Random Operator Equations Research Articles

Newton's method for random operator equations

Newton's method for random operator equations

Solution of random operator equations by random step-contractors

Solution of random operator equations by random step-contractors

Stochastic projectional schemes for random linear operator equations of the first and second kinds

In this paper we develop stochastic projectional schemes for constructing measurable approximations to solutions or least-squares solutions of linear random operator equations. Both operators with closed range and with non-closed range are considered.

Random nonlinear equations and monotonic nonlinearities

The question of existence of random solutions of nonlinear random operator equations involving a monotonic nonlinearity is discussed.

Measurability of Generalized Inverses of Random Linear Operators

Let $( {\Omega ,\mathcal{B}} )$ be a measurable space, ${\rm X},Y$ be separable Hilbert spaces. Let T be a random linear operator from $\Omega \times {\rm X}$ into Y. Let $T^\dag ( \omega )$ denote the generalized inverse of $T( \omega )$, for $\omega \in \Omega $. Questions of measurability of $T^\dag ( \omega )$ are investigated in this paper, and in particular the following results are established: (i) If T is bounded, then $T^\dag $ is a random operator. (ii) If T is a closed operator with dense domain and if $T^\dag $ is bounded, then $T^\dag $ is a random operator under some mild restriction on the domains of $T( \omega )$ and $T^ * ( \omega ),\omega \in \Omega $. The results are applied to the measurability of best approximate solutions of random linear operator equations.

Addendum: Linear Random Operator Equations in Mathematical Physics. III

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Linear Random Operator Equations in Mathematical Physics. II

This paper is a continuation of Paper I [J. Math. Phys. 11, 1069 (1970)]. A general expression is determined for the stochastic Green's function (SGF) for two-point correlation functions, and various useful relationships are determined between the stochastic Green's functions for various statistical measures and between the stochastic Green's functions, random Green's functions, and ordinary Green's functions. In the author's dissertation and earlier papers, SGF was given as an ensemble average of the product of random Green's functions. This random Green's function is now specified in terms of an ordinary Green's function for a deterministic operator and a resolvent kernel which can be calculated for the random part of the stochastic operator. Hence that SGF is determinable which yields the desired statistical measure of the solution process directly. Second, the two-point correlation function of the solution process is found for the perturbation case. It is also demonstrated that, in the event that perturbation theory is adequate to deal with the randomness involved, the correct two-point correlation of the solution process is easily specialized from the general expression, i.e., the results of perturbation theory are obtained from the SGF of Adomian when perturbation theory is applicable.

Linear Random Operator Equations in Mathematical Physics. III

The problem of wave motion in a stochastic medium is treated as an application of stochastic operator theory and as a generalization of Papers I and II (and previous work by the author) to the case of partial differential equations and random fields without monochromaticity assumptions and closure approximations. Connections to the theory of partial coherence are considered. The stochastic Green's function for the two-point correlation of the solution process can be determined so the correlation can be obtained. Spectral spreading in a ``hot'' medium is easily demonstrable and can be calculated.

Random Operator Equations in Mathematical Physics. I

For stochastic differential equations arising in physical problems, the objectives, limitations, and restrictive assumptions of the various methods are studied and some promising new methods are derived which eliminate various limitations and allow treatment of a wide class of applications in physics. (Among these, will be an adequate treatment of the propagation of an electromagnetic wave in a random continuum or a random d'Alembertian operator without assumptions of ``small randomness'' and other restrictions.)