Some comments on the solution of linear equations

Homogeneous, ill-conditioned and singular linear equations are considered and some methods for their solution are described.This research was supported in part by the National Aeronautics and Space Administration Grant No. NGR-33-015-013.     

Orthogonal projection regularization operators

Tikhonov regularization often is applied with a finite difference regularization operator that approximates a low-order derivative. This paper proposes the use of orthogonal projections as regularization operators, e.g., with the same null space as commonly used finite difference operators. Applications to iterative and SVD-based methods for Tikhonov regularization are described. Truncated iterative and SVD methods are also considered. Research of L. Reichel was supported in part by an OBR Research Challenge Grant. Research of F. Sgallari was supported in part by PRIN 2004 grant 2004014411-005.     

Interior-Point Gradient Method for Large-Scale Totally Nonnegative Least Squares Problems

We study an interior-point gradient method for solving a class of so-called totally nonnegative least-squares problems. At each iteration, the method decreases the residual norm along a diagonally-scaled negative gradient direction with a special scaling. We establish the global convergence of the method and present some numerical examples to compare the proposed method with a few similar methods including the affine scaling method.This author was supported in part by DOE/LANL Contract 03891-99-23This author was supported in part by NSF Grant DMS-0442065     

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.     

Analysis of decomposition algorithms via nonlinear splitting functions

This paper provides a structural analysis of decomposition algorithms using a generalization of linear splitting methods. This technique is used to identify explicitly the essential similarities and differences between several classical algorithms. Similar concepts can be used to analyze a large class of multilevel hierarchical structures.This research was supported in part by ONR Contract No. N00014-76-C-0346, in part by the US Department of Energy, Division of Electric Energy Systems, Contract No. ERDA-E(49-18)-2087 at the Massachusetts Institute of Technology, and in part by the Joint Services Electronics Program, Contract No. DAAG-29-78-C-0016.The authors would like to thank Dr. P. Varaiya, University of California at Berkeley, and Dr. D. Bertsekas for their comments and suggestions.     

Finite-termination schemes for solving semi-infinite satisfycing problems

The problem of finding a parameter which satisfies a set of specifications in inequality form is sometimes referred to as the satisfycing problem. We present a family of methods for solving, in a finite number of iterations, satisfycing problems stated in the form of semi-infinite inequalities. These methods range from adaptive uniform discretization methods to outer approximation methods.The research reported herein was sponsored in part by the National Science Foundation Grant ECS-8713334, the Air Force Office of Scientific Research Contract AFOSR-86-0116, and the State of California MICRO Program Grant 532410-19900.     

Schwarz alternating and iterative refinement methods for mixed formulations of elliptic problems,part II: Convergence theory

Summary In this paper we discuss bounds for the convergence rates of several domain decomposition algorithms to solve symmetric, indefinite linear systems arising from mixed finite element discretizations of elliptic problems. The algorithms include Schwarz methods and iterative refinement methods on locally refined grids. The implementation of Schwarz and iterative refinement algorithms have been discussed in part I. A discussion on the stability of mixed discretizations on locally refined grids is included and quantiative estimates for the convergence rates of some iterative refinement algorithms are also derived.Department of Mathematics, University of Wyoming, Laramie, WY 82071-3036. This work was supported in part by the National Science Foundation under Grant NSF-CCR-8903003, while the author was a graduate student at New York University, and in part by NSF Grant ASC 9003002, while the author was a Visiting, Assistant Researcher at UCLA.     

Approximation procedures based on the method of multipliers

In this paper, we consider a method for solving certain optimization problems with constraints, nondifferentiabilities, and other ill-conditioning terms in the cost functional by approximating them by well-behaved optimization problems. The approach is based on methods of multipliers. The convergence properties of the methods proposed can be inferred from corresponding properties of multiplier methods with partial elimination of constraints. A related analysis is provided in this paper.This work was supported in part by the Joint Services Electronics Program (US Army, US Navy, and US Air Force) under Contract No. DAAB-07-72-C-0259, and by the National Science Foundation under Grant No. ENG-74-19332.     

An algorithm for a piecewise linear model of trade and production with negative prices and bankruptcy

The general equilibrium model is approximated as a piecewise linear convex model and solved from the point of view of welfare economics using linear programming and fixed point methods.This research was supported in part by Army Research Office-Durham Contract DAAG-29-74-C-0032, NSF Grant MPS-72-04832-A03, and Energy Research and Development Administration Contract E(04-3)-326 PA#18.     

Superlinear and quadratic convergence of primal-dual interior-point methods for linear programming revisited

Recently, Zhang, Tapia, and Dennis (Ref. 1) produced a superlinear and quadratic convergence theory for the duality gap sequence in primal-dual interior-point methods for linear programming. In this theory, a basic assumption for superlinear convergence is the convergence of the iteration sequence; and a basic assumption for quadratic convergence is nondegeneracy. Several recent research projects have either used or built on this theory under one or both of the above-mentioned assumptions. In this paper, we remove both assumptions from the Zhang-Tapia-Dennis theory.Dedicated to the Memory of Magnus R. Hestenes, 1906–1991This research was supported in part by NSF Cooperative Agreement CCR-88-09615 and was initiated while the first author was at Rice University as a Visiting Member of the Center for Research in Parallel Computation.The authors thank Yinyu Ye for constructive comments and discussions concerning this material.This author was supported in part by NSF Grant DMS-91-02761 and DOE Grant DE-FG05-91-ER25100.This author was supported in part by AFOSR Grant 89-0363, DOE Grant DE-FG05-86-ER25017, and ARO Grant 9DAAL03-90-G-0093.     

Balancing Neumann-Neumann preconditioners for mixed approximations of heterogeneous problems in linear elasticity

Summary. Balancing Neumann-Neumann methods are extented to mixed formulations of the linear elasticity system with discontinuous coefficients, discretized with mixed finite or spectral elements with discontinuous pressures. These domain decomposition methods implicitly eliminate the degrees of freedom associated with the interior of each subdomain and solve iteratively the resulting saddle point Schur complement using a hybrid preconditioner based on a coarse mixed elasticity problem and local mixed elasticity problems with natural and essential boundary conditions. A polylogarithmic bound in the local number of degrees of freedom is proven for the condition number of the preconditioned operator in the constant coefficient case. Parallel and serial numerical experiments confirm the theoretical results, indicate that they still hold for systems with discontinuous coefficients, and show that our algorithm is scalable, parallel, and robust with respect to material heterogeneities. The results on heterogeneous general problems are also supported in part by our theory. Mathematics Subject Classification (1991):65N55, 65N30, 65N35, 65F10, 65Y05This work was supported by a scholarship of CNPq, of the Ministry for Science and Technology of Brazil, under process 201205/97-1. The work was developed in part at MCS/ANL-DOE, under a Givens Research Associate appointment in Summer 2001.This work was supported in part by the National Science Foundation under Grant NSF-CCR-9732208 and in part by MIUR.This work was supported in part by the National Science Foundation under Grants qNSF-CCR-9732208, and in part by the U.S. Department of Energy under contracts DE-FC02-01ER25482 and DE-FG02-92ER25127.     

Extrapolation with spline-collocation methods for two-point boundary-value problems I: Proposals and justifications

Asymptotic expansions for the error in some spline interpolation schemes are used to derive asymptotic expansions for the truncation errors in some spline-collocation methods for two-point boundary-value problems. This raises the possibility of using Richardson extrapolation or iterated deferred corrections to develop efficient high-order algorithms based on low-order collocation in analogy with similar codes based on low-order finite difference methods; some specific such procedures are proposed.This research was supported in part by the United States Office of Naval Research under Contract N00014-67-A-0126-0015.     

On the relation between quadratic termination and convergence properties of minimization algorithms

Summary It is shown that the theory developed in part I of this paper [22] can be applied to some well-known minimization algorithms with the quadratic termination property to prove theirn-step quadratic convergence. In particular, some conjugate gradient methods, the rank-1-methods of Pearson and McCormick (see Pearson [18]) and the large class of rank-2-methods described by Oren and Luenberger [16, 17] are investigated.This work was supported in part at Stanford University, Stanford, California, under Energy Research and Development Administration, Contract E(04-3) 326 PA No. 30, and National Science Foundation Grant DCR 71-01996 A04 and in part by the Deutsche Forschungsgemeinschaft     

On the convergence rate of Newton interior-point methods in the absence of strict complementarity

In the absence of strict complementarity, Monteiro and Wright [7] proved that the convergence rate for a class of Newton interior-point methods for linear complementarity problems is at best linear. They also established an upper bound of 1/4 for the Q ₁-factor of the duality gap sequence when the steplengths converge to one. In the current paper, we prove that the Q ₁ factor of the duality gap sequence is exactly 1/4. In addition, the convergence of the Tapia indicators is also discussed.This author was supported in part by NSF Coop. Agr. No. CCR-8809615 and AFOSR 89-0363 and the REDI Foundation.This author was supported in part by NSF Coop. Agr. No. CCR-8809615, AFOSR 89-0363, DOE DEFG05-86ER25017 and ARO 9DAAL03-90-G-0093.Visiting member of the Center for Research on Parallel Computations, Rice University, Houston, Texas, 77251-1892. This author was supported in part by DOE DE-FG02-93ER25171.     

Knot polynomials and Vassiliev's invariants

Summary A fundamental relationship is established between Jones' knot invariants and Vassiliev's knot invariants. Since Vassiliev's knot invariants have a firm grounding in classical topology, one obtains as a result a first step in understanding the Jones polynomial by topological methods.Oblatum 20-V-1991 & 10-VI-1992Research supported in part by NSF Grant DMS-88-055627.Research supported in part by NSF Grant DMS-90-04017.     

A product-form Cholesky factorization method for handling dense columns in interior point methods for linear programming

Cholesky factorization has become the method of choice for solving the symmetric system of linear equations arising in interior point methods (IPMs) for linear programming (LP), its main advantages being numerical stability and efficiency for sparse systems. However in the presence of dense columns there is dramatic loss of efficiency. A typical approach to remedy this problem is to apply the Sherman-Morrison-Woodbury (SMW) update formula to the dense part. This approach while being very efficient, is not numerically stable. Here we propose using product-form Cholesky factorization to handle dense columns. The proposed approach is essentially as stable as the original Cholesky factorization and nearly as efficient as the SMW approach. We demonstrate these properties both theoretically and computationally. A key part of our theoretical analysis is the proof that the elements of the Cholesky factors of the matrices that arise in IPMs for LP are uniformly bounded as the duality gap converges to zero.The doctoral research of this author was supported in part by an IBM Cooperative FellowshipResearch supported in part by NSF Grants DMS 91-06195, DMS 94-14438, DMS 95-27124, DMS 01-04282 and CDA 97-26385 and DOE Grant DE-FG02-92ER25126     

Relaxed variants of Karmarkar's algorithm for linear programs with unknown optimal objective value

Variants of Karmarkar's algorithm are given for solving linear programs with unknown optimal objective valuez ^*. These new methods combine the approach of Goldfarb and Mehrotra for relaxing the requirement that certain projections be computed exactly with the approach of Todd and Burrell for generating an improving sequence of lower bounds forz ^* using dual feasible solutions. These methods retain the polynomial-time complexity of Karmarkar's algorithm.This research was supported in part by NSF Grants DMS-85-12277 and CDR-84-21402, and ONR Contract N0014-87-K0214.     

On solution-containing ellipsoids in linear programming

Ellipsoids that contain all optimal dual slack solutions and those that contain all optimal primal solutions and that are independent of the algorithm used are derived. Based upon these ellipsoids, two criteria each for detecting optimal basic and nonbasic variables prior to optimality in interior-point methods are obtained. Using these results, we then derive a sufficient condition for a linear program to be feasible.This research was supported in part by NSF Grant DMS-85-12277, DMS-91-06195, CDR-84-21402 and ONR Contract N00014-87-K-0214.     

On iterative methods for solving nonlinear least squares problems over convex sets

Nonlinear least squares problems over convex sets inR ⁿ are treated here by iterative methods which extend the classical Newton, gradient and steepest descent methods and the methods studied recently by Pereyra and the author. Applications are given to nonlinear least squares problems under linear constraint, and to linear and nonlinear inequalities. Part of the research underlying this report was undertaken for the Office of Naval Research, Contract Nonr-1228(10), Project NR047-021, and for the U.S. Army Research Office — Durham, Contract No. DA-31-124-ARO-D-322 at Northwestern University. Reproduction of this paper in whole or in part is permitted for any purpose of the United States Government.     

Diffusion processes with identical bridges

It is shown that diffusions with identical bridges compose classes of stochastic processes for which the methods of calculation of distributions of functionals are identical as well. Some particular classes of such processes are considered. Bibliography: 8 titles.Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 294, 2002, pp. 29–42.This research was supported in part by the Russian Foundation for Basic Research, grants 02-01-00265, 00-15-96019, and 99-01-04027 (joint with DFG).Translated by V. N. Sudakov.