Fast Maximum Entropy Algorithm for Ill-Posed Problems

Abstract

The fast algorithm of the Maximum Entropy (MaxEnt) numerical solution of the linear inverse problem is described. The minimization of a general functional intrinsic to the MaxEnt approach is reduced to an iteration procedure with each step being a constrained least-squares problem (minimization of a quadratic functional with linear inequality constraints). The algorithm is structurally simple and can be assembled from blocks available in standard program libraries. The algorithm is tested on “toy” tasks with exponential kernel, as well as on practical problems of the recovery of the spectral density of strongly correlated quantum systems from the imaginary time Green’s functions obtained by Monte Carlo.

Shell model monte carlo methods

“…As an example, Fig. 2.2 shows the GamowTeller strengths in 54 Fe and 56 Fe calculated at various levels of truncation; there is clear convergence to the complete result obtained with the SMMC methods (see Section 8).…”

“…The dominant role is played by the isoscalar 1 + channel, which couples protons and neutrons in the same orbitals and in spin-orbit partners. As a consequence we find that, for 54,56 Fe, the proton pair matrix in the f 5/2 orbital, M 33 , is larger than in the p 3/2 orbital, although the latter is favored in energy. For 58 Fe, this ordering is inverted, as the increasing number of neutrons in the p 3/2 orbital increases the proton pairing in that orbital.…”

“…While this does not result in a complete solution to the many-body problem in the sense of giving all eigenvalues and eigenstates ofĤ, it can result in much useful information. For example, the expectation value of some observableΩ in the grand canonical ensemble can be obtained by adding toĤ a term −µ nN − µ pẐ , (µ n and µ p are the neutron and proton chemical potentials) and then calculating 54,56 Fe in a series of direct diagonalizations with decreasing level of truncation [52] with the complete results obtained with the SMMC method (solid symbols at t = 10) [55]. The open symbols at t = 10 show the extrapolated no-truncation results of [52].…”

“…7.2 we exemplify the g-extrapolation procedure for several observables, calculated again for 54 Fe. In all cases we use polynomial extrapolations from negative g-values to the physical case, g = 1.…”

“…for those nuclei in the mass range A = 48 − 56 for which direct diagonalization studies [51] or estimates on the basis of a series of truncation calculations [52] exist. An example for the nuclei 54,56 Fe is given in Fig. 2.2.…”