On Axelsson's perturbations

A nice perturbation technique was introduced by Axelsson and further developed by Gustafsson to prove that factorization iterative methods are able, under appropriate conditions, to reach a convergence rate larger by an order of magnitude than that of classical schemes. Gustafsson observed however that the perturbations introduced to prove this result seemed actually unnecessary to reach it in practice. In the present work, on the basis of eigenvalue bounds recently obtained by the author, we offer an alternative approach which brings a partial confirmation of Gustafsson's conjecture.     

Spectral convergence and nonlinear dynamics of reaction-diffusion equations under perturbations of the domain

In this paper we obtain the continuity of attractors for semilinear parabolic problems with Neumann boundary conditions relatively to perturbations of the domain. We show that, if the perturbations on the domain are such that the convergence of eigenvalues and eigenfunctions of the Neumann Laplacian is granted then, we obtain the upper semicontinuity of the attractors. If, moreover, every equilibrium of the unperturbed problem is hyperbolic we also obtain the continuity of attractors. We also give necessary and sufficient conditions for the spectral convergence of Neumann problems under perturbations of the domain.     

Precise evaluation of a polynomial at a point given in staggered correction format

The problem of the evaluation in floating-point arithmetic of a polynomial with floating-point coefficients at a point which is a finite sum of floating-point numbers is studied. The solution is obtained as an infinite convergent series of floating-point numbers. The algorithm requires a precise scalar product, but this can always be implemented by software in a high-level language without assembly language routines as we indicate. A convergence result is proved under a very weak restriction on the size of the degree of the polynomial in terms of the unit roundoff u; roughly speaking, the degree should not be larger than the square root of (1 + u)(2u). Even in the particular case when the point at which to evaluate the polynomial reduces to one floating-point number, we find a new simplified algorithm among the whole family that the preceding convergence result allows.

This problem occurs, among others, in the convergence of the Newton method to some real root of the given polynomial p. If we simply use the Horner scheme to evaluate the polynomial p in a neighbourhood of the root, in some cases the evaluation will contain no correct digits and will prevent us from getting convergence even to machine accuracy. The convergence of iterative methods, among which the Newton method, with added perturbations was the central theme of my talk given at the ICCAM'92. The second part will appear in a forthcoming paper. These added perturbations can represent for example forward or backward errors occurring in finite-precision computations.

The problem discussed here appears in validating some hypotheses of these general convergence results (see the forthcoming paper).     

Rates of decay and growth of solutions to linear stochastic differential equations with state-independent perturbations

The paper studies the almost sure asymptotic convergence to zero of solutions of perturbed linear stochastic differential equations, where the unperturbed equation has an equilibrium at zero, and all solutions of the unperturbed equation tend to zero, almost surely. The perturbation is present in the drift term, and both drift and diffusion coefficients are state‐dependent. We determine necessary and sufficient conditions for the almost sure convergence of solutions to the equilibrium of the unperturbed equation. In particular, a critical polynomial rate of decay of the perturbation is identified, such that solutions of equations in which the perturbation tends to zero more quickly that this rate are almost surely asymptotically stable, while solutions of equations with perturbations decaying more slowly that this critical rate are not asymptotically stable. As a result, the integrability or convergence to zero of the perturbation is not by itself sufficient to guarantee the asymptotic stability of solutions when the stochastic equation with the perturbing term is asymptotically stable. Rates of decay when the perturbation is subexponential are also studied, as well as necessary and sufficient conditions for exponential stability.     

Variational Convergence of Composed Convex Functions

In this paper we introduce a concept of variational convergence for mappings taking values in order topological vector spaces. This variational convergence notion is shown to be well adapted to the (epi)-convergence of composed convex functions, in the sense that it is preserved after composition with nondecreasing functions. It is proved how this stability result can be applied to the continuity of multipliers under perturbations associated with a family of constrained optimization problems. Other applications are also given.     

A highly sensitive mean-reverting process in finance and the Euler-Maruyama approximations

Empirical studies show that the most successful continuous-time models of the short-term rate in capturing the dynamics are those that allow the volatility of interest changes to be highly sensitive to the level of the rate. However, from the mathematics, the high sensitivity to the level implies that the coefficients do not satisfy the linear growth condition, so we can not examine its properties by traditional techniques. This paper overcomes the mathematical difficulties due to the nonlinear growth and examines its analytical properties and the convergence of numerical solutions in probability. The convergence result can be used to justify the method within Monte Carlo simulations that compute the expected payoff of financial products. For illustration, we apply our results compute the value of a bond with interest rate given by the highly sensitive mean-reverting process as well as the value of a single barrier call option with the asset price governed by this process.     

Wavelet Approximation in Weighted Sobolev Spaces of Mixed Order with Applications to the Electronic Schr?dinger Equation

We study the approximation of functions in weighted Sobolev spaces of mixed order by anisotropic tensor products of biorthogonal, compactly supported wavelets. As a main result, we characterize these spaces in terms of wavelet coefficients, which also enables us to explicitly construct approximations. In particular, we derive approximation rates for functions in exponentially weighted Sobolev spaces discretized on optimized general sparse grids. Under certain regularity assumptions, the rate of convergence is independent of the number of dimensions. We apply these results to the electronic Schr?dinger equation and obtain a convergence rate which is independent of the number of electrons; numerical results for the helium atom are presented.     

A framework for robustness to ambiguity of higher-order beliefs

Standard type spaces induce belief structures defined by precise beliefs. This paper proposes and analyzes simple procedures for constructing perturbations of such belief structures in which beliefs have a degree of ambiguity. Specifically, we construct ambiguous type spaces whose induced (ambiguous) belief hierarchies approximate the standard, precise, belief hierarchies corresponding to the initial type space. Based on a metric that captures the resulting approximation, two alternative procedures to construct such perturbations are introduced, and are shown to yield a simple and intuitive characterization of convergence to the initial unperturbed environment. As a special case, one of these procedures is shown to characterize the set of all finite perturbations. The introduced perturbations and their convergence properties provide conceptual foundations for the analysis of robustness to ambiguity of various solutions concepts, and for various decision rules under ambiguity.     

Stability of semilinear elliptic optimal control problems with pointwise state constraints

A semi-linear elliptic control problems with distributed control and pointwise inequality constraints on the control and the state is considered. The general optimization problem is perturbed by a certain class of perturbations, and we establish convergence of local solutions of the perturbed problems to a local solution of the unperturbed optimal control problem. This class of perturbations include finite element discretization as well as data perturbation such that the theory implies convergence of finite element approximation and stability w.r.t.?noisy data.     

Estimation of a temporally and spatially varying diffusion coefficient in a parabolic system by an augmented Lagrangian technique

Summary In this paper we apply a hybrid method to estimate a temporally and spatially varying diffusion coefficient in a parabolic system. This technique combines the output-least-squares- and the equation error method. The resulting optimization problem is solved by an augmented Lagrangian approach and convergence as well as rate of convergence proofs are provided. The stability of the estimated coefficient with respect to perturbations in the observation is guaranteed.Supported in part by the Fonds zur Förderung der wissenschaftlichen Forschung, Austria, under project S3206. K.K. also acknowledges support through AFOSR-F49620-86-C111     

Convergence and stability analysis of system of partial differential equations under Markovian structural perturbations–i: vector lyapunov–like functions

In this work we consider a system of partial differential equations of parabolic type under Markovian structural perturbations. Sufficient conditions for the stability and the convergence of the solution process of the system are given by developing a block comparison theorems in the context of Lyapunov-like functions. Moreover, an effort has been made to characterize the effects of random structural perturbations. In fact, it has been shown that the random structural perturbations are indeed the stabilizing agents. In addition, examples are given to illustrate the significance of the presented results     

The inexact, inexact perturbed, and quasi-Newton methods are equivalent models

A classical model of Newton iterations which takes into account some error terms is given by the quasi-Newton method, which assumes perturbed Jacobians at each step. Its high convergence orders were characterized by Dennis and Moré [Math. Comp. 28 (1974), 549-560]. The inexact Newton method constitutes another such model, since it assumes that at each step the linear systems are only approximately solved; the high convergence orders of these iterations were characterized by Dembo, Eisenstat and Steihaug [SIAM J. Numer. Anal. 19 (1982), 400-408]. We have recently considered the inexact perturbed Newton method [J. Optim. Theory Appl. 108 (2001), 543-570] which assumes that at each step the linear systems are perturbed and then they are only approximately solved; we have characterized the high convergence orders of these iterates in terms of the perturbations and residuals.

In the present paper we show that these three models are in fact equivalent, in the sense that each one may be used to characterize the high convergence orders of the other two. We also study the relationship in the case of linear convergence and we deduce a new convergence result.

     

Multiscale problems and homogenization for second-order Hamilton–Jacobi equations

We prove a general convergence result for singular perturbations with an arbitrary number of scales of fully nonlinear degenerate parabolic PDEs. As a special case we cover the iterated homogenization for such equations with oscillating initial data. Explicit examples, among others, are the two-scale homogenization of quasilinear equations driven by a general hypoelliptic operator and the n-scale homogenization of uniformly parabolic fully nonlinear PDEs.     

Algebraic domain decomposition solver for linear elasticity

We generalize the overlapping Schwarz domain decomposition method to problems of linear elasticity. The convergence rate independent of the mesh size, coarse-space size, Korn's constant and essential boundary conditions is proved here. Abstract convergence bounds developed here can be used for an analysis of the method applied to singular perturbations of other elliptic problems.     

Stability of Planar Waves in the Allen–Cahn Equation

We study the asymptotic stability of planar waves for the Allen–Cahn equation on ? ⁿ , where n ≥ 2. Our first result states that planar waves are asymptotically stable under any—possibly large—initial perturbations that decay at space infinity. Our second result states that the planar waves are asymptotically stable under almost periodic perturbations. More precisely, the perturbed solution converges to a planar wave as t → ∞. The convergence is uniform in ? ⁿ . Lastly, the existence of a solution that oscillates permanently between two planar waves is shown, which implies that planar waves are not asymptotically stable under more general perturbations.     

Analytic smoothing of geometric maps with applications to KAM theory

We show that finitely differentiable diffeomorphisms which are either symplectic, volume-preserving, or contact can be approximated with analytic diffeomorphisms that are, respectively, symplectic, volume-preserving or contact. We prove that the approximating functions are uniformly bounded on some complex domains and that the rate of convergence, in Cr-norms, of the approximation can be estimated in terms of the size of such complex domains and the order of differentiability of the approximated function. As an application to this result, we give a proof of the existence, the local uniqueness and the bootstrap of regularity of KAM tori for finitely differentiable symplectic maps. The symplectic maps considered here are not assumed either to be written in action-angle variables or to be perturbations of integrable systems. Our main assumption is the existence of a finitely differentiable parameterization of a maximal dimensional torus that satisfies a non-degeneracy condition and that is approximately invariant. The symplectic, volume-preserving and contact forms are assumed to be analytic.     

General random perturbations of hyperbolic and expanding transformations

Small random perturbations of a general form of diffeomorphisms having hyperbolic invariant sets and expanding maps are considered. The convergence of invariant measures of perturbations to the Sinaî-Bowen-Ruelle measure in the case of a hyperbolic attractor and to the smooth invariant measure in the expanding case are proved. The convergence of corresponding entropy characteristics and the approximation of the topological pressure by means of perturbations is considered as well.     

Preservation of the convergence of trajectories under small perturbations of hyperbolic mappings of manifolds with boundary

A theorem is proved regarding the preservation of the convergence of trajectories under small perturbations of hyperbolic mappings, possessing a strict Lyapunov function. This result is applied to some models in population genetics.Translated from Teoriya Funktsii, Funktsional'nyi Analiz i Ikh Prilozheniya, No. 50, pp. 82–86, 1988.