The treatment of nonhomogeneous Dirichlet boundary conditions by thep-version of the finite element method

Summary The paper addresses the problem of the implementation of nonhomogeneous essential Dirichlet type boundary conditions in thep-version of the finite element method.Partially supported by the Office of Naval Research under Grant N-00014-85-K-0169Research partially supported by the Air Force Office of Scientific Research, Air Force Systems Command, USAF, under Grant Number AFOSR 85-0322     

Interpolation problems of Pick-Nevanlinna and Loewner types for meromorphic matrix functions: Parametrization of the set of all solutions

We consider a general matrix version of a Pick-Loewner interpolation problem on the closed unit disk. Solutions are allowed to have a finite numberl of free poles in the open disk. We show that the smallestl for which a solution to the problem exists is the number of negative eigenvalues of an appropriately defined Pick matrix, and for this value ofl we obtain a linear fractional map parametrization of the class of all solutions. The idea is to adapt the Grassmannian approach involving Krein space geometry and invariant subspace representations of the authors; this was successful previously for the case where all interpolating points are inside the disk. Also an appendix includes an errata to earlier work together with simplified proofs.Research was supported in part by the National Science Foundation.Research was supported in part by the National Science Foundation, Office of Naval Research and the Air Force Office of Scientific Research.     

Identifying an optimal basis in linear programming

We propose a sufficient condition that allows an optimal basis to be identified from a central path point in a linear programming problem. This condition can be applied when there is a gap in the sorted list of slack values. Unlike previously known conditions, this condition is valid for real-number data and does not involve the number of bits in the data.This work is supported 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. Also supported in part by an NSF Presidential Young Investigator Award with matching funds received from AT&T and the Xerox Corporation. Part of this work was carried out while the author was visiting the Sandia National Laboratories, supported by the U.S. Department of Energy under Contract DE-AC04-76DP00789.The author is supported in part by NSF Grant DDM-9207347. Part of this work was carried out while the author was on a sabbatical leave from the University of Iowa and visiting the Cornell Theory Center, Cornell University, Ithaca, NY 14853, supported in part by the Cornell Center for Applied Mathematics and by the Advanced Computing Research Institute, a unit of the Cornell Theory Center, which receives major funding from the National Science Foundation and the IBM Corporation, with additional support from New York State and members of its Corporate Research Institute.     

Stable processes: moving averages versus fourier transforms

Summary We study the relationship between Fourier transforms and moving averages of stable processes. The two classes have a large intersection, however as soon as the noise is required to be bounded they become disjoint.Research supported by the Air Force Office of Scientific Research Contract No. F49620 85C 0144 and by the Office of Naval Research Grant No. N00014 86C 0227This work was done in part when the author was visiting the Center for Computational Statistics, George Mason University, Fairfax, VA 22030-4444     

Nonlinear programming and stationary equilibria in stochastic games

Stationary equilibria in discounted and limiting average finite state/action space stochastic games are shown to be equivalent to global optima of certain nonlinear programs. For zero sum limiting average games, this formulation reduces to a linear objective, nonlinear constraints program, which finds the best stationary strategies, even when-optimal stationary strategies do not exist, for arbitrarily small. The work of the first author was supported in part by the Air Force Office of Scientific Research, and by the National Science Foundation under Grant No ECS-8704954.The work of the third author was supported by The Netherlands Organization for Scientific Research NWO, project 10-64-10.     

Polynomial dual network simplex algorithms

We show how to use polynomial and strongly polynomial capacity scaling algorithms for the transshipment problem to design a polynomial dual network simplex pivot rule. Our best pivoting strategy leads to an O(m ² logn) bound on the number of pivots, wheren andm denotes the number of nodes and arcs in the input network. If the demands are integral and at mostB, we also give an O(m(m+n logn) min(lognB, m logn))-time implementation of a strategy that requires somewhat more pivots.Research supported by AFOSR-88-0088 through the Air Force Office of Scientific Research, by NSF grant DOM-8921835 and by grants from Prime Computer Corporation and UPS.Research supported by NSF Research Initiation Award CCR-900-8226, by U.S. Army Research Office Grant DAAL-03-91-G-0102, and by ONR Contract N00014-88-K-0166.Research supported in part by a Packard Fellowship, an NSF PYI award, a Sloan Fellowship, and by the National Science Foundation, the Air Force Office of Scientific Research, and the Office of Naval Research, through NSF grant DMS-8920550.     

Linear stochastic singular control problems

The explicit feedback control law for the singular linear-quadratic-gaussian stochastic control problem is derived. The interesting implication of the control law in terms of information pattern is discussed.The research reported in this document was done while the author was a Guggenheim Fellow at the Imperial College, London, England, and was supported in part by the U.S. Army Research Office, the U.S. Air Force Office of Scientific Research, and the U.S. Office of Naval Research under the Joint Services Electronics Program, Contracts Nos. N00014-67-A-0298-0006, 0005, and 0008. The author is indebted to various discussions with D. Q. Mayne and J. M. C. Clark of Imperial College which clarified several points.     

A martingale method for the convergence of a sequence of processes to a jump-diffusion process

Summary A convenient method for proving weak convergence of a sequence of non-Markovian processesx (·) to a jump-diffusion process is proved. Basically, it is shown that the limit solves the martingale problem of Strook and Varadhan. The proofs are relatively simple, and the conditions apparently weaker than required by other current methods (in particular, for limit theorems for a sequence of ordinary differential equations with random right hand sides). In order to illustrate the relative ease of applicability in many cases, a simpler proof of a known result on averaging is given.Brown University, Divisions of Applied Mathematics and Engineering and Lefschetz Center for Dynamical Systems. Research supported in part by the Air Force Office of Scientific Research under AFOSR AF-76-3063, by the National Science Foundation under NSF Eng. 73-03846A03 and in part by the Office of Naval Research ONR N00014-76-C-0279 P003     

Approximations and computational methods for Optimal Stopping and Stochastic Impulsive Control problems

The paper treats a computational method for the Optimal Stopping and Stochastic Impulsive Control problem for a diffusion. In the latter problem control acts only intermittently since there is a basic positive transaction cost to be paid at each instant that the control acts. For eachh > 0, a controlled Markov chain is constructed, whose continuous time interpolations are a natural approximation to the diffusion, for both the optimal stopping and impulsive control situations. The solutions to the optimal stopping and impulsive control problems for the chains are relatively easy to obtain by using standard procedures, and they converge to the solutions of the corresponding problems for the diffusion models ash0.Research was supported in part by the Air Force Office of Scientific Research AF-AFOSR 71-2078C, in part by the National Science Foundation GK 40493X and in part by the Office of Naval Research NONR N00014-67-A-0191-001804.     

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.     

Geometric convergence to e−z by rational functions with real poles

Summary In this paper, we show that there exists a sequence of rational functions of the formR _n(z)=p_n–1(z)/(1+z/n)ⁿ,n=1, 2, ..., with degp _n–1n–1, which converges geometrically toe ^–z in the uniform norm on [0, +), as well as on some infinite sector symmetric about the positive real axis. We also discuss the usefulness of such rational functions in approximating the solutions of heat-conduction type problems.Research supported in part by the Air Force Office of Scientific Research under Grant AFOSR-74-2688, and by the University of South Florida Research Council.Research supported in part by the Air Force Office of Scientific Research under Grant AFOSR-74-2729, and by the Energy Research and Development Administration (ERDA) under Grant E(11-1)-2075.     

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.     

Some connections between excursion theory and the discrete Schrödinger equation with random potentials

Summary We use here discrete excursion theory, and the Ray-Knight theorem, in order to study questions concerning the analyticity of the density of states at low disorder, as well as its smoothness properties, in one dimension.Research supported in part by the grants NSF-MCS-82-01599 and DAAG6-84-K-0155Research partially supported by grant AFOSR-85-0017 from the Air Force Office of Scientific Research     

On the average cost optimality equation and the structure of optimal policies for partially observable Markov decision processes

We consider partially observable Markov decision processes with finite or countably infinite (core) state and observation spaces and finite action set. Following a standard approach, an equivalent completely observed problem is formulated, with the same finite action set but with anuncountable state space, namely the space of probability distributions on the original core state space. By developing a suitable theoretical framework, it is shown that some characteristics induced in the original problem due to the countability of the spaces involved are reflected onto the equivalent problem. Sufficient conditions are then derived for solutions to the average cost optimality equation to exist. We illustrate these results in the context of machine replacement problems. Structural properties for average cost optimal policies are obtained for a two state replacement problem; these are similar to results available for discount optimal policies. The set of assumptions used compares favorably to others currently available.This research was supported in part by the Advanced Technology Program of the State of Texas, in part by the Air Force Office of Scientific Research under Grant AFOSR-86-0029, in part by the National Science Foundation under Grant ECS-8617860, and in part by the Air Force Office of Scientific Research (AFSC) under Contract F49620-89-C-0044.     

Degrees of freedom versus dimension for containment orders

Given a family of sets L, where the sets in L admit k degrees of freedom, we prove that not all (k+1)-dimensional posets are containment posets of sets in L. Our results depend on the following enumerative result of independent interest: Let P(n, k) denote the number of partially ordered sets on n labeled elements of dimension k. We show that log P(n, k)nk log n where k is fixed and n is large.Research supported in part by Allon Fellowship and by a grant from Bat Sheva de Rothschild Foundation.Research supported in part by the Office of Naval Research, contract number N00014-85-K0622.     

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.     

Geometric convergence of rational approximations toe −z in infinite sectors

Summary In this paper, we study the geometric convergence of rational approximations toe ^–z in infinite sectors symmetric about the positive real axis.Research supported in part by the Air Force Office of Scientific Research under Grant AFOSR-74-2688, and by the University of South Florida Research CouncilResearch supported in part by the Air Force Office of Scientific Research under Grant AFOSR-74-2729, and by the Atomic Energy Commission under Grant AT (11-1)-2075     

Conditions for finite moments of waiting times inG/G/1 queues

We find conditions for E(W ) to be finite whereW is the stationary waiting time random variable in a stableG/G/1 queue with dependent service and inter-arrival times.Supported in part by KBN under grant 640/2/9, and at the Center for Stochastic Processes, Department of Statistics at the University of North Carolina Chapel Hill by the Air Force Office of Scientific Research Grant No. 91-0030 and the Army Research Office Grant No. DAAL09-92-G-0008.     

Symbolic dynamics and nonlinear semiflows

Summary For a transverse homoclinic orbit of a mapping (not necessarily invertible) on a Banach space, it is shown that the mapping restricted to orbits near is equivalent to the shift automorphism on doubly infinite sequences on finitely many symbols. Implications of this result for the Poincaré map of semiflows are given.This work was supported by the Air Force Office of Scientific Research under Grant #81-0198, by the National Science Foundation under Grant #MCS-8205355 and by the Army Research Office under Grant ù DAAG-29-83-K-0029.     

Solution manifolds and submanifolds of parametrized equations and their discretization errors

Summary The paper concerns solution manifolds of nonlinear parameterdependent equations (1)F(u, )=y₀ involving a Fredholm operatorF between (infinite-dimensional) Banach spacesX=Z× andY, and a finitedimensional parameter space . Differntial-geometric ideas are used to discuss the connection between augmented equations and certain onedimensional submanifolds produced by numerical path-tracing procedures. Then, for arbitrary (finite) dimension of , estimates of the error between the solution manifold of (1) and its discretizations are developed. These estimates are shown to be applicable to rather general nonlinear boundaryvalue problems for partial differential equations.This work was in part supported by the U.S. Air Force Office of Scientific Research under Grant 80-0176, the National Science Foundation under Grant MCS-78-05299, and the Office of Naval Research under Contract N-00014-80-C-0455