Hausdorff moment problem and maximum entropy: A unified approach

Tagliani, Aldo

doi:10.1016/s0096-3003(98)10084-x

Cited by 62 publications

(36 citation statements)

References 9 publications

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“…The existence of such ME distribution is shown in [44] as soon as the vector of moments M belongs to the interior of the moment space M N (S min , S max ). This is a standard constrained optimization problem, leading to the following explicit representation of the ME approximate :…”

Section: Ndf Reconstruction Through the Maximum Entropy Formalismmentioning

confidence: 99%

A Robust Moment Method for Evaluation of the Disappearance Rate of Evaporating Sprays

Massot¹,

Laurent²,

Kah³

et al. 2010

SIAM J. Appl. Math.

131

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Abstract. In this paper we tackle a critical issue in the numerical modeling, by Eulerian moment methods, of polydisperse multiphase systems, constituted of dispersed particles or droplets, a general class of systems which include aerosols. Their modeling starts at a mesoscopic scale with an equation on the number density function NDF of particles/droplets which satisfies a population balance equation. (PBE, also called Williams equation in the spray community). In order to limit the computational cost, moment methods provide a system of conservation equation with an eventual closure problem which can be solved using quadrature methods in order to retrieve the unclosed terms from the considered set of moments. However, a drift velocity, that is, the rate of change due to continuous phenomena of the internal coordinate such as the size of the particles, has sometimes to be taken into account; it can be either positive like molecular growth, or negative such as for evaporation of droplets in aerosols or oxidation of soots. When negative, it leads to the disappearance of droplets/particles thus creating a negative flux at zero size. Its closure requires an evaluation of the reconstructed NDF at zero size from the knowledge of a given finite set of moments. The nature of this information, pointwise in internal coordinate, and its influence on moment dynamics results in a difficulty from both a modeling and a numerical point of view. We obtain in the present contribution a comprehensive solution to this important issue. Since we introduce some new tools in order to resolve the flux evaluation, we also introduce a new Eulerian type of description which will combine both the flexibility of Eulerian models for which the size phase space is discretized into "sections" (i.e. size intervals) and the efficiency of Direct Quadrature Method of Moments (DQMOM). It yields a precise and stable description of moment dynamics with a minimal number of variables which should lead to a low computational cost in multi-dimensional configurations.

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Section: Ndf Reconstruction Through the Maximum Entropy Formalismmentioning

confidence: 99%

A Robust Moment Method for Evaluation of the Disappearance Rate of Evaporating Sprays

Massot¹,

Laurent²,

Kah³

et al. 2010

SIAM J. Appl. Math.

131

View full text Add to dashboard Cite

show abstract

“…When the moments are assigned, the condition for existence of an MEP solution is discussed in [25]. It is identical to the condition that the corresponding truncated Hausdorff moment problem admits a solution, i.e., the Hankel determinants must be positive.…”

Section: Determination Of Admissible Range Of Momentmentioning

confidence: 99%

Reliability bound based on the maximum entropy principle with respect to the first truncated moment

Sung

Kwak

2010

J Mech Sci Technol

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Various reliability methods have been suggested in the literature, but the bound of an estimated reliability has received less attention. The maximum entropy principle is used to obtain the reliability bound with respect to the first moment truncated for the first time. Compared to the previous methods of probability bounding based on given moments, our method is demonstrated to generate a tight upper bound that is practically useful for engineering applications. Numerical examples have shown that a good upper bound of probability of failure is well obtained up to four given moments, but with more moments a divergence problem can occur.

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Section: Introductionmentioning

confidence: 99%

“…A standard formalism [41] transforms the constrained maximum entropy problem into the unconstrained minimization problem of the dual objective function. More details on theoretical aspects of the maximum entropy moment problem can be found in [8,15,14,38].Recently, the author developed new algorithms for the multidimensional momentconstrained maximum entropy problem [1,2,3]. While the method in [1] is somewhat primitive and is only capable of solving two-dimensional maximum entropy problems with moments of order up to 4, the improved algorithm in [2] uses a suitable orthonormal polynomial basis in the space of Lagrange multipliers to improve convergence of its iterative optimization process.…”

mentioning

confidence: 99%

The multidimensional maximum entropy moment problem: a review of numerical methods

Abramov¹

2010

Communications in Mathematical Sciences

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Abstract. Recently the author developed a numerical method for the multidimensional momentconstrained maximum entropy problem, which is practically capable of solving maximum entropy problems in the two-dimensional domain with moment constraints of order up to 8, in the threedimensional domain with moment constraints of order up to 6, and in the four-dimensional domain with moment constraints of order up to 4, corresponding to the total number of moment constraints of 44, 83 and 69, respectively. In this work, the author brings together key algorithms and observations from his previous works as well as other literature in an attempt to present a comprehensive exposition of the current methods and results for the multidimensional maximum entropy moment problem.Key words. maximum entropy problem, moment constraints, numerical methods.AMS subject classifications. 49M, 65K, 90C. IntroductionThe moment-constrained maximum entropy problem yields an estimate of a probability density with highest uncertainty among all densities satisfying supplied moment constraints. The moment constrained maximum entropy problem arises in a variety of settings in solid state physics [7,22,23,39], econometrics [34,40], statistical description of gas flows [27,33], geophysical applications such as weather and climate prediction [4,5,21,25,28,29,35,36], and many other areas. The approximation of the probability density itself is obtained by maximizing the Shannon entropy under the constraints established by measured moments (phase space-averaged monomials of problem variables) [32]. A standard formalism [41] transforms the constrained maximum entropy problem into the unconstrained minimization problem of the dual objective function. More details on theoretical aspects of the maximum entropy moment problem can be found in [8,15,14,38].Recently, the author developed new algorithms for the multidimensional momentconstrained maximum entropy problem [1,2,3]. While the method in [1] is somewhat primitive and is only capable of solving two-dimensional maximum entropy problems with moments of order up to 4, the improved algorithm in [2] uses a suitable orthonormal polynomial basis in the space of Lagrange multipliers to improve convergence of its iterative optimization process. It is capable of practically solving two-dimensional problems with moments of order up to 8, three-dimensional problems with moments of order up to 6, and four-dimensional problems of order up to 4, totalling 44, 83 and 69 moment constraints, respectively, not counting the normalization constraint for a probability density. In [3], further improvements were introduced to the algorithm for the multidimensional moment-constrained maximum entropy problem, such as multiple Broyden-Fletcher-Goldfarb-Shanno (BFGS) iterations [9,10,12,19,37] to adaptively progress between points of polynomial reorthonormalization, as opposed to single Newton steps introduced in [2], and suitable constraint rescaling to reduce difference in magnitude between high and low order moment constraints. ...

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Hausdorff moment problem and maximum entropy: A unified approach

Cited by 62 publications

References 9 publications

A Robust Moment Method for Evaluation of the Disappearance Rate of Evaporating Sprays

A Robust Moment Method for Evaluation of the Disappearance Rate of Evaporating Sprays

Reliability bound based on the maximum entropy principle with respect to the first truncated moment

The multidimensional maximum entropy moment problem: a review of numerical methods

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