Potential analysis for a class of diffusion equations: A Gaussian bounds approach

We axiomatically develop a potential analysis for a general class of hypoelliptic diffusion equations under the following basic assumptions: doubling condition and segment property for an underlying distance and Gaussian bounds of the fundamental solution. Our analysis is principally aimed to obtain regularity criteria and uniform boundary estimates for the Perron-Wiener solution to the Dirichlet problem. As an example of application, we also derive an exterior cone criterion of boundary regularity and scale-invariant Harnack inequality and Hölder estimate for an important class of operators in non-divergence form with Hölder continuous coefficients, modeled on Hörmander vector fields.     

Pointwise solution bounds for a class of singular diffusion problems in physiology

A numerical technique for obtaining pointwise bounds for the solution of a class of nonlinear boundary-value problems in physiology is presented. Simple analytic bounding functions are obtained using an integral representation for the solution. The computations are performed in interval arithmetic, thus obtaining lower and upper bounds simultaneously. The oxygen diffusion problem in spherical cells and a nonlinear heat-conduction model of the human head are presented as illustrative examples. For these examples, the present technique is computationally more efficient than the existing ones in that it yields sharper bounds with fewer integration steps.     

On lower bounds for a class of quadratic 0, 1 programs

We present a new method of obtaining lower bounds for a class of quadratic 0, 1 programs that includes the quadratic assignment problem. The method generates a monotonic sequence of lower bounds and may be interpreted as a Lagrangean dual ascent procedure. We report on a computational comparison of our bounds with earlier work in [2] based on subgradient techniques.     

A priori bounds for a class of semi-linear degenerate elliptic equations

In this paper,we mainly discuss a priori bounds of the following degenerate elliptic equation,a ij(x)■ij u+b i(x)■iu+f(x,u)=0,in ΩRn,(*)where aij■iφ■jφ=0 on■Ω,andφis the defining function of ■Ω.Imposing suitable conditions on the coefficients and f(x,u),one can get the L∞-estimates of(*)via blow up method.     

A class of problems where dual bounds beat underestimation bounds

We investigate the problem of minimizing a nonconvex function with respect to convex constraints, and we study different techniques to compute a lower bound on the optimal value: The method of using convex envelope functions on one hand, and the method of exploiting nonconvex duality on the other hand. We investigate which technique gives the better bound and develop conditions under which the dual bound is strictly better than the convex envelope bound. As a byproduct, we derive some interesting results on nonconvex duality.     

Analytic-numerical solutions with a priori error bounds for a class of strongly coupled mixed partial differential systems

This paper deals with the construction of analytic-numerical solutions with a priori error bounds for systems of the type u_t = Au_xx, u(0,t) + u_x(0,t) = 0, Bu(1,t) + Cu_x(1,t) = 0, 0 < x < 1, t > 0, u(x,0) = f(x). Here A, B, C are matrices for which no diagonalizable hypothesis is assumed. First an exact series solution is obtained after solving appropriate vector Sturm-Liouville-type problems. Given an admissible error ε and a bounded subdomain D, after appropriate truncation an approximate solution constructed in terms of data and approximate eigenvalues is given so that the error is less than the prefixed accuracy ε, uniformly in D.     

Renewal theorem for a class of stationary sequences

Summary A renewal theorem is obtained for stationary sequences of the form _n=(...,X _n-1,X _n,X _n+1...), whereX _n, , are i.i.d. r.v.s. valued in a Polish space. This class of processes is sufficiently broad to encompass functionals of recurrent Markov chains, functionals of stationary Gaussian processes, and functionals of one-dimensional Gibbs states. The theorem is proved by a new coupling construction.Research supported by the National Science Foundation     

Strong approximation for a class of stationary processes

Strong approximation for sums of a class of stationary processes with optimal bound is established. The main tools are m$m$-dependent approximation and block techniques. Some previous results are improved.     

A mixed-step algorithm for the approximation of the stationary regime of a diffusion

In some recent papers, some procedures based on some weighted empirical measures related to decreasing-step Euler schemes have been investigated to approximate the stationary regime of a diffusion (possibly with jumps) for a class of functionals of the process. This method is efficient but needs the computation of the function at each step. To reduce the complexity of the procedure (especially for functionals), we propose in this paper to study a new scheme, called the mixed-step scheme, where we only keep some regularly time-spaced values of the Euler scheme. Our main result is that, when the coefficients of the diffusion are smooth enough, this alternative does not change the order of the rate of convergence of the procedure. We also investigate a Richardson–Romberg method to speed up the convergence and show that the variance of the original algorithm can be preserved under a uniqueness assumption for the invariant distribution of the “duplicated” diffusion, condition which is extensively discussed in the paper. Finally, we conclude by giving sufficient “asymptotic confluence” conditions for the existence of a smooth solution to a discrete version of the associated Poisson equation, condition which is required to ensure the rate of convergence results.     

A property of eigenvalue bounds for a class of symmetric tridiagonal interval matrices

The eigenvalue bounds of interval matrices are often required in some mechanical and engineering fields. In this paper, we consider an interval eigenvalue problem with symmetric tridiagonal matrices. A theoretical result is obtained that under certain assumptions the upper and lower bounds of interval eigenvalues of the problem must be achieved just at some vertex matrices of the interval matrix. Then a sufficient condition is provided to guarantee the assumption to be satisfied. The conclusion is illustrated also by a numerical example. Copyright © 2010 John Wiley & Sons, Ltd.     

Karush-Kuhn-Tucker systems: regularity conditions,error bounds and a class of Newton-type methods

 We consider optimality systems of Karush-Kuhn-Tucker (KKT) type, which arise, for example, as primal-dual conditions characterizing solutions of optimization problems or variational inequalities. In particular, we discuss error bounds and Newton-type methods for such systems. An exhaustive comparison of various regularity conditions which arise in this context is given. We obtain a new error bound under an assumption which we show to be strictly weaker than assumptions previously used for KKT systems, such as quasi-regularity or semistability (equivalently, the R ₀-property). Error bounds are useful, among other things, for identifying active constraints and developing efficient local algorithms. We propose a family of local Newton-type algorithms. This family contains some known active-set Newton methods, as well as some new methods. Regularity conditions required for local superlinear convergence compare favorably with convergence conditions of nonsmooth Newton methods and sequential quadratic programming methods. Received: December 10, 2001 / Accepted: July 28, 2002 Published online: February 14, 2003 Key words. KKT system – regularity – error bound – active constraints – Newton method Mathematics Subject Classification (1991): 90C30, 65K05     

Periodic orbits for a class ofC 1 three-dimensional systems

We studyC ¹ perturbations of a reversible polynomial differential system of degree 4 in\(\mathbb{R}^3 \). We introduce the concept of strongly reversible vector field. If the perturbation is strongly reversible, the dynamics of the perturbed system does not change. For non-strongly reversible perturbations we prove the existence of an arbitrary number of symmetric periodic orbits. Additionally, we provide a polynomial vector field of degree 4 in\(\mathbb{R}^3 \) with infinitely many limit cycles in a bounded domain if a generic assumption is satisfied.     

A class of bounds for convex bodies in Hilbert space

We extend to infinite dimensions a class of bounds forL _p metrics of finite-dimensional convex bodies. A generalization to arbitrary increasing convex functions is done simultaneously. The main tool is the use of Gaussian measure to effect a normalization for varying dimension. At a point in the proof we also invoke a strong law of large numbers for random sets to produce a rotational averaging.Supported in part by ONR Grant N0014-90-J-1641 and NSF Grant DMS-9002665.     

On lower bounds for errors of prior density estimators

Lower bounds for errors of prior density estimators are presented for a wide class of conditional densities of observations. Translated fromStatisticheskie Metody Otsenivaniya i Proverki Gipotez, pp. 77–88, Perm, 1993.     

Density estimation for a class of stationary nonlinear processes

Let {X _t ;t∈ℤ be a strictly stationary nonlinear process of the formX _t =ε _t +∑ _r=1 ^∞ W _rt , whereW _rt can be written as a functiong _r (ε _t−1,...ε _t-r-q ), {ε _t ;t∈ℤ is a sequence of independent and identically distributed (i.i.d.) random variables withE|ε₁| ^g < ∞ for some γ>0 andq≥0 is fixed integer. Under certain mild regularity conditions ofg _r and {ε _t } we then show thatX ₁ has a density functionf and that the standard kernel type estimator baded on a realization {X ₁,...,X _n } from {X _t } is, asymptotically, normal and converges a.s. tof(x) asn→∞. The research of this author was partially carried out while he was a research scholar, on a sabbatical leave, at the Department of Statistics and Probability, Michigan State University.