On the matrix completion problem for multivariate filter bank construction

We survey the main techniques for the construction of multivariate filter banks and present new results about special matrices of order four and eight suitable for their construction. Qiuhui Chen: Supported in part by NSFC under grant 10201034 and project-sponsored by SRF for ROCS, SEM. Charles A. Micchelli: Supported in part by the US National Science Foundation under grant CCR-0407476. Yuesheng Xu: All correspondence to this author. Supported in part by the US National Science Foundation under grant CCR-0407476, by the Natural Science Foundation of China under grant 10371122 and by the Chinese Academy of Sciences under the program “One Hundred Distinguished Chinese Young Scientists”.     

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”.     

Multilevel augmentation methods for differential equations

We develop multilevel augmentation methods for solving differential equations. We first establish a theoretical framework for convergence analysis of the boundary value problems of differential equations, and then construct multiscale orthonormal bases in H₀^m(0,1) spaces. Finally, the multilevel augmentation methods in conjunction with the multiscale orthonormal bases are applied to two-point boundary value problems of both second-order and fourth-order differential equations. Theoretical analysis and numerical tests show that these methods are computationally stable, efficient and accurate. Dedicated to Professor Charles A. Micchelli on the occasion of his 60th birthday with friendship and esteem. Mathematics subject classifications (2000) 65J15, 65R20. Zhongying Chen: Supported in part by the Natural Science Foundation of China under grants 10371137 and 10201034, the Foundation of Doctoral Program of National Higher Education of China under grant 20030558008, Guangdong Provincial Natural Science Foundation of China under grant 1011170 and the Foundation of Zhongshan University Advanced Research Center. Yuesheng Xu: Corresponding author. Supported in part by the US National Science Foundation under grants 9973427 and 0312113, by NASA under grant NCC5-399, by the Natural Science Foundation of China under grant 10371122 and by the Chinese Academy of Sciences under the program of “One Hundred Distinguished Young Scientists”.     

Weighted median algorithms for L1 approximation

The weighted median problem arises as a subproblem in certain multivariate optimization problems, includingL ₁ approximation. Three algorithms for the weighted median problem are presented and the relationships between them are discussed. We report on computational experience with these algorithms and on their use in the context of multivariateL ₁ approximation.This work was supported in part by National Science Foundation Grant CCR-8713893 and in part by a grant from The City University of New York PSC-CUNY Research Award program.     

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.     

Vertical decomposition of arrangements of hyperplanes in four dimensions

We show that, for any collection ℋ ofn hyperplanes in ℜ⁴, the combinatorial complexity of thevertical decomposition of the arrangementA(ℋ) of ℋ isO(n ⁴ logn). The proof relies on properties of superimposed convex subdivisions of 3-space, and we also derive some other results concerning them. Work on this paper by Leonidas Guibas and Micha Sharir has been supported by a grant from the U.S.-Israeli Binational Science Foundation. Work by Leonidas Guibas was also supported by National Science Foundation Grant CCR-9215219. Work by Micha Sharir was also supported by National Science Foundation Grant CCR-91-22103, and by grants from the G.I.F.—the German Isreali Foundation for Scientific Research and Development, and the Fund for Basic Research administered by the Israeli Academy of Sciences. Work by Jiří Matouŝek was done while he was visiting Tel Aviv University, and its was partially supported by a Humboldt Research Fellowship. Work on this paper by Dan Halperin was carried out while he was at Tel Aviv University.     

Complementarity and nondegeneracy in semidefinite programming

Primal and dual nondegeneracy conditions are defined for semidefinite programming. Given the existence of primal and dual solutions, it is shown that primal nondegeneracy implies a unique dual solution and that dual nondegeneracy implies a unique primal solution. The converses hold if strict complementarity is assumed. Primal and dual nondegeneracy assumptions do not imply strict complementarity, as they do in LP. The primal and dual nondegeneracy assumptions imply a range of possible ranks for primal and dual solutionsX andZ. This is in contrast with LP where nondegeneracy assumptions exactly determine the number of variables which are zero. It is shown that primal and dual nondegeneracy and strict complementarity all hold generically. Numerical experiments suggest probability distributions for the ranks ofX andZ which are consistent with the nondegeneracy conditions. Supported in part by the U.S. National Science Foundation grant CCR-9625955. Supported in part by U.S. National Science Foundation grant CCR-9501941 and the U.S. Office of Naval Research grant N00014-96-1-0704. Supported in part by U.S. National Science Foundation grant CCR-9401119.     

On better heuristics for Steiner minimum trees

Finding a shortest network interconnecting a given set of points in a metric space is called the Steiner minimum tree problem. The Steiner ratio is the largest lower bound for the ratio between lengths of a Steiner minimum tree and a minimum spanning tree for the same set of points. In this paper, we show that in a metric space, if the Steiner ratio is less than one and finding a Steiner minimum tree for a set of size bounded by a fixed number can be performed in polynomial time, then there exists a polynomialtime heuristic for the Steiner minimum tree problem with performance ratio bigger than the Steiner ratio. It follows that in the Euclidean plane, there exists a polynomial-time heuristic for Steiner minimum trees with performance ratio bigger than . This solves a long-standing open problem.Part of this work was done while this author visited the Department of Computer Science, Princeton University, supported in part by DIMACS (Center for Discrete Mathematics and Theoretical Computer Science), a National Science Foundation Science and Technology Center, under NSF grant STC88-09648, supported in part by NSF grant No. CCR-8920505, and also supported in part by the National Natural Science Foundation of China.     

On the Surrogate Gradient Algorithm for Lagrangian Relaxation

When applied to large-scale separable optimization problems, the recently developed surrogate subgradient method for Lagrangian relaxation (Zhao et al.: J. Optim. Theory Appl. 100, 699–712, 1999) does not need to solve optimally all the subproblems to update the multipliers, as the traditional subgradient method requires. Based on it, the penalty surrogate subgradient algorithm was further developed to address the homogenous solution issue (Guan et al.: J. Optim. Theory Appl. 113, 65–82, 2002; Zhai et al.: IEEE Trans. Power Syst. 17, 1250–1257, 2002). There were flaws in the proofs of Zhao et al., Guan et al., and Zhai et al.: for problems with inequality constraints, projection is necessary to keep the multipliers nonnegative; however, the effects of projection were not properly considered. This note corrects the flaw, completes the proofs, and asserts the correctness of the methods. This work is supported by the NSFC Grant Nos. 60274011, 60574067, the NCET program (No. NCET-04-0094) of China. The third author was supported in part by US National Science Foundation under Grants ECS-0323685 and DMI-0423607.     

A note on the existence of the Alizadeh-Haeberly-Overton direction for semidefinite programming

This note establishes a new sufficient condition for the existence and uniqueness of the Alizadeh-Haeberly-Overton direction for semidefinite programming. The work of these authors was based on research supported by the National Science Foundation under grants INT-9600343 and CCR-970048 and the Office of Naval Research under grant N00014-94-1-0340.     

A B-spline approach for empirical mode decompositions

We propose an alternative B-spline approach for empirical mode decompositions for nonlinear and nonstationary signals. Motivated by this new approach, we derive recursive formulas of the Hilbert transform of B-splines and discuss Euler splines as spline intrinsic mode functions in the decomposition. We also develop the Bedrosian identity for signals having vanishing moments. We present numerical implementations of the B-spline algorithm for an earthquake signal and compare the numerical performance of this approach with that given by the standard empirical mode decomposition. Finally, we discuss several open mathematical problems related to the empirical mode decomposition. Dedicated to Professor Charles A. Micchelli on the occasion of his 60th birthday with friendship and esteem Mathematics subject classification (2000) 94A12. Supported in part by National Aeronautics and Space Administration under grant NAG5-5364, and National Science Foundation under grants NSF0314742 and NSF0312113. Yuesheng Xu: Corresponding author. Supported in part by Natural Science Foundation of China under grant 10371122.     

Quadrature formulae on half-infinite intervals

We develop two classes of quadrature rules for integrals extended over the positive real axis, assuming given algebraic behavior of the integrand at the origin and at infinity. Both rules are expressible in terms of Gauss-Jacobi quadratures. Numerical examples are given comparing these rules among themselves and with recently developed quadrature formulae based on Bernstein-type operators.Work supported, in part, by the National Science Foundation under grant CCR-8704404.     

An algorithm for nonlinear optimization using linear programming and equality constrained subproblems

This paper describes an active-set algorithm for large-scale nonlinear programming based on the successive linear programming method proposed by Fletcher and Sainz de la Maza [10]. The step computation is performed in two stages. In the first stage a linear program is solved to estimate the active set at the solution. The linear program is obtained by making a linear approximation to the ₁ penalty function inside a trust region. In the second stage, an equality constrained quadratic program (EQP) is solved involving only those constraints that are active at the solution of the linear program. The EQP incorporates a trust-region constraint and is solved (inexactly) by means of a projected conjugate gradient method. Numerical experiments are presented illustrating the performance of the algorithm on the CUTEr [1, 15] test set.This author was supported by Air Force Office of Scientific Research grant F49620-00-1-0162, Army Research Office Grant DAAG55-98-1-0176, and National Science Foundation grant INT-9726199.This author was supported in part by the EPSRC grant GR/R46641.These authors were supported by National Science Foundation grants CCR-9987818, ATM-0086579 and CCR-0219438 and Department of Energy grant DE-FG02-87ER25047-A004.Report OTC 2002/4, Optimization Technology CenterTo Roger Fletcher, with respect and admiration     

Necessary and Sufficient Conditions for Weak No-Arbitrage in Securities Markets with Frictions

In this paper we consider a financial market model with frictions which include transaction costs, bid-ask spread and taxes. By using optimization, linear and nonlinear programming and convex programming techniques, several necessary and sufficient conditions are derived for the weak no-arbitrage. Some results on state prices are also provided. The results of this paper can provide at least some theoretical insight to the problem. This research was supported in part by a grant of the National Excellent Ph.D. Thesis Project of China (No. 200267), a grant of the National Natural Science Foundation of China (No. 10171115), a “Tenth Five-Year Plan” project of Ministry of Education of China (No. 01JA630009), a grant of the Natural Science Foundation of Guangdong Province (No. 011193), two grants from the Research Grants Council of Hong Kong (CityU 1081/02E, HKU 7139/01H), and a research grant of the University of Hong Kong.     

On the Characterization of Preinvex Functions

In Refs. [J. Math. Anal. Appl. 258:287–308, [2001]; J. Math. Anal. Appl. 256:229–241, [2001]], Yang and Li presented a characterization of preinvex functions and semistrictly preinvex functions under a certain set of conditions. In this note, we show that the same results or even more general ones can be obtained under weaker assumptions; we also give a characterization of strictly preinvex functions under mild conditions. This research was supported by the National Natural Science Foundation of China under Grants 70671064 and 60673177, and the Education Department Foundation of Zhejiang Province Grant 20070306. The authors thank Professor F. Giannessi for valuable comments on the original version of this paper.     

Collusive game solutions via optimization

A Nash-based collusive game among a finite set of players is one in which the players coordinate in order for each to gain higher payoffs than those prescribed by the Nash equilibrium solution. In this paper, we study the optimization problem of such a collusive game in which the players collectively maximize the Nash bargaining objective subject to a set of incentive compatibility constraints. We present a smooth reformulation of this optimization problem in terms of a nonlinear complementarity problem. We establish the convexity of the optimization problem in the case where each player's strategy set is unidimensional. In the multivariate case, we propose upper and lower bounding procedures for the collusive optimization problem and establish convergence properties of these procedures. Computational results with these procedures for solving some test problems are reported. It is with great honor that we dedicate this paper to Professor Terry Rockafellar on the occasion of his 70th birthday. Our work provides another example showing how Terry's fundamental contributions to convex and variational analysis have impacted the computational solution of applied game problems. This author's research was partially supported by the National Science Foundation under grant ECS-0080577. This author's research was partially supported by the National Science Foundation under grant CCR-0098013.     

Local and parallel finite element algorithms for the stokes problem

Based on two-grid discretizations, some local and parallel finite element algorithms for the Stokes problem are proposed and analyzed in this paper. These algorithms are motivated by the observation that for a solution to the Stokes problem, low frequency components can be approximated well by a relatively coarse grid and high frequency components can be computed on a fine grid by some local and parallel procedure. One technical tool for the analysis is some local a priori estimates that are also obtained in this paper for the finite element solutions on general shape-regular grids. Y. He was partially subsidized by the NSF of China 10671154 and the National Basic Research Program under the grant 2005CB321703; A. Zhou was partially supported by the National Science Foundation of China under the grant 10425105 and the National Basic Research Program under the grant 2005CB321704; J. Li was partially supported by the NSF of China under the grant 10701001. J. Xu was partially supported by Alexander von Humboldt Research Award for Senior US Scientists, NSF DMS-0609727 and NSFC-10528102.     

Tannakian Categories, Linear Differential Algebraic Groups, and Parametrized Linear Differential Equations

We provide conditions for a category with a fiber functor to be equivalent to the category of representations of a linear differential algebraic group. This generalizes the notion of a neutral Tannakian category used to characterize the category of representations of a linear algebraic group [18], [9]. This work was partially supported by NSF grant CCR-0096842 and by the Russian Foundation for Basic Research, project no. 05-01-00671.     

Complementarity formulations and existence of solutions of dynamic multi-rigid-body contact problems with coulomb friction

In this paper, we study the problem of predicting the acceleration of a set of rigid, 3-dimensional bodies in contact with Coulomb friction. The nonlinearity of Coulomb's law leads to a nonlinear complementarity formulation of the system model. This model is used in conjunction with the theory of quasi-variational inequalities to prove for the first time that multi-rigid-body systems with all contacts rolling always has a solution under a feasibility-type condition. The analysis of the more general problem with sliding and rolling contacts presents difficulties that motivate our consideration of a relaxed friction law. The corresponding complementarity formulations of the multi-rigid-body contact problem are derived and existence of solutions of these models is established. The research of this author was based on work supported by the National, Science Foundation under grants DDM-9104078 and CCR-9213739. The research of this author was partially supported by the National Science Foundation under grant IRI-9304734, by the Texas Advanced Research Program grant 999903-078, and by the Texas Advanced Technology Program under grant 999903-095.     

The superconvergence of Newton–Cotes rules for the Hadamard finite-part integral on an interval

We study the general (composite) Newton–Cotes rules for the computation of Hadamard finite-part integral with the second-order singularity and focus on their pointwise superconvergence phenomenon, i.e., when the singular point coincides with some a priori known point, the convergence rate is higher than what is globally possible. We show that the superconvergence rate of the (composite) Newton–Cotes rules occurs at the zeros of a special function and prove the existence of the superconvergence points. Several numerical examples are provided to validate the theoretical analysis. The work of J. Wu was partially supported by the National Natural Science Foundation of China (No. 10671025) and a grant from the Research Grants Council of the Hong Kong Special Administrative Region, China (No. CityU 102507). The work of W. Sun was supported in part by a grant from the Research Grants Council of the Hong Kong Special Administrative Region, China (No. City U 102507) and the National Natural Science Foundation of China (No. 10671077).