Maximum entropy summation of divergent perturbation series

Bender, Carl M.; Mead, Lawrence R.; Papanicolaou, N. I.

doi:10.1063/1.527593

Cited by 24 publications

(35 citation statements)

References 6 publications

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“…Asymptotic properties of E 0 (*) are rather clear as * tends to zero along the negative real axis * Ä 0& [38,39] (and references therein). As to the behavior of E 0 (*) when * Ä 0+ along the positive real axis, the problem persists in spite of the effort put into its solution [40,41,42,43,44].…”

Section: Appendix: the Range Of Small *mentioning

confidence: 99%

On the Nature of Energy Levels of Anharmonic Oscillators

Chebotarev¹

1999

Annals of Physics

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Section: Appendix: the Range Of Small *mentioning

confidence: 99%

On the Nature of Energy Levels of Anharmonic Oscillators

Chebotarev¹

1999

Annals of Physics

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“…In particular, several cubature formula behave very well for exponentials of polynomials as shown in e.g. Bender et al [5]. An alternative with no cubature formula is also proposed in [15].…”

Section: Estimating the Density G(y)mentioning

confidence: 97%

“…In addition, in the latter case the max-entropy estimation has been shown to be very efficient in several examples in the literature; see e.g. [5,23,24]. However, in view of the present status of SDP solvers, if no sparsity or symmetry is taken into account as proposed in e.g.…”

Section: Introductionmentioning

confidence: 99%

A “Joint+Marginal” Approach to Parametric Polynomial Optimization

Lasserre¹

2010

SIAM J. Optim.

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Abstract. Given a compact parameter set Y ⊂ R p , we consider polynomial optimization problems (Py) on R n whose description depends on the parameter y ∈ Y. We assume that one can compute all moments of some probability measure ϕ on Y, absolutely continuous with respect to the Lebesgue measure (e.g. Y is a box or a simplex and ϕ is uniformly distributed). We then provide a hierarchy of semidefinite relaxations whose associated sequence of optimal solutions converges to the moment vector of a probability measure that encodes all information about all global optimal solutions x * (y) of Py, as y ∈ Y. In particular, one may approximate as closely as desired any polynomial functional of the optimal solutions, like e.g. their ϕ-mean. In addition, using this knowledge on moments, the measurable function y → x * k (y) of the k-th coordinate of optimal solutions, can be estimated, e.g. by maximum entropy methods. Also, for a boolean variable x k , one may approximate as closely as desired its persistency ϕ({y : x * k (y) = 1}, i.e. the probability that in an optimal solution x * (y), the coordinate x * k (y) takes the value 1. At last but not least, from an optimal solution of the dual semidefinite relaxations, one provides a sequence of polynomial (resp. piecewise polynomial) lower approximations with L 1 (ϕ) (resp. almost uniform) convergence to the optimal value function. IntroductionRoughly speaking, given a set parameters Y and an optimization problem whose description depends on y ∈ Y (call it P y ), parametric optimization is concerned with the behavior and properties of the optimal value as well as primal (and possibly dual) optimal solutions of P y , when y varies in Y. This a quite challenging problem and in general one may obtain information locally around some nominal value y 0 of the parameter. There is a vast and rich literature on the topic and for a detailed treatment, the interested reader is referred to e.g. Bonnans and Shapiro [4] and the many references therein. Sometimes, in the context of optimization with data uncertainty, some probability distribution ϕ on the parameter set Y is available and in this context one is also interested in e.g. the distribution of the optimal value, optimal solutions, all viewed as random variables. In particular, for discrete optimization problems where cost coefficients are random variables with joint distribution ϕ, some bounds on the expected optimal value have been obtained. More recently Natarajan et al. [17] extended the earlier work in [3] to even 1991 Mathematics Subject Classification. 65 D15, 65 K05, 46 N10, 90 C22.

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“…MaxEnt extrapolation [ 26,27,35 ] may be seen as a generalization of Borel extrapolation. That is, instead of Equation (8), we consider approximants

f_{N} (x)

for

f (x)

of Equation (9) of the more general form

f_{N} (x) = \int_{D} d t ρ (t) K (t x)

where the density

ρ (z)

is semipositive in the domain D , i.e.,

ρ (z) ⩾ 0

for

z \in D

.…”

Section: Maxent Extrapolationmentioning

confidence: 99%

Constrained Extrapolation Problem and Order‐Dependent Mappings

Wellenhofer

Phillips

Schwenk

2021

Physica Status Solidi (b)

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The problem of extrapolating the perturbation series for the dilute Fermi gas in three dimensions to the unitary limit of infinite scattering length and into the BEC region is considered, using the available strong‐coupling information to constrain the extrapolation problem. In this constrained extrapolation problem (CEP), the goal is to find classes of approximants that give well‐converged results already for low perturbative truncation orders. First, it is shown that the standard Padé and Borel methods are too restrictive to give satisfactory results for this CEP. A generalization of Borel extrapolation is given by the so‐called maximum‐entropy (MaxEnt) extrapolation method. However, it is shown that MaxEnt requires extensive elaborations to be applicable to the dilute Fermi gas and is, thus, not practical for the CEP in this case. Instead, order‐dependent‐mapping extrapolation (ODME) as a simple, practical, and general method for the CEP is proposed. It is found that the ODME approximants for the ground‐state energy of the dilute Fermi gas are robust with respect to changes of the mapping choice and agree with results from quantum Monte Carlo simulations within uncertainties.

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Maximum entropy summation of divergent perturbation series

Cited by 24 publications

References 6 publications

On the Nature of Energy Levels of Anharmonic Oscillators

On the Nature of Energy Levels of Anharmonic Oscillators

A “Joint+Marginal” Approach to Parametric Polynomial Optimization

Constrained Extrapolation Problem and Order‐Dependent Mappings

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