Additive Schwarz Methods for Elliptic Mortar Finite Element Problems

Summary. Two variants of the additive Schwarz method for solving linear systems arising from the mortar finite element discretization on nonmatching meshes of second order elliptic problems with discontinuous coefficients are designed and analyzed. The methods are defined on subdomains without overlap, and they use special coarse spaces, resulting in algorithms that are well suited for parallel computation. The condition number estimate for the preconditioned system in each method is proportional to the ratio H/h, where H and h are the mesh sizes, and it is independent of discontinuous jumps of the coefficients. For one of the methods presented the choice of the mortar (nonmortar) side is independent of the coefficients.This work has been supported in part by the Norwegian Research Council, grant 113492/420This work has been supported in part by the National Science Foundation, grant NSF-CCR-9732208 and in part by the Polish Science Foundation, grant 2P03A02116 Mathematics Subject Classification (2000):65N55     

A Neumann-Neumann algorithm for a mortar discretization of elliptic problems with discontinuous coefficients

Summary. Neumann-Neumann algorithm have been well developed for standard finite element discretization of elliptic problems with discontinuous coefficients. In this paper, an algorithm of this kind is designed and analyzed for a mortar finite element discretization of problems in three dimensions. It is established that its rate of convergence is independent of the discretization parameters and jumps of coefficients between subregions. The algorithm is well suited for parallel computations.Mathematics Subject Classification (1991): 65N55, 65N10, 65N30, 65N22.The work was supported in part by the U.S. Department of Energy under contract DE-FG02-92ER25127 and in part by Polish Science Foundation under grant 2P03A00524.AcknowledgmentThe author would like to thank Olof Widlund for many fruitful discussions and valuable remarks and suggestions on how to improve the presentation of our results.     

A Dual-Primal FETI method for incompressible Stokes equations

In this paper, a dual-primal FETI method is developed for incompressible Stokes equations approximated by mixed finite elements with discontinuous pressures. The domain of the problem is decomposed into nonoverlapping subdomains, and the continuity of the velocity across the subdomain interface is enforced by introducing Lagrange multipliers. By a Schur complement procedure, the solution of an indefinite Stokes problem is reduced to solving a symmetric positive definite problem for the dual variables, i.e., the Lagrange multipliers. This dual problem is solved by the conjugate gradient method with a Dirichlet preconditioner. In each iteration step, both subdomain problems and a coarse level problem are solved by a direct method. It is proved that the condition number of this preconditioned dual problem is independent of the number of subdomains and bounded from above by the square of the product of the inverse of the inf-sup constant of the discrete problem and the logarithm of the number of unknowns in the individual subdomains. Numerical experiments demonstrate the scalability of this new method. This work is based on a doctoral dissertation completed at Courant Institute of Mathematical Sciences, New York University. This work was supported in part by the National Science Foundation under Grants NSF-CCR-9732208, and in part by the U.S. Department of Energy under contract DE-FG02-92ER25127.     

An analysis of discontinuous Galerkin methods for elliptic problems

We study global and local behaviors for three kinds of discontinuous Galerkin schemes for elliptic equations of second order. We particularly investigate several a posteriori error estimations for the discontinuous Galerkin schemes. These theoretical results are applied to develop local/parallel and adaptive finite element methods, based on the discontinuous Galerkin methods. Dedicated to Dr. Charles A. Micchelli on the occasion of his 60th birthday with friendship and esteem Mathematics subject classifications (2000) 65N12, 65N15, 65N30. Aihui Zhou: Subsidized by the Special Funds for Major State Basic Research Projects, and also partially supported by National Science Foundation of China. Reinhold Schneider: Supported in part by DFG Sonderforschungsbereich SFB 393. Yuesheng Xu: Correspondence author. Supported in part by the US National Science Foundation under grants DMS-9973427 and CCR-0312113, by Natural Science Foundation of China under grant 10371122 and by the Chinese Academy of Sciences under program “Hundreds Distinguished Young Chinese Scientists”.     

Quadratic programming with one negative eigenvalue is NP-hard

We show that the problem of minimizing a concave quadratic function with one concave direction is NP-hard. This result can be interpreted as an attempt to understand exactly what makes nonconvex quadratic programming problems hard. Sahni in 1974 [8] showed that quadratic programming with a negative definite quadratic term (n negative eigenvalues) is NP-hard, whereas Kozlov, Tarasov and Haijan [2] showed in 1979 that the ellipsoid algorithm solves the convex quadratic problem (no negative eigenvalues) in polynomial time. This report shows that even one negative eigenvalue makes the problem NP-hard.This author's work supported by the Applied Mathematical Sciences Program (KC-04-02) of the Office of Energy Research of the U.S. Department of Energy under grant DE-FG02-86ER25013. A000 and in part by the National Science Foundation, the Air Force Office of Scientific Research, and the Office of Naval Research, through NSF grant DMS 8920550.     

Almost optimal interior penalty discontinuous approximations of symmetric elliptic problems on non-matching grids

Summary. We consider an interior penalty discontinuous approximation for symmetric elliptic problems of second order on non-matching grids in this paper. The main result is an almost optimal error estimate for the interior penalty approximation of the original problem based on partitioning of the domain into a finite number of subdomains. Further, an error analysis for the finite element approximation of the penalty formulation is given. Finally, numerical experiments on a series of model second order problems are presented. Mathematics Subject Classification (2000):65F10, 65N20, 65N30The work of the first and the second authors has been partially supported by the National Science Foundation under Grant DMS-9973328. The work of the last author was performed under the auspices of the U. S. Department of Energy by University of California Lawrence Livermore National Laboratory under contract W-7405-Eng-48.     

Adaptive polynomial preconditioning for hermitian indefinite linear systems

This paper explores the use of polynomial preconditioned CG methods for hermitian indefinite linear systems,Ax=b. Polynomial preconditioning is attractive for several reasons. First, it is well-suited to vector and/or parallel architectures. It is also easy to employ, requiring only matrix-vector multiplication and vector addition. To obtain an optimum polynomial preconditioner we solve a minimax approximation problem. The preconditioning polynomial,C(), is optimum in that it minimizes a bound on the condition number of the preconditioned matrix,C(A)A. We also characterize the behavior of this minimax polynomial, which makes possible a thorough understanding of the associated CG methods. This characterization is also essential to the development of an adaptive procedure for dynamically determining the optimum polynomial preconditioner. Finally, we demonstrate the effectiveness of polynomial preconditioning in a variety of numerical experiments on a Cray X-MP/48. Our results suggest that high degree (20–50) polynomials are usually best.This research was supported in part by the Applied Mathematical Sciences subprogram of the Office of Energy Research, U.S. Dept. of Energy, by Lawrence Livermore National Laboratory under contract W-7405-ENG-48.This research was supported in part by the Dept. of Energy and the National Science Foundation under grant DMS 8704169.This research was supported in part by U.S. Dept. of Energy grant DEFG02-87ER25026 and by National Science Foundation grant DMS 8703226.     

Performance Study and Analysis of Parallel Multilevel Preconditioners

The significant gap between peak and realized performance of parallel systems motivates the need for performance analysis. In order to predict the performance of a class of parallel multilevel ILU preconditioner (PBILUM), we build two performance prediction models for both the preconditioner construction phase and the solution phase. These models combine theoretical features of the preconditioners with estimates on computation cost, communications overhead, etc. Experimental simulations show that our model predication based on certain reasonable assumptions is close to the simulation results. The models may be used to predict the performance of this class of parallel preconditioners.*The research work of the authors was supported in part by the U.S. National Science Foundation under grants CCR-9988165, CCR-0092532, ACR-0202934, and ACR-234270, by the U.S. Department of Energy Office of Science under grant DE-FG02-02ER45961, by the Kentucky Science & Engineering Foundation under grant KSEF-02-264-RED-002.     

An analysis of reduced Hessian methods for constrained optimization

We study the convergence properties of reduced Hessian successive quadratic programming for equality constrained optimization. The method uses a backtracking line search, and updates an approximation to the reduced Hessian of the Lagrangian by means of the BFGS formula. Two merit functions are considered for the line search: the ₁ function and the Fletcher exact penalty function. We give conditions under which local and superlinear convergence is obtained, and also prove a global convergence result. The analysis allows the initial reduced Hessian approximation to be any positive definite matrix, and does not assume that the iterates converge, or that the matrices are bounded. The effects of a second order correction step, a watchdog procedure and of the choice of null space basis are considered. This work can be seen as an extension to reduced Hessian methods of the well known results of Powell (1976) for unconstrained optimization.This author was supported, in part, by National Science Foundation grant CCR-8702403, Air Force Office of Scientific Research grant AFOSR-85-0251, and Army Research Office contract DAAL03-88-K-0086.This author was supported by the Applied Mathematical Sciences subprogram of the Office of Energy Research, U.S. Department of Energy, under contracts W-31-109-Eng-38 and DE FG02-87ER25047, and by National Science Foundation Grant No. DCR-86-02071.     

A domain decomposition discretization of parabolic problems

In recent years, domain decomposition methods have attracted much attention due to their successful application to many elliptic and parabolic problems. Domain decomposition methods treat problems based on a domain substructuring, which is attractive for parallel computation, due to the independence among the subdomains. In principle, domain decomposition methods may be applied to the system resulting from a standard discretization of the parabolic problems or, directly, be carried out through a discretization of parabolic problems. In this paper, a direct domain decomposition method is introduced to discretize the parabolic problems. The stability and convergence of this algorithm are analyzed. This work was supported in part by Polish Sciences Foundation under grant 2P03A00524. This work was supported in part by the US Department of Energy under Contracts DE-FG02-92ER25127 and by the Director, Office of Science, Advanced Scientific Computing Research, U.S. Department of Energy under contract DE-AC02-05CH11231.     

An analysis of the composite step biconjugate gradient method

Summary The composite step biconjugate gradient method (CSBCG) is a simple modification of the standard biconjugate gradient algorithm (BCG) which smooths the sometimes erratic convergence of BCG by computing only a subset of the iterates. We show that 2×2 composite steps can cure breakdowns in the biconjugate gradient method caused by (near) singularity of principal submatrices of the tridiagonal matrix generated by the underlying Lanczos process. We also prove a best approximation result for the method. Some numerical illustrations showing the effect of roundoff error are given.The work of this author was supported by the Office of Naval Research under contract N00014-89J-1440.The work of this author was supported by the Office of Naval Research under contracts N00014-90-J-1695 and N00014-92-J-1890, the Department of Energy under, contract DE-FG03-87ER25307, the National Science Foundation under contracts ASC 90-03002 and ASC 92-01266, and the Army Research Office under contract DAAL03-91-G-0150. Part of this work was completed during a visit to the Computer Science Dept. The Chinese University of Hong Kong.     

Mixed finite element methods for stationary incompressible magneto–hydrodynamics

Summary. A new mixed variational formulation of the equations of stationary incompressible magneto–hydrodynamics is introduced and analyzed. The formulation is based on curl-conforming Sobolev spaces for the magnetic variables and is shown to be well-posed in (possibly non-convex) Lipschitz polyhedra. A finite element approximation is proposed where the hydrodynamic unknowns are discretized by standard inf-sup stable velocity-pressure space pairs and the magnetic ones by a mixed approach using Nédélecs elements of the first kind. An error analysis is carried out that shows that the proposed finite element approximation leads to quasi-optimal error bounds in the mesh-size. Mathematics Subject Classification (2000):65N30This work was partially supported by the Swiss National Science Foundation under Project 2100-068126.02.     

Approximation algorithms for indefinite quadratic programming

We consider-approximation schemes for indefinite quadratic programming. We argue that such an approximation can be found in polynomial time for fixed andt, wheret denotes the number of negative eigenvalues of the quadratic term. Our algorithm is polynomial in 1/ for fixedt, and exponential int for fixed. We next look at the special case of knapsack problems, showing that a more efficient (polynomial int) approximation algorithm exists.Part of this work was done while the author was visiting Sandia National Laboratories, Albuquerque, New Mexico, supported by the U.S. Department of Energy under contract DE-AC04-76DP00789. Part of this work was also supported by the Applied Mathematical Sciences Program (KC-04-02) of the Office of Energy Research of the U.S. Department of Energy under grant DE-FG02-86ER25013.A000 and in part by the National Science Foundation, the Air Force Office of Scientific Research, and the Office of Naval Research, through NSF grant DMS 8920550.     

Quasi-variational inequalities, generalized Nash equilibria, and multi-leader-follower games

The noncooperative multi-leader-follower game can be formulated as a generalized Nash equilibrium problem where each player solves a nonconvex mathematical program with equilibrium constraints. Two major deficiencies exist with such a formulation: One is that the resulting Nash equilibrium may not exist, due to the nonconvexity in each players problem; the other is that such a nonconvex Nash game is computationally intractable. In order to obtain a viable formulation that is amenable to practical solution, we introduce a class of remedial models for the multi-leader-follower game that can be formulated as generalized Nash games with convexified strategy sets. In turn, a game of the latter kind can be formulated as a quasi-variational inequality for whose solution we develop an iterative penalty method. We establish the convergence of the method, which involves solving a sequence of penalized variational inequalities, under a set of modest assumptions. We also discuss some oligopolistic competition models in electric power markets that lead to multi-leader-follower games.Jong-Shi Pang: The work of this authors research was partially supported by the National Science Foundation under grant CCR-0098013 and ECS-0080577 and by the Office of Naval Research under grant N00014-02-1-0286.Masao Fukushima: The work of this authors research was partially supported by a Grant-in-Aid for Scientific Research from the Ministry of Education, Science, Culture and Sports of Japan.     

A nonconforming finite element method for a two-dimensional curl–curl and grad-div problem

A numerical method for a two-dimensional curl–curl and grad-div problem is studied in this paper. It is based on a discretization using weakly continuous P ₁ vector fields and includes two consistency terms involving the jumps of the vector fields across element boundaries. Optimal convergence rates (up to an arbitrary positive ) in both the energy norm and the L ₂ norm are established on graded meshes. The theoretical results are confirmed by numerical experiments. The work of the first author was supported in part by the National Science Foundation under Grant No. DMS-03-11790 and by the Humboldt Foundation through her Humboldt Research Award. The work of the third author was supported in part by the National Science Foundation under Grant No. DMS-06-52481.     

Dines—Fourier—Motzkin quantifier elimination and an application of corresponding transfer principles over ordered fields

A constructive procedure using Dines—Fourier—Motzkin elimination is given for eliminating quantifiers in a linear first order formula over ordered fields. An ensuing transfer principle is illustrated by showing that a locally one-to-one affine map is globally one-to-one and onto all over ordered fields.This research is based on work supported in part by the National Science Foundation under Grant DMS-86-03232, by the Department of Energy grant DE-FG03-87ER25028 and by the United States-Israel Binational Science Foundation Grant 85-00295.     

A globally and quadratically convergent affine scaling method for linearℓ 1 problems

Recently, various interior point algorithms related to the Karmarkar algorithm have been developed for linear programming. In this paper, we first show how this interior point philosophy can be adapted to the linear ₁ problem (in which there are no feasibility constraints) to yield a globally and linearly convergent algorithm. We then show that the linear algorithm can be modified to provide aglobally and ultimatelyquadratically convergent algorithm. This modified algorithm appears to be significantly more efficient in practise than a more straightforward interior point approach via a linear programming formulation: we present numerical results to support this claim.This paper was presented at the Third SIAM Conference on Optimization, in Boston, April 1989.Research partially supported by the Applied Mathematical Sciences Research Program (KC-04-02) of the Office of Energy Research of the U.S. Department of Energy under grant DE-FG02-86ER25013.A000, by the U.S. Army Research Office through the Mathematical Sciences Institute, Cornell University, and by the Computational Mathematics Program of the National Science Foundation under grant DMS-8706133.Research partially supported by the U.S. Army Research Office through the Mathematical Sciences Institute, Cornell University and by the Computational Mathematics Program of the National Science Foundation under grant DMS-8706133.     

On the convergence of interior-reflective Newton methods for nonlinear minimization subject to bounds

We consider a new algorithm, an interior-reflective Newton approach, for the problem of minimizing a smooth nonlinear function of many variables, subject to upper and/or lower bounds on some of the variables. This approach generatesstrictly feasible iterates by using a new affine scaling transformation and following piecewise linear paths (reflection paths). The interior-reflective approach does not require identification of an activity set. In this paper we establish that the interior-reflective Newton approach is globally and quadratically convergent. Moreover, we develop a specific example of interior-reflective Newton methods which can be used for large-scale and sparse problems.Research partially supported by the Applied Mathematical Sciences Research Program (KC-04-02) of the Office of Energy Research of the U.S. Department of Energy under grant DE-FG02-86ER25013.A000, and in part by NSF, AFOSR, and ONR through grant DMS-8920550, and by the Advanced Computing Research Institute, a unit of the Cornell Theory Center which receives major funding from the National Science Foundation and IBM Corporation, with additional support from New York State and members of its Corporate Research Institute.Corresponding author.     

Practical stabilizability of uncertain dynamical systems: Application to robotic tracking

The tracking problem for a robotic manipulator withn controlled degrees of freedom and uncertain dynamics is considered. Based on the deterministic theory of Refs. 1–15 and requiring only knowledge of bounds on the system uncertainty, a classC of continuous feedback controls (adapted from Ref. 11) is proposed with respect to which the uncertain tracking system is practically stabilizable in the sense that, given a feasible path to be tracked and an arbitrarily small neighborhood of the origin in the appropriate error space, there exists a control inC such that the tracking error for the feedback controlled uncertain system is ultimately bounded with respect to . The theory is illustrated in a numerical example of a robot with two degrees of freedom.This paper is based on research supported by the National Science Foundation and the Air Force Office of Scientific Research under Grant ECS-8210324 and by the UK Science and Engineering Research Council under Grants GR/B/32047 and GR/C/43658. This research was carried out, in part, while G. Leitmann was the recipient of a US Senior Scientist Award of the Alexander von Humboldt Foundation.     

A class of “onto” multifunctions

A convex valued, upper continuous multifunction with pointed closed convex domain is proved to be onto, if there is a selection with certain coercive properties and if the selection for each point lies in the affine hull of the smallest face of the domain containing the point. The proof uses a piecewise affine homotopy construction. We apply the onto theorem to an approximation of a network of servers and show that arbitrary congestion levels can be realized with appropriate arrival levels.This research is based on work supported in part by the National Science Foundation Grant DMS 92-07409, by the Department of Energy Grant DE-FG03-87-ER-25028, by the United States—Israel Binational Science Foundation Grant 85-00295 and by ONR Grant N00014-92-J1142.