the perturbed displacement field because of small perturbations in the interface of the inclusion. We extend these techniques to determine a relationship between traction‐displacement measurements and the shape of the object and derive an asymptotic expansion for the perturbation in the elastic moment tensors because of the presence of small changes in the interface of the inclusion. Copyright © 2016 John Wiley & Sons, Ltd.     

Numerical solution of boundary value problems by using an optimized two-step block method

Numerical Algorithms - This paper aims at the application of an optimized two-step hybrid block method for solving boundary value problems with different types of boundary conditions. The proposed...     

OpTiX-II: A software environment for the parallel solution of nonlinear optimization problems

The intention of the paper is to give an introduction to the OpTiX-II Software Environment, which supports the parallel and distributed solution of mathematical nonlinear programming problems. First, a brief summary of nonsequential solution concepts for nonlinear optimization on multiprocessor systems will be given. The focus of attention will be put on coarse-grained parallelization and its implementation on multi-computer clusters. The conceptual design objectives for the OpTiX-II Software Environment will be presented as well as the implementation on a workstation cluster, a transputer system and a multiprocessor workstation (shared memory). The OpTiX-II system supports the steps from the formulation of nonlinear optimization problems to their solution on networks of (parallel) computers. In order to demonstrate the use of OpTiX-II, the solution of a nonlinear optimization problem from the field of structural design is discussed and some numerical test results are supplied.     

Parallel collocation solution of index-1 BVP-DAEs arising from constrained optimal control problems

The indirect solution of constrained optimal control problems gives rise to two-point boundary value problems (BVPs) that involve index-1 differential-algebraic equations (DAEs) and inequality constraints. This paper presents a parallel collocation algorithm for the solution of these inequality constrained index-1 BVP-DAEs. The numerical algorithm is based on approximating the DAEs using piecewise polynomials on a nonuniform mesh. The collocation method is realized by requiring that the BVP-DAE be satisfied at Lobatto points within each interval of the mesh. A Newton interior-point method is used to solve the collocation equations, and maintain feasibility of the inequality constraints. The implementation of the algorithm involves: (i) parallel evaluation of the collocation equations; (ii) parallel evaluation of the system Jacobian; and (iii) parallel solution of a boarded almost block diagonal (BABD) system to obtain the Newton search direction. Numerical examples show that the parallel implementation provides significant speedup when compared to a sequential version of the algorithm.     

Geometric multigrid algorithms for elliptic interface problems using structured grids

Numerical Algorithms - In this work, we develop geometric multigrid algorithms for the immersed finite element methods for elliptic problems with interface (Chou et al. Adv. Comput. Math. 33,...     

Numerical solution of scattering problems using pseudo-differential impedance operators

Direct numerical solutions of scattering problems based on boundary-integral equations are computationally costly at high frequencies. A numerical method is presented that efficiently computes accurate approximations to unknown surface quantities given known surface data (an approximate Dirichlet-to-Neumann map). The method is based on a pseudo-differential impedance operator (PIO) numerically implemented using rational approximations. An example of a PIO is the so-called on-surface radiation condition (OSRC) method. For a convex obstacle, the method can be viewed as applying a parabolic equation directly on the surface of a scatterer. In contrast to past OSRCs, the use of rational approximations provides accuracy for wide scattering angles which is needed for grazing angles of incidence. Generalization to impedance operators for two-dimensional acoustic scatterers with concave parts is presented. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

The solution of contact problems using boundary element method

The formulation of contact problems is extended to the case of moving punches and to the case when the state of the systems being investigated depends on the history of the change in the external actions. The quasi-static contact problem for a moving rigid rough punch and a single linearly deformable body is considered. A new iterational process is proposed for solving contact problems, taking friction in the contact area into account, and its convergence is proved. An algorithm of the solution, based on the boundary element method, is developed. Solutions of specific problems are given and analysed. Estimates of the difference of the solutions due to the difference in the impenetrability conditions and the difference in the steps of the loading parameter are obtained.     

A direct method for accurate solution and gradient computations for elliptic interface problems

Numerical Algorithms - Recently, an augmented IIM (Li et al. SIAM J. Numer. Anal. 55, 570–597 2017) has been developed and analyzed for some interface problems. The augmented IIM can provide...     

Immersed interface methods for moving interface problems

A second order difference method is developed for the nonlinear moving interface problem of the form $$u_t + \lambda uu_x = \left( {\beta u_x } \right)_x - f\left( {x,t} \right),x \in \left[ {\left. {0,\alpha } \right) \cup \left( {\left. {\alpha ,1} \right]} \right.,} \right.$$ $$\frac{{d\alpha }}{{dt}} = w\left( {t,\alpha ;u,u_x } \right),$$ , where α (t) is the moving interface. The coefficient β(x,t) and the source term f(x,t) can be discontinuous across α (t) and moreover, f(x,t) may have a delta or/and delta-prime function singularity there. As a result, although the equation is parabolic, the solution u and its derivatives may be discontinuous across α (t). Two typical interface conditions are considered. One condition occurs in Stefan-like problems in which the solution is known on the interface. A new stable interpolation strategy is proposed. The other type occurs in a one-dimensional model of Peskin’s immersed boundary method in which only jump conditions are given across the interface. The Crank-Nicolson difference scheme with modifications near the interface is used to solve for the solution u(x,t) and the interface α (t) simultaneously. Several numerical examples, including models of ice-melting and glaciation, are presented. Second order accuracy on uniform grids is confirmed both for the solution and the position of the interface.     

On the iterative solution of KKT systems in potential reduction software for large-scale quadratic problems

Iterative solvers appear to be very promising in the development of efficient software, based on Interior Point methods, for large-scale nonlinear optimization problems. In this paper we focus on the use of preconditioned iterative techniques to solve the KKT system arising at each iteration of a Potential Reduction method for convex Quadratic Programming. We consider the augmented system approach and analyze the behaviour of the Constraint Preconditioner with the Conjugate Gradient algorithm. Comparisons with a direct solution of the augmented system and with MOSEK show the effectiveness of the iterative approach on large-scale sparse problems. Work partially supported by the Italian MIUR FIRB Project Large Scale Nonlinear Optimization, grant no. RBNE01WBBB.     

Numerical solution of singular control problems using multiple shooting techniques

Algorithms for calculating the junction points between optimal nonsingular and singular subarcs of singular control problems are developed. The algorithms consist in formulating appropriate initialvalue and boundary-value problems; the boundary-value problems are solved with the method of multiple shooting. Two examples are detailed to illustrate the proposed numerical methods.The author would like to thank Professor Dr. R. Bulirsch, who stimulated and encouraged this work, which is part of the author's dissertation.     

Parallel complexity of sorting problems

The paper discusses three basic sorting problems using a parallel model, the selection problem, the merging problem and the full sorting problem. The merging problem is completely solved, for the other two upper and lower bounds for the cost functions are derived.     

An approach to the solution of recognition problems using neural networks

Conditions are determined under which, for pattern recognition problems with standard information (Ω-regular problems), a correct algorithm and a six-level spatial neural network reproducing the calculations performed by the correct algorithm can be constructed. The proposed approach to constructing the neural network is not related to the traditional approach based on minimizing a functional.