Olavi Nevanlinna

Convergence of iterations for linear equations

1. Motivation, problem and notation.- 1.1 Motivation.- 1.2 Problem formulation.- 1.3 Usual tools.- 1.4 Notation for polynomial acceleration.- 1.5 Minimal error and minimal residual.- 1.6 Approximation of the solution operator.- 1.7 Location of zeros.- 1.8 Heuristics.- Comments to Chapter 1.- 2. Spectrum, resolvent and power boundedness.- 2.1 The spectrum.- 2.2 The resolvent.- 2.3 The spectral mapping theorem.- 2.4 Continuity of the spectrum.- 2.5 Equivalent norms.- 2.6 The Yosida approximation.- 2.7 Power bounded operators.- 2.8 Minimal polynomials and algebraic operators.- 2.9 Quasialgebraic operators.- 2.10 Polynomial numerical hull.- Comments to Chapter 2.- 3. Linear convergence.- 3.1 Preliminaries.- 3.2 Generating functions and asymptotic convergence factors.- 3.3 Optimal reduction factor.- 3.4 Greens function for G?.- 3.5 Optimal polynomials for.- 3.6 Simply connected G?(L).- 3.7 Stationary recursions.- 3.8 Simple examples.- Comments to Chapter 3.- 4. Sublinear convergence.- 4.1 Introduction.- 4.2 Convergence of Lk(L?1).- 4.3 Splitting into invariant subspaces.- 4.4 Uniform convergence.- 4.5 Nonisolated singularity and successive approximation.- 4.6 Nonisolated singularity and polynomial acceleration.- 4.7 Fractional powers of operators.- 4.8 Convergence of iterates.- 4.9 Convergence with speed.- Comments to Chapter 4.- 5. Superlinear convergence.- 5.1 What is superlinear.- 5.2 Introductory examples.- 5.3 Order and type.- 5.4 Finite termination.- 5.5 Lower and upper bounds for optimal polynomials.- 5.6 Infinite products.- 5.7 Almost algebraic operators.- 5.8 Estimates using singular values.- 5.9 Multiple clusters.- 5.10 Approximation with algebraic operators.- 5.11 Locally superlinear implies superlinear.- Comments to Chapter 5.- References.- Definitions.

Convergence of dynamic iteration methods for initial value problems

In VLSI-simulation there has recently been interest in an iterative technique called the waveform relaxation method. In this paper we set up a theoretical framework to analyze the convergence of such methods. We restrict the discussion to linear systems and do not consider the effects of time discretization, but assume that the initial value problems are solved exactly.

On resolvent conditions and stability estimates

As many numerical processes for time discretization of evolution equations can be formulated as analytic mappings of the generator, they can be represented in terms of the resolvent. To obtain stability estimates for time discretizations, one therefore would like to carry known estimates on the resolvent back to the time domain. For different types of bounds of the resolvent of a linear operator, bounds for the norm of the powers of the operator and for their sum are given. Under similar bounds for the resolvent of the generator, some new stability bounds for one-step and multistep discretizations of evolution equations are then obtained.

Remarks on Picard-Lindelo¨f iteration: part II

The paper discusses Picard-Lindelof iteration for systems of autonomous linear equations on finite intervals, as well as its numerical variants. Most of the discussion is under a model assumption which roughly says that the coupling terms are of moderate size compared with the slow time scales in the problem. It is shown that the speed of convergence is quite independent of the step sizes already for very large time steps. This makes it possible to design strategies in which the mesh gets gradually refined during the iteration in such a way that the iteration error stays essentially on the level of discretization error.

Remarks on Picard-Lindelöf iteration: Part II

The paper discusses Picard-Lindelöf iteration for systems of autonomous linear equations on finite intervals, as well as its numerical variants. Most of the discussion is under a model assumption which roughly says that the coupling terms are of moderate size compared with the slow time scales in the problem. It is shown that the speed of convergence is quite independent of the step sizes already for very large time steps. This makes it possible to design strategies in which the mesh gets gradually refined during the iteration in such a way that the iteration error stays essentially on the level of discretization error.

Accelerating with rank-one updates

Abstract Consider the iteration xk + 1 = xk + H(b> − Axk) for solving Ax = b (A is n x n nonsingular). We discuss rank-one updates to improve H as an approximation to A−1 during the iteration. The update kills and reduces singular values of I − AH and thus speeds up the convergence. The algorithm terminates after at most n sweeps, and if all n sweeps are needed, then A−1 has been computed.

On the growth of the resolvent operators for power bounded operators

Outline. In this paper I discuss some quantitative aspects related to power bounded operators T and to the decay of T (T − 1). For background I refer to two recent surveys J. Zemánek [1994], C. J. K. Batty [1994]. Here I try to complement these two surveys in two different directions. First, if the decay of T (T − 1) is as fast as O(1/n) then quite strong conclusions can be made. The situation can be thought of as a discrete version of analytic semigroups; I try to motivate this in Section 1 by demonstrating the similarity and lack of it between power boundedness of T and uniform boundedness of et(cT−1) where c is a constant of modulus 1 and t > 0. Section 2 then contains the main result in this direction. I became interested in studying the quantitative aspects of the decay of T (T − 1) since it can be used as a simple model for what happens in the early phase of an iterative method (O. Nevanlinna [1993]). Secondly, the so called Kreiss matrix theorem relates bounds for the powers to bounds for the resolvent. The estimate is proportional to the dimension of the space and thus has as such no generalization to operators. However, qualitatively such a result holds in Banach spaces e.g. for Riesz operators: if the resolvent satisfies the resolvent condition, then the operator is power bounded (but without an estimate). I introduce in Section 3 a growth function for bounded operators. This allows one to obtain a result of the form: if the resolvent condition holds and if the growth function is finite at 1, then the powers are bounded and can be estimated. In Section 4 in addition to the Kreiss matrix theorem, two other applications of the growth function are given.

Multiplier techniques for linear multistep methods

A theory is developed for the fixed-h stability of integration schemes based on A(α)-stable formulas when applied to nonlinear parabolic-like stiff equations. The theory is based on a general multiplier technique whose properties we fully develop. Assuming that a few simply checkable criteria which are related to monotonicity properties of the nonlinearity around the computed solution are satisfied, we obtain various error bounds and boundedness results for that solution. Some practical implications of the theory are also given.

Stability of Two-Step Methods for Variable Integration Steps

Two of the most commonly used methods, the trapezoidal rule and the two-step backward differentiation method, both have drawbacks when applied to difficult stiff problems. The trapezoidal rule does not sufficiently damp the stiff components and the backward differentiation method is unstable for certain stable variable-coefficient problems with variable-steps. In this paper we show that there exists a one-parameter family of two-step, second-order one-leg methods which are stable for any dissipative nonlinear system and for any test problem of the form

Linear acceleration of Picard-Lindelöf iteration

\dot x = \lambda (t)x