The maximum entropy method applied to stationary density computation

Abstract

The maximum entropy method (maxent) is widely used in the context of the moment problem which appears naturally in many branches of physics and engineering; it is used to numerically recover the density with least bias from finitely many known moments. We introduce the basic idea behind this method and apply this method to approximating fixed densities of Markov operators in stochastic analysis and Frobenius–Perron operators in ergodic theory of chaotic dynamics.

Introduction

The concept of entropy was first introduced by Clausius into thermodynamics in the middle of the nineteenth century, and later used in a different form by L. Boltzmann in his pioneering work on the kinetic theory of gases in 1866 [9]. It is a measure of the amount of information required to specify the state of a thermodynamic system. The famous Second Law of Thermodynamics says that in an isolated system, the (thermodynamic) entropy never decreases.

A modern concept of entropy was established in information theory by C.E. Shannon in 1948. The Shannon entropy defined for all finite sample spaces with events w₁, w₂, … , w_n with probabilities p₁, p₂, … , p_n (discrete information sources) is

$$H(p_1\text{,}{p}_2\text{,}\dots \text{,}{p}_n)=-\sum _{i=1}^n{p}_i\log p_i$$

which is a measure of uncertainty of an information source.

Based on the Shannon entropy, the Kolmogorov–Sinai–Ornstein entropy defined for measurable transformations of probability spaces was successfully used in solving the problem of isomorphism of dynamical systems. And in 1965 R. Adler, A. Konhein, and M. McAndrew introduced the topological entropy for continuous transformations of compact Hausdorff spaces (see e.g., [5]).

The principle of maximum entropy was introduced in the context of information theory in 1957 by Jaynes. Since his seminal paper [8], the maximum entropy method has been widely used in many areas of science and technology. Basically this numerical scheme is used to numerically recover the required density function with least bias among all the possible candidates which satisfy given constraints. The maximum entropy method has diverse applications in physics and engineering (see [12], [13] and references therein).

In this paper, we will introduce the basic idea of the maximum entropy method and present some of the recent progresses of the method in solving some type of operator equations. In Section 2 we give the formation of the method and study their properties. Another formulation based on the Galerkin projection will also be presented. In Section 3 we apply the method to solve the Markov operator fixed density problem which is important in the stochastic analysis of deterministic dynamical systems. Some error analysis and numerical experiments are contained in Section 4. We conclude in Section 5.