We present a technique for entropy optimization to calculate a distribution from its moments. The technique is based upon maximizing a discretized form of the Shannon entropy functional by mapping the problem onto a dual space where an optimal solution can be constructed iteratively. We demonstrate the performance and stability of our algorithm with several tests on numerically difficult functions. We then consider an electronic structure application, the electronic density of states of amorphous silica and study the convergence of Fermi level with increasing number of moments.PACS numbers: 71.23. Cq, 71.55.Jv, 02.30.Zz One of the fixed themes of physics is the solution of inverse problems. A ubiquitous example in theoretical physics is the "Classical Moment Problem" (CMP), in which only a finite set of power moments of a nonnegative distribution function p is known, and the full distribution is needed [1]. It is obvious that the solution for p is not unique for a finite set of moments. This non-uniqueness suggests the need for a "best guess" for p, based upon the available information. With its ultimate roots in nineteenth century statistical mechanics and a subsequent strong justification based upon probability theory, the "maximum entropy" (maxent) method has provided an extremely successful variational principle to address this type of inverse problem [2]. Collins and Wragg used the maxent method to solve the CMP for a modest number of moments [3]. In a comprehensive paper with seminal applications, Mead and Papanicolaou [4] solved the CMP with maximum entropy techniques and proposed the first practical numerical scheme to solve the moment problem for up to 15 moments. In a host of subsequent papers, the utility of the method as an unbiased and surprisingly efficient (rapidly convergent) solution of the CMP has been established. The principle has been used extensively in a number of diverse applications ranging from image construction to spectral analysis, large-scale electronic structure problems [5,6], series extrapolation and analytic continuation [7], quantum electronic transport [8], ligand-binding distribution in polymers [9], and transport planning [10].There exist a number of maximum entropy algorithms [4,6,11,12,13] that have been developed over the last two decades. Many of the algorithms (but not all) are constrained by the number of moments that it can deal with and become unreliable when the number of constraints exceeds a problem-dependent upper limit. As the number of moments increases, the calculation of moments (particularly the power moments) becomes more sensitive to machine precision and the optimization problem becomes ill-conditioned. It has been observed that implementation of a maxent algorithm with more than 20 power moments is notoriously difficult even with extended precision arithmetic and it rarely gives any further information on the nature of the distribution. The use of orthogonal polynomials as basis set significantly improves the accuracy and remedies most of the problems...

Some problems associated with the use of the maximum entropy principle, namely, (i) possible divergence of the series that is exponentiated, (ii) input-dependent asymptotic behaviour of the density function resulting from the truncation of the said series, and (iii) non-vanishing of the density function at the boundaries of a finite domain are pointed out. Prescriptions for remedying the aforesaid problems are put forward. Pilot calculations involving the ground quantum eigenenergy states of the quartic oscillator, the particle-in-a-box model, and the classical Maxwellian speed and energy distributions lend credence to our approach.

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ABSTRACT:We exploit the interrelation among the parameters embedded in the maximum entropy ansatz to develop a scheme for obtaining accurate estimates of the ground-state energy and wave function of systems for which the potential is represented by a rational function. Our scheme reduces an N-parameter optimization problem to a two-parameter one, leading to considerable simplification of the prevalent strategy. An indirect route for the study of excited states is also sketched. Test calculations on hydrogenic systems subject to strong or superstrong radial magnetic fields with and without electric field reveal the advantages of our approach. Additional studies on 1-D anharmonic oscillators affirm its workability and generality.