Mesh Selection for a Nearly Singular Boundary Value Problem

In this paper, we investigate the numerical solution of a model equation u _xx = exp(– ) (and several slightly more general problems) when 1 using the standard central difference scheme on nonuniform grids. In particular, we are interested in the error behaviour in two limiting cases: (i) the total mesh point number N is fixed when the regularization parameter 0, and (ii) is fixed when N. Using a formal analysis, we show that a generalized version of a special piecewise uniform mesh 12 and an adaptive grid based on the equidistribution principle share some common features. And the optimal meshes give rates of convergence bounded by |log()| as 0 and N is given, which are shown to be sharp by numerical tests.     

Almost surely continuous solutions of a nonlinear stochastic integral equation

A nonlinear stochastic integral equation of the Hammerstein type in the formx(t; ) = h(t, x(t; )) + _s k(t, s; )f(s, x(s; ); )d(s) is studied wheret S, a measure space with certain properties, , the supporting set of a probability measure space (,A, P), and the integral is a Bochner integral. A random solution of the equation is defined to be an almost surely continuousm-dimensional vector-valued stochastic process onS which is bounded with probability one for eacht S and which satisfies the equation almost surely. Several theorems are proved which give conditions such that a unique random solution exists. AMS (MOS) subject classifications (1970): Primary; 60H20, 45G99. Secondary: 60G99.     

Directional differentiation of marginal functions in mathematical programming problems

We consider regular mathematical programming problems of the form f(x, y) inf, y F(x), x Rⁿ, where F(x) = {y R^m h_i| (x, y) 0, , h_i (x, y) = 0, . The directional derivatives offunctions (x) = inf{f(x, y)|y F(x)} are estimated.Translated from Kibernetika i Sistemnyi Analiz, No. 6, pp. 70–77, November–December, 1991.     

The existence of solutions,stability, and linearization of volterra systems

LetB be a Banach space ofR _n valued continuous functions on [0, ) withfB. Consider the nonlinear Volterra integral equation (^*)x(t)+ _o ^t K(t,s,x(s))ds. We use the implicit function theorem to give sufficient conditions onB andK (t,s,x) for the existence of a unique solutionxB to (^*) for eachf B with f ^B sufficiently small. Moreover, there is a constantM>0 independent off with Mf^B.Part of this work was done while the author was visiting at Wright State University.     

Duality in Quantum Information Geometry

Let H be a separable Hilbert space. We consider the manifold M consisting of density operators on H such that ^p is of trace class for some p (0, 1). We say M is nearby if there exists C > 1 such that C ^–1C. We show that the space of nearby points to can be furnished with the two flat connections known as the (±)-affine structures, which are dual relative to the BKM metric. We furnish M with a norm making it into a Banach manifold.     

The complexity of computing maximal word functions

Maximal word functions occur in data retrieval applications and have connections with ranking problems, which in turn were first investigated in relation to data compression [21]. By the maximal word function of a languageL ^*, we mean the problem of finding, on inputx, the lexicographically largest word belonging toL that is smaller than or equal tox.In this paper we present a parallel algorithm for computing maximal word functions for languages recognized by one-way nondeterministic auxiliary pushdown automata (and hence for the class of context-free languages).This paper is a continuation of a stream of research focusing on the problem of identifying properties others than membership which are easily computable for certain classes of languages. For a survey, see [24].     

Counting connected components of a semialgebraic set in subexponential time

Let a semialgebraic set be given by a quantifier-free formula of the first-order theory of real closed fields withk atomic subformulae of the typef _i0 for 1ik, where the polynomialsf _i[X ₁,...,X _n] have degrees deg(f _i)<d and the absolute value of each (integer) coefficient off _i is at most 2M. An algorithm is exhibited which counts the number of connected components of the semialgebraic set in time (M (kd)^{n 20})^{O (1)}. Moreover, the algorithm allows us to determine whether any pair of points from the set are situated in the same connected component.     

Multi-time scale zero-sum differential games with perfect state measurements

We obtain necessary and sufficient conditions for the existence of approximate saddle-point solutions in linear-quadratic zero-sum differential games when the state dynamics are defined on multiple (three) time scales. These different time scales are characterized in terms of two small positive parameters ₁, and ₂, and the terminology approximate saddle-point solution is used to refer to saddle-point policies that do not depend on ₁ and ₂, while providing cost levels withinO(₁) of the full-order game. It is shown in the paper that, under perfect state measurements, the original game can be decomposed into three subgames-slow, fast and fastest, the composite saddle-point solution of which make up the approximate saddle-point solution of the original game. Specifically, for the minimizing player, it is necessary to use a composite policy that uses the solutions of all three subgames, whereas for the maximizing player, it is sufficient to use a slow policy. In the finite-horizon case this slow policy could be a feedback policy, whereas in the infinite-horizon case it has to be chosen as an open-loop policy that is generated from the solution and dynamics of the slow subgame. These results have direct applications in theH -optimal control of three-time scale singularly perturbed linear systems under perfect state measurements.Research supported in part by the National Science Foundation under Grant ECS 91-13153, and in part by the U.S. Department of Energy under Grant DE-FG-02-88-ER-13939. This paper is dedicated to the memory of John Breakwell.     

Bounding Perturbations in Zeros of Nonlinear Systems

Algorithms for determining computationally rigorous componentwise bounds on the solutions x R ⁿ of equations F(x, t) = 0 R ^m containing parameters t R ^l due to small perturbations in t when m n and when F is at most twice continuously differentiable in x and in t are described. Numerical results which illustrate the behaviour of the algorithms are presented.     

On a random Volterra integral equation

Summary Tsokos [12] showed the existence of a unique random solution of the random Volterra integral equation (*)x(t; ) = h(t; ) + _o ^t k(t, ; )f(, x(; )) d, where , the supporting set of a probability measure space (,A, P). It was required thatf must satisfy a Lipschitz condition in a certain subset of a Banach space. By using an extension of Banach's contraction-mapping principle, it is shown here that a unique random solution of (*) exists whenf is (, )-uniformly locally Lipschitz in the same subset of the Banach space considered in [12].     

Polynomial-Time Decomposition Algorithms for Support Vector Machines

This paper studies the convergence properties of a general class of decomposition algorithms for support vector machines (SVMs). We provide a model algorithm for decomposition, and prove necessary and sufficient conditions for stepwise improvement of this algorithm. We introduce a simple rate certifying condition and prove a polynomial-time bound on the rate of convergence of the model algorithm when it satisfies this condition. Although it is not clear that existing SVM algorithms satisfy this condition, we provide a version of the model algorithm that does. For this algorithm we show that when the slack multiplier C satisfies 1/2 C mL, where m is the number of samples and L is a matrix norm, then it takes no more than 4LC ² m ⁴/ iterations to drive the criterion to within of its optimum.     

Algorithmic properties of maximal orders in simple algebras over Q

This paper addresses an algorithmic problem related to associative algebras. We show that the problem of deciding if the index of a given central simple algebra over an algebraic number field isd, whered is a given natural number, belongs to the complexity classN P co N P. As consequences, we obtain that the problem of deciding if is isomorphic to a full matrix algebra over the ground field and the problem of deciding if is a skewfield both belong toN P co N P. These results answer two questions raised in [25]. The algorithms and proofs rely mostly on the theory of maximal orders over number fields, a noncommutative generalization of algebraic number theory. Our results include an extension to the noncommutative case of an algorithm given by Huang for computing the factorization of rational primes in number fields and of a method of Zassenhaus for testing local maximality of orders in number fields.     

Fast exponentiation using the truncation operation

Given an integerk, and anarbitrary integer greater than , we prove a tight bound of on the time required to compute with operations {+, –, *, /, ·, }, and constants {0, 1}. In contrast, when the floor operation is not available this computation requires (k) time. Using the upper bound, we give an time algorithm for computing log loga, for alln-bit integersa. This upper bound matches the lower bound for computing this function given by Mansour, Schieber, and Tiwari. To the best of our knowledge these are the first non-constant tight bounds for computations involving the floor operation.     

PAC Learning Axis-aligned Rectangles with Respect to Product Distributions from Multiple-Instance Examples

We describe a polynomial-time algorithm for learning axis-aligned rectangles in Q ^d with respect to product distributions from multiple-instance examples in the PAC model. Here, each example consists of n elements of Q^d together with a label indicating whether any of the n points is in the rectangle to be learned. We assume that there is an unknown product distribution D over Q ^d such that all instances are independently drawn according to D. The accuracy of a hypothesis is measured by the probability that it would incorrectly predict whether one of n more points drawn from D was in the rectangle to be learned. Our algorithm achieves accuracy with probability 1- in O (d⁵ n¹²/²⁰ log² nd/ time.     

Stabilizability of linear systems defined overC *-algebras

LetF andG be elements of aC ^*-algebraA. Assume that, for each irreducible^*-representation ofA on a Hilbert space210B; , there is a bounded linear operatorL B( ) such that the spectrum of(F) –(G)L is contained in the open left half plane. We prove that there is then an elementL A such that the spectrum ofF — GL is contained in the open left half plane. That is, if the system (F, G) is locally stabilizable, then it is stabilizable. We also consider the analogous problem with the open left half plane replaced by the open unit disk.This paper was supported in part by the National Science Foundation under Grant NSF-MCS-8002138.     

Improving known solutions is hard

In this paper, we study the complexity of computing better solutions to optimization problems given other solutions. We use a model of computation suitable for this purpose, the counterexample computation model. We first prove that, if PH _P ³ , polynomial time transducers cannot compute optimal solutions for many problems, even givenn ^1– non-trivial solutions, for any >0. These results are then used to establish sharp lower bounds for several problems in the counterexample model. We extend the model by defining probabilistic counterexample computations and show that our results hold even in the presence of randomness.     

On Bounding Solutions of Underdetermined Systems

Sufficient conditions for the existence and uniqueness of a solution x* D (R ⁿ ) of Y(x) = 0 where : R ⁿ R ^m (m n) with C ²(D) where D R ⁿ is an open convex set and Y = (x)⁺ are given, and are compared with similar results due to Zhang, Li and Shen (Reliable Computing 5(1) (1999)). An algorithm for bounding zeros of f (·) is described, and numerical results for several examples are given.     

Directional derivatives of the maximum function

Directional differentiability of the function (x) = sup{f(x, u), u U} is proved for a class of smooth functions f. The result is applied to study the directional differentiability of the function (x) = sup{f(x, y), y F(x)}, where F is a multivalued mapping.Translated from Kibernetika i Sistemnyi Analiz, No. 2, pp. 171–173, March–April, 1992.     

On the Axioms of Scale Space Theory

We consider alternative scale space representations beyond the well-established Gaussian case that satisfy all reasonable axioms. One of these turns out to be subject to a first order pseudo partial differential equation equivalent to the Laplace equation on the upper half plane {(x, s) ^d × | s > 0}. We investigate this so-called Poisson scale space and show that it is indeed a viable alternative to Gaussian scale space. Poisson and Gaussian scale space are related via a one-parameter class of operationally well-defined intermediate representations generated by a fractional power of (minus) the spatial Laplace operator.     

An Interval Linear Tolerance Problem

This paper is concerned with a tolerance problem for an interval linear system A x = b requiring inner estimation of the admissible solution set {x ⁿ | (A A)(Ax b)} formed by vectors x for which the product Ax remains within b for any possible A A. Methods for verifying the emptiness and nonemptiness of admissible solution sets are developed. Formulas for the dimensions of the interval solution of a tolerance problem with known center are derived.