SOLVING SYMMETRIC MONOTONE LINEAR VARIATIONAL INEQUALITIES BY SOME MODIFIED LEVITIN-POLYAK PROJECTION METHODS

Some modified Levitin-Polyak projection methods are proposed in this paper for solving monotone linear variational inequality x∈Ω，（x′-x)^T(Hx c)≤0，for any x′∈Ω.It is pointed out that there are similar methods for solving a general linear variational inequality.     

Inertial manifolds and linear multi-step methods

We determine the existence and C ¹ convergence of an inertial manifold for a strongly A(α) stable, pth order, p≧1, linear multi-step method approximating a sectorial evolution equation that satisfies a gap condition. This inertial manifold gives rise to a one-step method that C ¹ approximates the inertial form of the evolution equation and yields further approximation properties of the multi-step method. This revised version was published online in June 2006 with corrections to the Cover Date.     

Asynchronous block-iterative primal-dual decomposition methods for monotone inclusions

Mathematical Programming - We propose new primal-dual decomposition algorithms for solving systems of inclusions involving sums of linearly composed maximally monotone operators. The principal...     

Maximal monotone differential inclusions with memory

In this paper we study maximal monotone differential inclusions with memory. First we establish two existence theorems; one involving convex-valued orientor fields and the other nonconvex valued ones. Then we examine the dependence of the solution set on the data that determine it. Finally we prove a relaxation theorem.     

Continuous first-order methods for monotone inclusions in a Hilbert space

Equations in a Hilbert space that involve multivalued monotone mappings are examined. Solutions to such equations are understood in the inclusion sense. A continuous first-order method and its regularized version are constructed on the basis of the resolvent of the maximal monotone operator, and sufficient conditions for them to converge strongly are obtained.     

Asymptotic mean-square stability of linear multi-step methods for SODEs

In this article we present results of a linear stability analysis of stochastic linear multi-step methods for stochastic ordinary differential equations. As in deterministic numerical analysis we use a linear time-invariant test equation and study when the numerical approximation shares asymptotic properties in the mean-square sense of the exact solution of that test equation. Sufficient conditions for asymptotic mean-square stability of stochastic linear two-step-Maruyama methods are obtained with the aide of Lyapunov-type functionals. In particular we study the asymptotic mean-square stability of stochastic counterparts of two-step Adams-Bashforth- and Adams-Moulton-methods and the BDF method. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Improved linear multi-step methods for stochastic ordinary differential equations

We consider linear multi-step methods for stochastic ordinary differential equations and study their convergence properties for problems with small noise or additive noise. We present schemes where the drift part is approximated by well-known methods for deterministic ordinary differential equations. In previous work, we considered Maruyama-type schemes, where only the increments of the driving Wiener process are used to discretize the diffusion part. Here, we suggest the improvement of the discretization of the diffusion part by also taking into account mixed classical-stochastic integrals. We show that the relation of the applied step sizes to the smallness of the noise is essential in deciding whether the new methods are worthwhile. Simulation results illustrate the theoretical findings.     

Differential inclusions with nonstationary maximal monotone operators

Institute for Control Problems. Translated from Funktsional'nyi Analiz i Ego Prilozheniya, Vol. 24, No. 4, pp. 14–24, October–December, 1990.     

Systems of differential inclusions with maximal monotone terms

In this paper, we establish the existence of solutions to systems of second order differential inclusions with maximal monotone terms. Our proofs rely on the theory of maximal monotone operators and the Schauder degree theory. A notion of solution-tube to these problems is introduced. This notion generalizes the notion of upper and lower solutions of second order differential equations.     

Interior hybrid proximal extragradient methods for the linear monotone complementarity problem

We present new infeasible path-following methods for linear monotone complementarity problems based on Auslender, Teboulle and Ben-Tiba’s log-quadratic barrier functions. The central paths associated with these barriers are always well defined and, for those problems which have a solution, convergent to a pair of complementary solutions. Starting points in these paths are easy to compute. The theoretical iteration-complexity of these new path-following methods is derived and improved by a strategy which uses relaxed hybrid proximal-extragradient steps to control the quadratic term. Encouraging preliminary numerical experiments are presented.     

On linear monotone iteration and Schwarz methods for nonlinear elliptic PDEs

Summary The Schwarz Alternating Method can be used to solve elliptic boundary value problems on domains which consist of two or more overlapping subdomains. The solution is approximated by an infinite sequence of functions which results from solving a sequence of elliptic boundary value problems in each subdomain. In this paper, proofs of convergence of some Schwarz Alternating Methods for nonlinear elliptic problems which are known to have solutions by the monotone method (also known as the method of subsolutions and supersolutions) are given. In particular, an additive Schwarz method for scalar as well some coupled nonlinear PDEs are shown to converge to some solution on finitely many subdomains, even when multiple solutions are possible. In the coupled system case, each subdomain PDE is linear, decoupled and can be solved concurrently with other subdomain PDEs. These results are applicable to several models in population biology. This work was in part supported by a grant from the RGC of HKSAR, China (HKUST6171/99P)     

Spline monotone approximation with linear differential operators

Let f∈C³[a,b] and L be a linear differential operator such that L(f)≥0. Then there exists a sequence Q_n, n≥1, of polynomial splines with equally spaced knots, such that Q^(r), approximates f^(r), 0≤r≤s, simultaneously in the uniform norm. This approximation is given through inequalities with rates, involving a measure of smoothness to f^(s); so that L (Q_n)≥0. The encountered cases are the continuous, periodic and discrete.     

Non-convex perturbations of maximal monotone differential inclusions

We give an existence result for \(\dot x \in -- Ax + F(x)\) whereA is a maximal monotone map andF is a set-valued map, with images not necessarily convex.     

Systems of first order differential inclusions with maximal monotone terms

In this paper, we establish the existence of solutions to systems of first order differential inclusions with maximal monotone terms satisfying the periodic boundary condition. Our proofs rely on the theory of maximal monotone operators, and the Schauder and the Kakutani fixed point theorems. A notion of solution-tube to these problems is introduced. This notion generalizes the notion of upper and lower solutions of first order differential equations.     

Periodic solutions of monotone differential inclusions with fast and slow variables

We study the solvability of a periodic problem for monotone differential inclusions and the behavior of its solutions as the parameter changes.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 5, pp. 694–703, May, 1993.     

Underlying paths in interior point methods for the monotone semidefinite linear complementarity problem

An interior point method defines a search direction at each interior point of the feasible region. The search directions at all interior points together form a direction field, which gives rise to a system of ordinary differential equations (ODEs). Given an initial point in the interior of the feasible region, the unique solution of the ODE system is a curve passing through the point, with tangents parallel to the search directions along the curve. We call such curves off-central paths. We study off-central paths for the monotone semidefinite linear complementarity problem (SDLCP). We show that each off-central path is a well-defined analytic curve with parameter μ ranging over (0, ∞) and any accumulation point of the off-central path is a solution to SDLCP. Through a simple example we show that the off-central paths are not analytic as a function of and have first derivatives which are unbounded as a function of μ at μ = 0 in general. On the other hand, for the same example, we can find a subset of off-central paths which are analytic at μ = 0. These “nice” paths are characterized by some algebraic equations. This research was done during the author’s PhD study at the Department of Mathematics, NUS and as a Research Engineer at the NUS Business School.     

On nonconvex perturbations of maximal monotone differential inclusions

The paper is concerned with the evolution inclusionxAx+F(t,x), whereA generates a contractive semigroup andF is a lower semicontinuous multifunction. Constructing a suitable directionally continuous selection fromF, we prove the existence of solutions on a closed domain and the connectedness of the set of trajectories.     

Lower semicontinuous perturbations of maximal monotone differential inclusions

We prove existence of solutions to 1 $$\dot x \in - Ax + F\left( {t,x} \right),x\left( a \right) = x^0 ,$$ whereA is a maximal monotone operator inR ⁿ andF is a multifunction measurable in (t, x) and l.s.c. inx, satisfying a sublinear growth condition.     

Mean-square convergence of stochastic multi-step methods with variable step-size

We study mean-square consistency, stability in the mean-square sense and mean-square convergence of drift-implicit linear multi-step methods with variable step-size for the approximation of the solution of Itô stochastic differential equations. We obtain conditions that depend on the step-size ratios and that ensure mean-square convergence for the special case of adaptive two-step-Maruyama schemes. Further, in the case of small noise we develop a local error analysis with respect to the h$h$–ε$\epsilon$ approach and we construct some stochastic linear multi-step methods with variable step-size that have order 2 behaviour if the noise is small enough.